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note: 全部 active 概念的紧凑汇总；不含整书原文合并包。
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# 全量 active 概念跨书汇总包（1,170 个概念）

> 本文件是**紧凑汇总**（已内嵌各书首次出现锚点、已审定义原文摘录与显式依赖组状态），不含整书原文合并包。需要完整证据链或小节上下文推导时，请按条目给出的卡片 URL 或小节 URL 二跳读取。

## 极限点(limit point)

- UUID: `0009d7dc-5f4f-4928-a8cc-fa12def8161c` · okf_url: https://limbo137.top/okf/concepts/0009d7dc-5f4f-4928-a8cc-fa12def8161c.md
- labels: en = limit point；zh-CN = 极限点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-3f83681b3847dbd4e0e1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 余挠张量(contorsion tensor)

- UUID: `00480ba4-7e17-50d9-9a65-1629cbacbde4` · okf_url: https://limbo137.top/okf/concepts/00480ba4-7e17-50d9-9a65-1629cbacbde4.md
- labels: en = contorsion tensor；zh-CN = 余挠张量
- aliases: 扭度张量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §7.2.6 (pdf_page=273, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §7.2.6 (7.35), contorsion tensor K. Preserve the original OCR word contour and the displayed component expression verbatim; do not silently correct their indices. Exclude the following Levi-Civita definition. · 原文: `$$ K _ {\mu \nu} ^ {\kappa} \equiv \frac {1}{2} \left(T _ {\mu \nu} ^ {\kappa} + T _ {\mu \nu} ^ {\kappa} + T _ {\nu \mu} ^ {\kappa}\right). \tag {7.35} $$`
  - nakahara-geometry-topology-physics/second-edition-local §7.2.6 (pdf_page=273, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §7.2.6 (7.35), contorsion tensor K. Preserve the original OCR word contour and the displayed component expression verbatim; do not silently correct their indices. Exclude the following Levi-Civita definition. · 原文: `The second term of the last expression of (7.34) is called the contour, denoted by $K_{\mu \nu}^{\kappa}$ :`
  - nakahara-geometry-topology-physics/second-edition-local §7.4.1 (pdf_page=280, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §7.2.6 (7.35)余挠张量K；OCR contour按用户及§7.4.1完整术语contorsion tensor校正，原字词保持在证据中。 · 原文: `where $\left\{\begin{array}{c}\kappa \\ \mu\nu\end{array}\right\}$ is the Christoffel symbol and $K$ the contorsion tensor. It was shown in exercise 7.5 that $\Gamma^{\kappa}_{\mu\nu} \equiv \widetilde{\Gamma}^{\kappa}_{\mu\nu} + t^{\kappa}_{\mu\nu}$ is another connection coefficient if $t$ is a tensor field of type (1, 2). Now we choose $t^{\kappa}_{\mu\...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 双重覆盖(double cover)

- UUID: `004c393d-b3c2-4027-957a-d8cea3b92491` · okf_url: https://limbo137.top/okf/concepts/004c393d-b3c2-4027-957a-d8cea3b92491.md
- labels: en = double cover；zh-CN = 双重覆盖
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-double` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-double` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 仿射联络(affine connection)

- UUID: `008abe3e-4b89-4b4f-9096-b685aa35da43` · okf_url: https://limbo137.top/okf/concepts/008abe3e-4b89-4b4f-9096-b685aa35da43.md
- labels: en = affine connection；zh-CN = 仿射联络
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-6200db44beab0a364b80-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-6200db44beab0a364b80-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 商向量空间(quotient vector space)

- UUID: `009d7313-1b0f-44a9-b25f-fa3b3005458d` · okf_url: https://limbo137.top/okf/concepts/009d7313-1b0f-44a9-b25f-fa3b3005458d.md
- labels: en = quotient vector space；zh-CN = 商向量空间
- aliases: quotient space；商空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-quotient-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 爱因斯坦张量(Einstein tensor)

- UUID: `00d268ec-05fa-4942-85d1-a3faa750cee7` · okf_url: https://limbo137.top/okf/concepts/00d268ec-05fa-4942-85d1-a3faa750cee7.md
- labels: en = Einstein tensor；zh-CN = 爱因斯坦张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.6 · `zh/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-af05dfd43d5da4ca19aa-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.6 · `en/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-af05dfd43d5da4ca19aa-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/06-the-riemann-tensor-and-its-symmetries.md

## 置换对称群(symmetric group)

- UUID: `013665e1-3304-480f-81d6-f8f99a56381e` · okf_url: https://limbo137.top/okf/concepts/013665e1-3304-480f-81d6-f8f99a56381e.md
- labels: en = symmetric group；zh-CN = 置换对称群
- aliases: 对称群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-symmetric-group` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §2.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `If $X$ is a finite set ofcardinality $n$ then the transformations of $X$ are called permutations of the elements of $X$. The group ofpermutations of $X = \{ 1 , 2 , \ldots , n \}$ is called the symmetric group of order $n$, denoted $S _ { n }$ . Any subgroup of $S _ { n }$ is called a permutation group. A permutation $\pi$ on $n$ elements can be represent...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 鞍点(saddle point)

- UUID: `01ba5fba-f97d-4088-b58f-f176a5eaa6ce` · okf_url: https://limbo137.top/okf/concepts/01ba5fba-f97d-4088-b58f-f176a5eaa6ce.md
- labels: en = saddle point；zh-CN = 鞍点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-saddle-point` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 同伦(homotopy)

- UUID: `01bd4cf1-b466-569c-871b-340494649998` · okf_url: https://limbo137.top/okf/concepts/01bd4cf1-b466-569c-871b-340494649998.md
- labels: en = homotopy；zh-CN = 同伦
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=165, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.10：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ F \left(s _ {1}, \dots , s _ {n}, 1\right) = \beta \left(s _ {1}, \dots , s _ {n}\right) \tag {4.37b} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=165, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.10：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$F$ is called a homotopy between $\alpha$ and $\beta$ .`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.3 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.4：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.4. Let $\alpha, \beta: I \to X$ be loops at $x_0$ . They are said to be homotopic, written as $\alpha \sim \beta$ , if there exists a continuous map $F: I \times I \to X$ such that`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.3 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.4：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ F (s, 0) = \alpha (s), \quad F (s, 1) = \beta (s) \quad \forall s \in I \tag {4.3} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=165, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.10：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ F \left(s _ {1}, \dots , s _ {n}, t\right) = x _ {0} \quad \text {f o r} (s _ {1}, \dots , s _ {n}) \in \partial I ^ {n}, t \in I. \tag {4.37c} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=165, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.10：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ F \left(s _ {1}, \dots , s _ {n}, 0\right) = \alpha \left(s _ {1}, \dots , s _ {n}\right) \tag {4.37a} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.3 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.4：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ F (0, t) = F (1, t) = x _ {0} \qquad \forall t \in I. $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=165, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.10：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.10. Let $X$ be a topological space and $\alpha, \beta : I^n \to X$ be $n$ -loops at $x_0 \in X$ . The map $\alpha$ is homotopic to $\beta$ , denoted by $\alpha \sim \beta$ , if there exists a continuous map $F: I^n \times I \to X$ such that`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.3 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.4：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `It is helpful to represent a homotopy as figure 4.3(a). The vertical edges of the square $I \times I$ are mapped to $x_0$ . The lower edge is $\alpha(s)$ while the upper edge is $\beta(s)$ . In the space $X$ , the image is continuously deformed as in figure 4.3(b).`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.3 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.4：同伦。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The connecting map $F$ is called a homotopy between $\alpha$ and $\beta$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 李变换群(Lie group of transformations)

- UUID: `01dc6c65-7574-49f1-9a55-20c52d99f5f7` · okf_url: https://limbo137.top/okf/concepts/01dc6c65-7574-49f1-9a55-20c52d99f5f7.md
- labels: en = Lie group of transformations；zh-CN = 李变换群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-bedb7954e065af2f56d7-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.4 · `zh/chapter-19/04-lie-groups-of-transformations#concept-19-lie-group-of-transformations` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md

## 化学势(chemical potential)

- UUID: `01ff489c-58d1-4d75-87b2-d1cd999d36bd` · okf_url: https://limbo137.top/okf/concepts/01ff489c-58d1-4d75-87b2-d1cd999d36bd.md
- labels: en = chemical potential；zh-CN = 化学势
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.4 · `zh/chapter-14/04-quantum-statistical-mechanics#first-f00a4551bf40bd071302-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.4 · `en/chapter-14/04-quantum-statistical-mechanics#first-f00a4551bf40bd071302-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/04-quantum-statistical-mechanics.md

## 舒尔引理(Schur’s lemma)

- UUID: `023d3b84-da9a-4667-8674-d5b9ccbed948` · okf_url: https://limbo137.top/okf/concepts/023d3b84-da9a-4667-8674-d5b9ccbed948.md
- labels: en = Schur’s lemma；zh-CN = 舒尔引理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#first-6d7eb6cc7322de42f7e0-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §4.4 · `en/chapter-04/04-introduction-to-group-representation-theory#first-6d7eb6cc7322de42f7e0-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §2.1 · `books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory#first-e575a0c33aad56892410-zh` · translation)；woit-qtgr/2025-10-20 (en · §2.1 · `books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory#first-e575a0c33aad56892410-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md

## 局域性(locality)

- UUID: `0255717c-bfab-4555-82a5-f56059eba261` · okf_url: https://limbo137.top/okf/concepts/0255717c-bfab-4555-82a5-f56059eba261.md
- labels: en = locality；zh-CN = 局域性
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §38.1 · `books/woit-qtgr/2025-10-20/zh/chapter-38/01-unitary-transformations-on-h1#first-3bf7536f24586540a632-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/01-unitary-transformations-on-h1.md

## 直积(direct product)

- UUID: `0267f40f-cd9e-43de-8086-9a9f751a68b8` · okf_url: https://limbo137.top/okf/concepts/0267f40f-cd9e-43de-8086-9a9f751a68b8.md
- labels: en = direct product；zh-CN = 直积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-direct-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 毛雷尔–卡当关系(Maurer–Cartan relations)

- UUID: `02b57e3f-b649-49e8-b81d-4fff8a450e33` · okf_url: https://limbo137.top/okf/concepts/02b57e3f-b649-49e8-b81d-4fff8a450e33.md
- labels: en = Maurer–Cartan relations；zh-CN = 毛雷尔–卡当关系
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-5e0706bf00f7e8cf4ebb-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-maurer-cartan-relations` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 实射影 3-空间(real projective 3-space)

- UUID: `02cfe956-6666-4db6-bee4-19ea46011cfe` · okf_url: https://limbo137.top/okf/concepts/02cfe956-6666-4db6-bee4-19ea46011cfe.md
- labels: en = real projective 3-space；zh-CN = 实射影 3-空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-rp` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-rp` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 对易关系(commutation relations)

- UUID: `02db5117-4ecf-4ac2-9cd8-bf87eb334922` · okf_url: https://limbo137.top/okf/concepts/02db5117-4ecf-4ac2-9cd8-bf87eb334922.md
- labels: en = commutation relations；zh-CN = 对易关系
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.2 · `books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices#first-a153ea14d84e467474f4-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.2 · `books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices#first-a153ea14d84e467474f4-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/02-commutation-relations-for-pauli-matrices.md

## 同调群(homology group)

- UUID: `02f7a498-5c17-4366-9479-8666d5f9bb2e` · okf_url: https://limbo137.top/okf/concepts/02f7a498-5c17-4366-9479-8666d5f9bb2e.md
- labels: en = homology group；zh-CN = 同调群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-10254fb45e81b1ade885-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-10254fb45e81b1ade885-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H _ {r} (K) \equiv \{[ z ] | z \in Z _ {r} (K) \} \tag {3.29} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined. The group $H_r(K)$ is the set of equivalence classes of $r$ -cycles,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `write $H_{r}(K;\mathbb{Z})$ . We may also define the homology groups with $\mathbb{R}$ -coefficients, $H_{r}(K;\mathbb{R})$ or those with $\mathbb{Z}_2$ -coefficients, $H_{r}(K;\mathbb{Z}_{2})$ .]`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `[Remarks: If necessary, we define $H_r(K) = 0$ for $r > n$ or $r < 0$ . If we want to stress that the group structure is defined with integer coefficients, we`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.5. Let $K$ be an $n$ -dimensional simplicial complex. The $r$ th homology group $H_{r}(K)$ , $0 \leq r \leq n$ , associated with $K$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §— (pdf_page=125, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 本章主要采用整数系数；$0\le r\le n$。不能与实系数同调空间按定义完全等同。 · 原文: `Definition 3.5. Let $K$ be an $n$ -dimensional simplicial complex. The $r$ th homology group $H_{r}(K)$ , $0 \leq r \leq n$ , associated with $K$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where each equivalence class $[z]$ is called a homology class. Two $r$ -cycles $z$ and $z'$ are in the same equivalence class if and only if $z - z' \in B_r(K)$ , in which case $z$ is said to be homologous to $z'$ and denoted by $z \sim z'$ or $[z] = [z']$ . Geometrically $z - z'$ is a boundary of some space. By definition, any boundary $b \in B_r(K)$ is ...`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Szekeres 用光滑奇异单形的实系数链；整数系数的挠信息不能由该入口抹去。
- 显式依赖组(dependency_groups): nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , w...`；nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \partial_{r+1}d$ for some $d \in C_{r+1}(K)$ . Then we find $\partial_r c ...`
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 定向(orientation)

- UUID: `036e9782-c5c9-47d4-90c7-f64cdc08cbed` · okf_url: https://limbo137.top/okf/concepts/036e9782-c5c9-47d4-90c7-f64cdc08cbed.md
- labels: en = orientation；zh-CN = 定向
- aliases: orientations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.5 · `en/chapter-08/05-oriented-vector-spaces#first-af21f7e697993cf21a73-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.5 · `zh/chapter-08/05-oriented-vector-spaces#first-63755a3ff7166d63516c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/05-oriented-vector-spaces.md

## 平衡态(equilibrium state)

- UUID: `03d3ce88-0cb0-4510-9333-51e320b269f4` · okf_url: https://limbo137.top/okf/concepts/03d3ce88-0cb0-4510-9333-51e320b269f4.md
- labels: en = equilibrium state；zh-CN = 平衡态
- aliases: equilibrium states
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-0bfc777db849d2331419-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-c3ca3ab14ad5a4523b1c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 闭子群(closed subgroup)

- UUID: `03eb86c3-86f8-40cb-8e28-d7d053422767` · okf_url: https://limbo137.top/okf/concepts/03eb86c3-86f8-40cb-8e28-d7d053422767.md
- labels: en = closed subgroup；zh-CN = 闭子群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-0be35a567ad49081ee79-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.3 · `en/chapter-19/03-lie-subgroups#first-0be35a567ad49081ee79-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/03-lie-subgroups.md

## Hilbert空间的酉算符(unitary operator on a Hilbert space)

- UUID: `040e751e-8f47-4e5e-afb7-41c459e5d622` · okf_url: https://limbo137.top/okf/concepts/040e751e-8f47-4e5e-afb7-41c459e5d622.md
- labels: en = unitary operator on a Hilbert space；zh-CN = Hilbert空间的酉算符
- aliases: unitary；unitary operator；unitary operators；幺正；幺正的；幺正算符；酉；酉的；酉算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-d2b31bf1671157421fee-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-unitary` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §13.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Hilbert空间算符可逆且逆等于伴随；不把非满射等距嵌入视作酉算符 · 原文: `An operator $U : \mathcal { H } \to \mathcal { H }$ is called unitary if $$ \langle U u \mid U v \rangle = \langle u \mid v \rangle \quad \text { for all } u, v \in \mathcal {H}. $$ Since this implies $\langle U ^ { * } U u \mid v \rangle = \langle u \mid v \rangle$ , an operator $U$ is unitary if and only if $U ^ { - 1 } = U ^ { * }$ Every unitary operat...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 完备向量场(complete vector field)

- UUID: `042328bd-d5e2-4fe7-8a36-2a21890d5e6f` · okf_url: https://limbo137.top/okf/concepts/042328bd-d5e2-4fe7-8a36-2a21890d5e6f.md
- labels: en = complete vector field；zh-CN = 完备向量场
- aliases: complete
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-d4248df8d36a72045755-vector-field-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-complete-vector-field` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md

## 上链(cochain)

- UUID: `04adfa20-e5d3-4dd7-910d-d1fc29b4820b` · okf_url: https://limbo137.top/okf/concepts/04adfa20-e5d3-4dd7-910d-d1fc29b4820b.md
- labels: en = cochain；zh-CN = 上链
- aliases: cochains
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-66320d0b2eb1c12f93aa-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-e2e602e31df1891319af-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 体积元(volume element)

- UUID: `04c44987-c5e6-455f-8d1e-5c61314c5314` · okf_url: https://limbo137.top/okf/concepts/04c44987-c5e6-455f-8d1e-5c61314c5314.md
- labels: en = volume element；zh-CN = 体积元
- aliases: volume elements
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.5 · `en/chapter-08/05-oriented-vector-spaces#first-b9751310623f9ea86f0e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.5 · `zh/chapter-08/05-oriented-vector-spaces#first-2f4a87dbc2cc1e409752-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/05-oriented-vector-spaces.md

## 超几何(supergeometry)

- UUID: `05169104-3155-481a-ad02-24789d58afd7` · okf_url: https://limbo137.top/okf/concepts/05169104-3155-481a-ad02-24789d58afd7.md
- labels: en = supergeometry；zh-CN = 超几何
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §30.1 · `books/woit-qtgr/2025-10-20/en/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-afcafe8545b3a518f2b9-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §30.1 · `books/woit-qtgr/2025-10-20/zh/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-afcafe8545b3a518f2b9-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators.md

## 非退化内积(non-singular inner product)

- UUID: `05370b36-295b-4888-ab89-d433d246e4eb` · okf_url: https://limbo137.top/okf/concepts/05370b36-295b-4888-ab89-d433d246e4eb.md
- labels: en = non-singular inner product；zh-CN = 非退化内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-non-singular-inner-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 凯莱定理(Cayley’s theorem)

- UUID: `057330d9-9ff5-427f-a15c-73c814aa8a20` · okf_url: https://limbo137.top/okf/concepts/057330d9-9ff5-427f-a15c-73c814aa8a20.md
- labels: en = Cayley’s theorem；zh-CN = 凯莱定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §2.4 · `en/chapter-02/04-homomorphisms-and-isomorphisms#first-93da16ef5eea70dc3859-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#first-93da16ef5eea70dc3859-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 费米子泊松括号(fermionic Poisson bracket)

- UUID: `05c92274-bda7-4225-b24d-846ec64b2dbc` · okf_url: https://limbo137.top/okf/concepts/05c92274-bda7-4225-b24d-846ec64b2dbc.md
- labels: en = fermionic Poisson bracket；zh-CN = 费米子泊松括号
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §30.2 · `books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-81dcf2b2a80924f1fd88-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §30.2 · `books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-81dcf2b2a80924f1fd88-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket.md

## 李群表示(Lie group representation)

- UUID: `05d312b4-8c04-40c4-a051-d508a402b79f` · okf_url: https://limbo137.top/okf/concepts/05d312b4-8c04-40c4-a051-d508a402b79f.md
- labels: en = Lie group representation；zh-CN = 李群表示
- aliases: Lie group representations
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-9a44009b5c8fae6ca409-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-83c093b381f8adbcecae-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 位置算符(position operator)

- UUID: `05d69eba-f971-4eda-9264-d2a2b9cb71b1` · okf_url: https://limbo137.top/okf/concepts/05d69eba-f971-4eda-9264-d2a2b9cb71b1.md
- labels: en = position operator；zh-CN = 位置算符
- aliases: Position operator
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-position-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-173cf8d568657fd566fb-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §12.1 · `books/woit-qtgr/2025-10-20/en/chapter-12/01-the-position-operator#first-f489c633c2151c5cffa6-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §12.1 · `books/woit-qtgr/2025-10-20/zh/chapter-12/01-the-position-operator#first-e07a1c04cc2adbb5c323-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §12.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The position operator $Q$ is given by $$ Q \psi (q) = q \psi (q) $$`
  - woit-qtgr/2025-10-20 §12.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `位置算符 $Q$ 由下式给出 $$ Q \psi (q) = q \psi (q) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-12/01-the-position-operator.md

## 矩阵群(matrix group)

- UUID: `05f42bf3-d38b-49e0-bed7-1d0452d9ac3d` · okf_url: https://limbo137.top/okf/concepts/05f42bf3-d38b-49e0-bed7-1d0452d9ac3d.md
- labels: en = matrix group；zh-CN = 矩阵群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-matrix-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-fc558e266a7590e7bf39-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 准静态过程(quasi-static process)

- UUID: `065a4491-0bf8-406c-8c15-8091c5689490` · okf_url: https://limbo137.top/okf/concepts/065a4491-0bf8-406c-8c15-8091c5689490.md
- labels: en = quasi-static process；zh-CN = 准静态过程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-e0da582ef63dce2d1047-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 康托尔集(Cantor set)

- UUID: `06fabb9a-f268-4d2b-8732-263ac9213a4d` · okf_url: https://limbo137.top/okf/concepts/06fabb9a-f268-4d2b-8732-263ac9213a4d.md
- labels: en = Cantor set；zh-CN = 康托尔集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-827e033327040d2a58e4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## Berezin积分(Berezin integral)

- UUID: `078de077-8b79-46b3-8955-b2642c83034f` · okf_url: https://limbo137.top/okf/concepts/078de077-8b79-46b3-8955-b2642c83034f.md
- labels: en = Berezin integral；zh-CN = Berezin积分
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §30.1 · `books/woit-qtgr/2025-10-20/zh/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-9621fd28ca8f30e5b4a4-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §30.1 · `books/woit-qtgr/2025-10-20/en/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-9621fd28ca8f30e5b4a4-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators.md

## 电磁场张量(electromagnetic field tensor)

- UUID: `0794f89e-ba01-4e3a-843d-85c9ef760fd4` · okf_url: https://limbo137.top/okf/concepts/0794f89e-ba01-4e3a-843d-85c9ef760fd4.md
- labels: en = electromagnetic field tensor；zh-CN = 电磁场张量
- aliases: electromagnetc field
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-38f06d43171660f25939-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 范畴论(category theory)

- UUID: `07b47fe2-da51-4781-9b26-57197986cbf7` · okf_url: https://limbo137.top/okf/concepts/07b47fe2-da51-4781-9b26-57197986cbf7.md
- labels: en = category theory；zh-CN = 范畴论
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-f7f9ff150163b04e55aa-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 对偶向量空间(Dual vector space)

- UUID: `07fed5dc-6aa1-4ef6-a20a-58fa52a971f7` · okf_url: https://limbo137.top/okf/concepts/07fed5dc-6aa1-4ef6-a20a-58fa52a971f7.md
- labels: en = Dual vector space；zh-CN = 对偶向量空间
- aliases: dual vector space
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-9224a02242ce94330051-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.2 · `books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces#first-9224a02242ce94330051-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（[对偶向量空间(dual vector space)](/concepts/linear-algebra/dual-space)）。 对于域 $k$ 上的向量空间 $V$， 对偶向量空间 $V ^ { * }$ 是所有从 $V$ 到 $k$ 的[线性映射(linear maps)](/concepts/linear-algebra/linear-map)构成的向量空间， 即， $$ V ^ {*} = \{l: V \rightarrow k \text { such that } l (\alpha v + \beta w) = \alpha l (v) + \beta l (w) \} $$ 其中 $\alpha,\beta\in k$ 且...`
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Dual vector space). For $V$ a vector space over a field $k$, the dual vector space $V ^ { * }$ is the vector space of all linear maps $V \to k$, i.e., $$ V ^ {*} = \{l: V \rightarrow k \text { such that } l (\alpha v + \beta w) = \alpha l (v) + \beta l (w) \} $$ for $\alpha , \beta \in k , \ v , w \in V$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md

## 右作用(right action)

- UUID: `0809d0aa-e86c-401e-a94a-b029224f7535` · okf_url: https://limbo137.top/okf/concepts/0809d0aa-e86c-401e-a94a-b029224f7535.md
- labels: en = right action；zh-CN = 右作用
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-right-action` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md

## 互补(complementary subspaces)

- UUID: `084c39fe-d9c8-4937-9eca-f4baca436916` · okf_url: https://limbo137.top/okf/concepts/084c39fe-d9c8-4937-9eca-f4baca436916.md
- labels: en = complementary subspaces；zh-CN = 互补
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-complementary-subspaces` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 缓增分布(tempered distribution)

- UUID: `0851ac8c-a0a1-496c-8033-9592bf23ad8c` · okf_url: https://limbo137.top/okf/concepts/0851ac8c-a0a1-496c-8033-9592bf23ad8c.md
- labels: en = tempered distribution；zh-CN = 缓增分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.3 · `zh/chapter-12/03-fourier-transforms#first-4206424924c7958f3fe1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/03-fourier-transforms.md

## 格林函数(Green’s function)

- UUID: `087cfd6b-1fc8-40c8-aaf7-ce2bfb1f24bf` · okf_url: https://limbo137.top/okf/concepts/087cfd6b-1fc8-40c8-aaf7-ce2bfb1f24bf.md
- labels: en = Green’s function；zh-CN = 格林函数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.4 · `zh/chapter-12/04-green-s-functions#first-2969e73e476a1a734ea7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/04-green-s-functions.md

## 罗伯逊–沃克模型(Robertson–Walker model)

- UUID: `08c2620a-3da7-4383-ba7c-e217b3d97685` · okf_url: https://limbo137.top/okf/concepts/08c2620a-3da7-4383-ba7c-e217b3d97685.md
- labels: en = Robertson–Walker model；zh-CN = 罗伯逊–沃克模型
- aliases: Robertson–Walker models
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-b4ad1643f2dc9ff3a6a0-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-48ea5465a6f2b90dd55b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 几乎处处(almost everywhere)

- UUID: `08d376ac-f1ed-48fd-b44d-f1fd161e5431` · okf_url: https://limbo137.top/okf/concepts/08d376ac-f1ed-48fd-b44d-f1fd161e5431.md
- labels: en = almost everywhere；zh-CN = 几乎处处
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-6c4daa982c873b223e0c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 单连通空间(simply connected space)

- UUID: `0908b89b-b21e-43a5-9f3d-ad71014d364d` · okf_url: https://limbo137.top/okf/concepts/0908b89b-b21e-43a5-9f3d-ad71014d364d.md
- labels: en = simply connected space；zh-CN = 单连通空间
- aliases: simply connected；单连通的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：单连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.8. (a) A topological space $X$ is connected if it cannot be written as $X = X_{1} \cup X_{2}$ , where $X_{1}$ and $X_{2}$ are both open and $X_{1} \cap X_{2} = \emptyset$ . Otherwise $X$ is called disconnected.`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：单连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(b) A topological space $X$ is called arcwise connected if, for any points $x, y \in X$ , there exists a continuous map $f: [0,1] \to X$ such that $f(0) = x$ and $f(1) = y$ . With a few pathological exceptions, arcwise connectedness is practically equivalent to connectedness. (c) A loop in a topological space $X$ is a continuous map $f: [0,1] \to X$ such ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 正规子群(normal subgroup)

- UUID: `092b1d87-e948-49ed-b814-4d97167d0db4` · okf_url: https://limbo137.top/okf/concepts/092b1d87-e948-49ed-b814-4d97167d0db4.md
- labels: en = normal subgroup；zh-CN = 正规子群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-normal-subgroup` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 关系的传递性(transitive relation)

- UUID: `09554153-3c86-4e74-bed8-77f8ade887be` · okf_url: https://limbo137.top/okf/concepts/09554153-3c86-4e74-bed8-77f8ade887be.md
- labels: en = transitive relation；zh-CN = 关系的传递性
- aliases: transitive；传递；传递的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-16cddd3179afedd4e486-relation-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §1.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 对象是二元关系R；aRb且bRc蕴含aRc，不表示群作用只有一个轨道。 · 原文: `$R$ is transitive if ($aRb$ and $b R c ) \Rightarrow a R c$ for all $a$, $b , c \in S .$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 群 $U(1)$(U(1) group)

- UUID: `098437c0-f74c-41ec-82b2-7cdffbbe6feb` · okf_url: https://limbo137.top/okf/concepts/098437c0-f74c-41ec-82b2-7cdffbbe6feb.md
- labels: en = U(1) group；zh-CN = 群 $U(1)$
- aliases: The group U(1)；群 $U(1)$(group U(1))
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §2.2 · `books/woit-qtgr/2025-10-20/zh/chapter-02/02-the-group-u-1-and-its-representations#first-6ee16d64b9e7dd3d80db-zh` · translation)；woit-qtgr/2025-10-20 (en · §2.2 · `books/woit-qtgr/2025-10-20/en/chapter-02/02-the-group-u-1-and-its-representations#first-8ec13ed0164c6883c5c9-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §2.2 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (The group $U(1)$). The elements of the group $U(1)$ are points on the unit circle, which can be labeled by a unit complex number $e^{i\theta}$, or an angle $\theta\in\mathbf{R}$ with $\theta$ and $\theta+N2\pi$ labeling the same group element for $N\in\mathbf{Z}$. Multiplication of group elements is complex multiplication, which by the propert...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/02-the-group-u-1-and-its-representations.md

## 全对称的(totally symmetric)

- UUID: `09932e00-f8f2-44fe-9842-4c9fe5aa15dd` · okf_url: https://limbo137.top/okf/concepts/09932e00-f8f2-44fe-9842-4c9fe5aa15dd.md
- labels: en = totally symmetric；zh-CN = 全对称的
- aliases: 全对称
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-02fa9e816bfba2f0393f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 开邻域(open neighbourhood)

- UUID: `09ac0761-d818-449e-99c7-82395e89dbeb` · okf_url: https://limbo137.top/okf/concepts/09ac0761-d818-449e-99c7-82395e89dbeb.md
- labels: en = open neighbourhood；zh-CN = 开邻域
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-bb101d48f6c982eb4018-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 微分形式(differential form)

- UUID: `09b09776-2ccb-431b-b988-a10df4323428` · okf_url: https://limbo137.top/okf/concepts/09b09776-2ccb-431b-b988-a10df4323428.md
- labels: en = differential form；zh-CN = 微分形式
- aliases: differential；differential forms
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.1 · `en/chapter-16/01-differential-forms-and-exterior-derivative#first-f70ba2f2ffbea9175df1-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.1 · `zh/chapter-16/01-differential-forms-and-exterior-derivative#first-e4e7dd1afdff7a14b160-zh` · translation)；woit-qtgr/2025-10-20 (en · §30.1 · `books/woit-qtgr/2025-10-20/en/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-370d8aecf498723703b7-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §30.1 · `books/woit-qtgr/2025-10-20/zh/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-370d8aecf498723703b7-zh` · machine_draft)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $V_{i} \in T_{p}M$ . If $q + r > m$ , $\omega \wedge \xi$ vanishes identically. With this product, we define an algebra`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \Omega_ {p} ^ {*} (M) \equiv \Omega_ {p} ^ {0} (M) \oplus \Omega_ {p} ^ {1} (M) \oplus \dots \oplus \Omega_ {p} ^ {m} (M). \tag {5.66} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \alpha_ {\mu \nu} = \omega_ {[ \mu \nu ]} \equiv \frac {1}{2} \left(\omega_ {\mu \nu} - \omega_ {\nu \mu}\right). \tag {5.64b} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Since there are $\binom{m}{r}$ choices of the set $(\mu_1, \mu_2, \ldots, \mu_r)$ out of $(1, 2, \ldots, m)$ in (5.62), the dimension of the vector space $\Omega_p^r(M)$ is`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=215, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let us define the wedge product $\wedge$ of $r$ one-forms by the totally anti-symmetric tensor product`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For later convenience we define $\Omega_p^0 (M) = \mathbb{R}$ . Clearly $\Omega_p^1 (M) = T_p^* M$ . If $r$ in (5.62) exceeds $m$ , it vanishes identically since some index appears at least twice in the anti-symmetrized summation. The equality $\binom{m}{r}=\binom{m}{m-r}$ implies $\dim \Omega_p^r (M) = \dim \Omega_p^{m - r}(M)$ . Since $\Omega_p^r (M)$ i...`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \omega = \frac {1}{r !} \omega_ {\mu_ {1} \mu_ {2} \dots \mu_ {r}} d x ^ {\mu_ {1}} \wedge d x ^ {\mu_ {2}} \wedge \dots \wedge d x ^ {\mu_ {r}} \tag {5.63} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \sigma_ {\mu v} = \omega_ {(\mu v)} \equiv \frac {1}{2} \left(\omega_ {\mu v} + \omega_ {v \mu}\right) \tag {5.64a} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=215, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For example,`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If we denote the vector space of $r$ -forms at $p \in M$ by $\Omega_p^r (M)$ , the set of $r$ -forms (5.62) forms a basis of $\Omega_p^r (M)$ and an element $\omega \in \Omega_p^r (M)$ is expanded as`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Define the exterior product of a $q$ -form and an $r$ -form $\wedge : \Omega_p^q (M) \times \Omega_p^r (M) \to \Omega_p^{q + r}(M)$ by a trivial extension. Let $\omega \in \Omega_p^q (M)$ and $\xi \in \Omega_p^r (M)$ , for example. The action of the $(q + r)$ -form $\omega \wedge \xi$ on $q + r$ vectors is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$\Omega_p^*(M)$ is the space of all differential forms at $p$ and is closed under the exterior product.`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Observe that $\sigma_{\mu \nu}\mathrm{d}x^{\mu}\wedge \mathrm{d}x^{\nu} = 0$ , while $\alpha_{\mu \nu}\mathrm{d}x^{\mu}\wedge \mathrm{d}x^{\nu} = \omega_{\mu \nu}\mathrm{d}x^{\mu}\wedge \mathrm{d}x^{\nu}$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=215, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \begin{array}{l} \mathrm {d} x ^ {\mu} \wedge \mathrm {d} x ^ {\nu} = \mathrm {d} x ^ {\mu} \otimes \mathrm {d} x ^ {\nu} - \mathrm {d} x ^ {\nu} \otimes \mathrm {d} x ^ {\mu} \\ \mathrm {d} x ^ {\lambda} \wedge \mathrm {d} x ^ {\mu} \wedge \mathrm {d} x ^ {\nu} = \mathrm {d} x ^ {\lambda} \otimes \mathrm {d} x ^ {\mu} \otimes \mathrm {d} x ^ {\nu} + \...`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \begin{array}{l} (\omega \wedge \xi) (V _ {1}, \dots , V _ {q + r}) \\ = \frac {1}{q ! r !} \sum_ {P \in S _ {q + r}} \operatorname {s g n} (P) \omega \left(V _ {P (1)}, \dots , V _ {P (q)}\right) \xi \left(V _ {P (q + 1)}, \dots , V _ {P (q + r)}\right) \tag {5.65} \\ \end{array} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `It is readily verified that the wedge product satisfies the following.`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $\mathrm{d}x^{\mu_1}\wedge \ldots \wedge \mathrm{d}x^{\mu_r} = 0$ if some index $\mu$ appears at least twice. (ii) $\mathrm{d}x^{\mu_1}\wedge \ldots \wedge \mathrm{d}x^{\mu_r} = \operatorname {sgn}(P)\mathrm{d}x^{\mu_{P(1)}}\wedge \ldots \wedge \mathrm{d}x^{\mu_{P(r)}}.$ (iii) $\mathrm{d}x^{\mu_1}\wedge \ldots \wedge \mathrm{d}x^{\mu_r}$ is linear in ...`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $\omega_{\mu_1\mu_2\dots \mu_r}$ are taken totally anti-symmetric, reflecting the anti-symmetry of the basis. For example, the components of any second-rank tensor $\omega_{\mu \nu}$ are decomposed into the symmetric part $\sigma_{\mu \nu}$ and the anti-symmetric part $\alpha_{\mu \nu}$ :`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=216, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \left( \begin{array}{c} m \\ r \end{array} \right) = \frac {m !}{(m - r) ! r !}. $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=215, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.4. A differential form of order $r$ or an $r$ -form is a totally anti-symmetric tensor of type $(0, r)$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.1 (pdf_page=215, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.4：微分形式。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \mathrm {d} x ^ {\mu_ {1}} \wedge \mathrm {d} x ^ {\mu_ {2}} \wedge \dots \wedge \mathrm {d} x ^ {\mu_ {r}} = \sum_ {P \in S _ {r}} \operatorname {s g n} (P) \mathrm {d} x ^ {\mu_ {P (1)}} \wedge \mathrm {d} x ^ {\mu_ {P (2)}} \wedge \dots \wedge \mathrm {d} x ^ {\mu_ {P (r)}}. \tag {5.62} $$`
  - szekeres/source-snapshot-bc3404156f6e §16.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `A differential $r$-form $\alpha$ on an open subset $U \subseteq M$ is an $r$-form field, or assignment of an $r \mathrm { - f o r m ~ } \alpha _ { p }$ at every point $p \in U$ , such that the function $\alpha ( X _ { 1 } , X _ { 2 } , \ldots , X _ { r } ) ( p ) \equiv \alpha _ { p } ( ( X _ { 1 } ) _ { p } , ( X _ { 2 } ) _ { p } , \ldots , ( X _ { r } )...`
  - woit-qtgr/2025-10-20 §30.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Digression (Diferential forms). Readers may have already seen the Grassmann algebra in the context of diferential forms on $\mathbf { R } ^ { n }$ . These are known to physicists as “antisymmetric tensor $\it { f i e l d s } ^ { \prime \prime }$ , and given by taking elements of the exterior algebra $\boldsymbol { \Lambda } ^ { * } ( \mathbf { R } ^ { n }...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/01-differential-forms-and-exterior-derivative.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators.md

## 齐次接触变换(homogeneous contact transformation)

- UUID: `0a11fd94-c905-4961-b844-391fb60ef19f` · okf_url: https://limbo137.top/okf/concepts/0a11fd94-c905-4961-b844-391fb60ef19f.md
- labels: en = homogeneous contact transformation；zh-CN = 齐次接触变换
- aliases: homogeneous contact transformations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-d0e8dc219fbd62239b1a-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-785dd1593c508739385e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 内积(inner product)

- UUID: `0a507b65-dfea-4bc8-ada8-442de630a818` · okf_url: https://limbo137.top/okf/concepts/0a507b65-dfea-4bc8-ada8-442de630a818.md
- labels: en = inner product；zh-CN = 内积
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=98, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `Exercise 2.11. Let $V$ be a vector space over $\mathbb{C}$ . The inner product of two vectors $\pmb{v}_1$ and $\pmb{v}_2$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `$$ g: v ^ {j} \rightarrow g _ {i j} v ^ {j}. \tag {2.14} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=98, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `We require that the matrix $(g_{ij})$ be positive definite so that the inner product $g(\pmb{v},\pmb{v})$ has the meaning of the squared norm of $\pmb{v}$ . We also require that the metric be symmetric: $g_{ij} = g_{ji}$ so that $g(\pmb{v}_1,\pmb{v}_2) = g(\pmb{v}_2,\pmb{v}_1)$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `$$ g \left(\boldsymbol {v} _ {1}, \boldsymbol {v} _ {2}\right) \equiv \langle g \boldsymbol {v} _ {1}, \boldsymbol {v} _ {2} \rangle . \tag {2.15} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `$$ g \left(\boldsymbol {v} _ {1}, \boldsymbol {v} _ {2}\right) = v _ {1} ^ {i} g _ {j i} v _ {2} ^ {j}. \tag {2.16} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=98, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `$$ g \left(\boldsymbol {v} _ {1}, \boldsymbol {v} _ {2}\right) = \bar {v} _ {1} ^ {i} g _ {i j} v _ {2} ^ {j} \tag {2.20} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `Let $V = V(m, K)$ be a vector space with a basis $\{e_i\}$ and let $g$ be a vector space isomorphism $g: V \to V^*$ , where $g$ is an arbitrary element of $\mathrm{GL}(m, K)$ . The component representation of $g$ is`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=98, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `where $\bar{\cdot}$ denotes the complex conjugate. From the positivity and symmetry of the inner product, $g(\pmb{v}_1,\pmb{v}_2) = \overline{g(\pmb{v}_2,\pmb{v}_1)}$ , the vector space isomorphism $g:V\to V^{*}$ is required to be a positive-definite Hermitian matrix. Let $f:V\rightarrow W$ be a (complex) linear map and $G:W\to W^{*}$ be a vector space is...`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `Let us assume that the field $K$ is a real number $\mathbb{R}$ . For definiteness. Then equation (2.15) has a component expression,`
  - nakahara-geometry-topology-physics/second-edition-local §2.2.4 (pdf_page=97, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 正定实/复内积：同一向量空间V上的标量值配对，实数域为对称双线性，复数域为共轭对称半双线性，且非零向量的自配对为正实数。Nakahara§2.2.4明确要求实矩阵正定，练习2.11要求正定Hermitian矩阵；保留原UUID。不是V*×V自然配对、链与微分形式的积分配对，也不包括Szekeres/Woit允许的非正定实内积。既有书内误绑须逐节点重审，不因转正自动认可。 · 原文: `Once this isomorphism is given, we may define the inner product of two vectors $\pmb{v}_1, \pmb{v}_2 \in V$ by`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 快速下降函数(rapidly decreasing function)

- UUID: `0af8418c-bf2a-4a4f-9b66-1dc813c55557` · okf_url: https://limbo137.top/okf/concepts/0af8418c-bf2a-4a4f-9b66-1dc813c55557.md
- labels: en = rapidly decreasing function；zh-CN = 快速下降函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.3 · `zh/chapter-12/03-fourier-transforms#first-23b30995156bac1fe86b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/03-fourier-transforms.md

## 科苏尔联络(Koszul connection)

- UUID: `0b12bccf-e600-4e8b-8f23-c741396d237f` · okf_url: https://limbo137.top/okf/concepts/0b12bccf-e600-4e8b-8f23-c741396d237f.md
- labels: en = Koszul connection；zh-CN = 科苏尔联络
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-77aefb7fa1e7bce3b19b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-77aefb7fa1e7bce3b19b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 群平均(group average)

- UUID: `0b6ae429-3502-4988-a9be-fb41373ed628` · okf_url: https://limbo137.top/okf/concepts/0b6ae429-3502-4988-a9be-fb41373ed628.md
- labels: en = group average；zh-CN = 群平均
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#concept-5-group-average` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md

## 傅里叶反演(Fourier inversion)

- UUID: `0b8e66a2-7e78-4510-8c4c-61afabaae089` · okf_url: https://limbo137.top/okf/concepts/0b8e66a2-7e78-4510-8c4c-61afabaae089.md
- labels: en = Fourier inversion；zh-CN = 傅里叶反演
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §11.2 · `books/woit-qtgr/2025-10-20/en/chapter-11/02-the-group-r-and-the-fourier-transform#first-44829821f911a73fa156-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §11.2 · `books/woit-qtgr/2025-10-20/zh/chapter-11/02-the-group-r-and-the-fourier-transform#first-da5867a3b26200f282bb-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-11/02-the-group-r-and-the-fourier-transform.md

## 同伦等价(homotopy equivalence)

- UUID: `0bbe257e-1ad9-50b1-a816-f88f2c2992ae` · okf_url: https://limbo137.top/okf/concepts/0bbe257e-1ad9-50b1-a816-f88f2c2992ae.md
- labels: en = homotopy equivalence；zh-CN = 同伦等价
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=147, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.6：同伦等价。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。原书f与g复合式的左右空间下标疑似互换，保留逐字原文并标缺陷。 · 原文: `Definition 4.6. Let $X$ and $Y$ be topological spaces. $X$ and $Y$ are of the same homotopy type, written as $X \simeq Y$ , if there exist continuous maps $f: X \to Y$ and $g: Y \to X$ such that $f \circ g \sim \mathrm{id}_Y$ and $g \circ f \sim \mathrm{id}_X$ . The map $f$ is called the homotopy equivalence and $g$ , its homotopy inverse. [Remark: If $X$...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 外积(exterior product)

- UUID: `0bdd838b-0d49-44e7-8d2c-0f932b9b6db5` · okf_url: https://limbo137.top/okf/concepts/0bdd838b-0d49-44e7-8d2c-0f932b9b6db5.md
- labels: en = exterior product；zh-CN = 外积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-exterior-product` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-e88e50463654a4bd38b3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md

## 斯通–冯·诺依曼(Stone–von Neumann)

- UUID: `0bec019e-a424-4b09-b588-ac9e80189f9b` · okf_url: https://limbo137.top/okf/concepts/0bec019e-a424-4b09-b588-ac9e80189f9b.md
- labels: en = Stone–von Neumann；zh-CN = 斯通–冯·诺依曼
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.3 · `books/woit-qtgr/2025-10-20/zh/chapter-13/03-the-schr-odinger-representation#first-52f8320111586a59b4ae-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.3 · `books/woit-qtgr/2025-10-20/en/chapter-13/03-the-schr-odinger-representation#first-786105299f4be2a006a8-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/03-the-schr-odinger-representation.md

## 超雅可比恒等式(super-Jacobi identity)

- UUID: `0bec1410-0ea3-4aa7-a217-f21c439ebd4a` · okf_url: https://limbo137.top/okf/concepts/0bec1410-0ea3-4aa7-a217-f21c439ebd4a.md
- labels: en = super-Jacobi identity；zh-CN = 超雅可比恒等式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §30.2 · `books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-e9c0807aa4c7be488beb-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §30.2 · `books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-e9c0807aa4c7be488beb-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket.md

## 代数结构(algebraic structure)

- UUID: `0c15fecc-5d6e-48bc-9312-d07a837d2de9` · okf_url: https://limbo137.top/okf/concepts/0c15fecc-5d6e-48bc-9312-d07a837d2de9.md
- labels: en = algebraic structure；zh-CN = 代数结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-68dbb7a1028756971a4e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 李群的伴随表示(adjoint group representation)

- UUID: `0c30dd70-94da-4cce-81b0-03aa524fedef` · okf_url: https://limbo137.top/okf/concepts/0c30dd70-94da-4cce-81b0-03aa524fedef.md
- labels: en = adjoint group representation；zh-CN = 李群的伴随表示
- aliases: Adjoint representation；adjoint representation；伴随表示
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.1 · `books/woit-qtgr/2025-10-20/en/chapter-05/01-lie-algebras#first-039c1d6da9ac735ff8a5-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.1 · `books/woit-qtgr/2025-10-20/zh/chapter-05/01-lie-algebras#first-039c1d6da9ac735ff8a5-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.4 (pdf_page=243, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.17：伴随表示。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。此处先定义群的共轭表示；与诱导到李代数的线性伴随表示按上下文区分。 · 原文: `$$ \mathrm {a d} _ {a}: g \mapsto a g a ^ {- 1}. \tag {5.161} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.4 (pdf_page=243, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.17：伴随表示。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。此处先定义群的共轭表示；与诱导到李代数的线性伴随表示按上下文区分。 · 原文: `Definition 5.17. Take any $a \in G$ and define a homomorphism $\operatorname{ad}_a : G \to G$ by the conjugation,`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.4 (pdf_page=243, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.17：伴随表示。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。此处先定义群的共轭表示；与诱导到李代数的线性伴随表示按上下文区分。 · 原文: `This homomorphism is called the adjoint representation of $G$ .`
  - woit-qtgr/2025-10-20 §5.1 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Adjoint representation). The adjoint representation $(Ad,\mathfrak{g})$ is given by the homomorphism $$ \operatorname{Ad}: g\in G\to\operatorname{Ad}(g)\in GL(\mathfrak{g}) $$ where $A d ( g )$ acts on $X \in { \mathfrak { g } }$ by $$ (\operatorname{Ad}(g))(X)=gXg^{-1} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/01-lie-algebras.md

## 传播子，克莱因-戈尔登理论(Propagator, Klein-Gordon theory)

- UUID: `0c4ac77c-7bcc-4bfd-b1f8-352a07b796a8` · okf_url: https://limbo137.top/okf/concepts/0c4ac77c-7bcc-4bfd-b1f8-352a07b796a8.md
- labels: en = Propagator, Klein-Gordon theory；zh-CN = 传播子，克莱因-戈尔登理论
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §43.5 · `books/woit-qtgr/2025-10-20/en/chapter-43/05-the-scalar-field-propagator#first-ebedcb1fa39a4067a312-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §43.5 · `books/woit-qtgr/2025-10-20/zh/chapter-43/05-the-scalar-field-propagator#first-4d0d1092e0854a7c2195-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §43.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `相对论标量场论的传播子是振幅 $$ U (t _ {2}, \mathbf {x} _ {2}, t _ {1}, \mathbf {x} _ {1}) = \langle 0 | \widehat {\Phi} (t _ {2}, \mathbf {x} _ {2}) \widehat {\Phi} (t _ {1}, \mathbf {x} _ {1}) | 0 \rangle $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-43/05-the-scalar-field-propagator.md

## 四维电流(4-current)

- UUID: `0c5d8ec6-fa25-4c54-a29c-8c236fd5327f` · okf_url: https://limbo137.top/okf/concepts/0c5d8ec6-fa25-4c54-a29c-8c236fd5327f.md
- labels: en = 4-current；zh-CN = 四维电流
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-ca2f900bb43289184e8d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-ca2f900bb43289184e8d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 自旋表示(spin representation)

- UUID: `0c8937be-5b52-431f-a556-31c2c5f610f0` · okf_url: https://limbo137.top/okf/concepts/0c8937be-5b52-431f-a556-31c2c5f610f0.md
- labels: en = spin representation；zh-CN = 自旋表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.1 · `books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation#first-57f0c39fb5f6b90ccf1a-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.1 · `books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation#first-57f0c39fb5f6b90ccf1a-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 有限阶分布(distribution of order)

- UUID: `0cd6ae01-9496-4b3f-b107-1404b29acdb4` · okf_url: https://limbo137.top/okf/concepts/0cd6ae01-9496-4b3f-b107-1404b29acdb4.md
- labels: en = distribution of order；zh-CN = 有限阶分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-f6ce2c6788bb87b9b04d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 左不变微分形式(left-invariant differential form)

- UUID: `0d343b9b-e689-498e-ae9b-3760b06ca3c1` · okf_url: https://limbo137.top/okf/concepts/0d343b9b-e689-498e-ae9b-3760b06ca3c1.md
- labels: en = left-invariant differential form；zh-CN = 左不变微分形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-left-invariant-differential-form` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 闵可夫斯基空间傅里叶变换(Minkowski space Fourier transform)

- UUID: `0d3a1f8c-f2c8-418d-ab0f-9dc587e42313` · okf_url: https://limbo137.top/okf/concepts/0d3a1f8c-f2c8-418d-ab0f-9dc587e42313.md
- labels: en = Minkowski space Fourier transform；zh-CN = 闵可夫斯基空间傅里叶变换
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.3 · `books/woit-qtgr/2025-10-20/zh/chapter-40/03-the-fourier-transform-in-minkowski-space#first-df1e42a04f5b1645aba9-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §40.3 · `books/woit-qtgr/2025-10-20/en/chapter-40/03-the-fourier-transform-in-minkowski-space#first-c3a6c1b9a158929878e4-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §40.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [Fourier transform](/concepts/analysis/minkowski-space-fourier-transform) of a function $f$ on Minkowski space is given by $$ \begin{aligned} \widetilde {f} (p) &= \frac {1}{(2 \pi) ^ {2}} \int_ {M ^ {4}} e ^ {- i p \cdot x} f (x) d ^ {4} x \\[0.45em] &= \frac {1}{(2 \pi) ^ {2}} \int_ {M ^ {4}} e ^ {- i (- p _ {0} x _ {0} + p _ {1} x _ {...`
  - woit-qtgr/2025-10-20 §40.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `闵可夫斯基空间上函数 $f$ 的[傅里叶变换(Fourier transform)](/concepts/analysis/minkowski-space-fourier-transform)由下式给出 $$ \begin{aligned} \widetilde {f} (p) &= \frac {1}{(2 \pi) ^ {2}} \int_ {M ^ {4}} e ^ {- i p \cdot x} f (x) d ^ {4} x \\[0.45em] &= \frac {1}{(2 \pi) ^ {2}} \int_ {M ^ {4}} e ^ {- i (- p _ {0} x _ {0} + p _ {1} x _ {1} + p _ {2} x _ {2} + p ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/03-the-fourier-transform-in-minkowski-space.md

## $S$ 的临界点(Critical point for $S$)

- UUID: `0d9aff98-9f23-43f1-bc7b-b6bef79e2962` · okf_url: https://limbo137.top/okf/concepts/0d9aff98-9f23-43f1-bc7b-b6bef79e2962.md
- labels: en = Critical point for $S$；zh-CN = $S$ 的临界点
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-906c6327cfd126d06f83-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-ab1dad379cfc8e5c368f-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A path $\gamma$ is a [critical point](/concepts/calculus/critical-point-of-a-functional) of $S[\gamma]$ if $$ \delta S(\gamma)\equiv\left.\frac{d}{ds}S(\gamma_s)\right|_{s=0}=0, $$ where $\gamma_s:[t_1,t_2]\to\mathbb R^d$ is a smooth family of paths parametrized by $s\in(-\epsilon,\epsilon)$, with $\gamma_0=\gamma$.`
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `路径 $\gamma$ 是泛函 $S[\gamma]$ 的[临界点(critical point)](/concepts/calculus/critical-point-of-a-functional)，若 $$ \delta S(\gamma)\equiv\left.\frac{d}{ds}S(\gamma_s)\right|_{s=0}=0, $$ 其中 $\gamma_s:[t_1,t_2]\to\mathbb R^d$ 是由 $s\in(-\epsilon,\epsilon)$ 参数化的光滑路径族，且 $\gamma_0=\gamma$。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 偏序集(partially ordered set)

- UUID: `0dec91e9-170a-4043-84b7-74ce4d1ab250` · okf_url: https://limbo137.top/okf/concepts/0dec91e9-170a-4043-84b7-74ce4d1ab250.md
- labels: en = partially ordered set；zh-CN = 偏序集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-4c69fe0545865206bb51-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 中心函数(central function)

- UUID: `0e089f2b-e34d-4704-b872-f7d22b1e323f` · okf_url: https://limbo137.top/okf/concepts/0e089f2b-e34d-4704-b872-f7d22b1e323f.md
- labels: en = central function；zh-CN = 中心函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#concept-5-central-function` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md

## 理想(ideal)

- UUID: `0e0a5217-4c69-4190-ab6a-89df2645a4ff` · okf_url: https://limbo137.top/okf/concepts/0e0a5217-4c69-4190-ab6a-89df2645a4ff.md
- labels: en = ideal；zh-CN = 理想
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-ideal` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-492be4f952780e60d4c3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 湮灭算符(annihilation operator)

- UUID: `0e19be62-7905-43d9-ada6-eec68787c24a` · okf_url: https://limbo137.top/okf/concepts/0e19be62-7905-43d9-ada6-eec68787c24a.md
- labels: en = annihilation operator；zh-CN = 湮灭算符
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-annihilation-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-6aefa4d8f41acfacb7c7-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 纠缠(entanglement)

- UUID: `0e2b3187-27c9-43e4-998b-30db8c236b8a` · okf_url: https://limbo137.top/okf/concepts/0e2b3187-27c9-43e4-998b-30db8c236b8a.md
- labels: en = entanglement；zh-CN = 纠缠
- aliases: Entanglement；entangled
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-5ad21e2a28f7a280348e-zh` · translation)；woit-qtgr/2025-10-20 (en · §9.3 · `books/woit-qtgr/2025-10-20/en/chapter-09/03-indecomposable-vectors-and-entanglement#first-1332b482d7c855850508-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §9.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `An indecomposable state in the tensor product state space $\mathcal { H } _ { T } = \mathcal { H } _ { 1 } \otimes \mathcal { H } _ { 2 }$ is called an entangled state.`
  - woit-qtgr/2025-10-20 §9.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `张量积状态空间$\mathcal H_T=\mathcal H_1\otimes\mathcal H_2$中的不可分解态称为纠缠态。`
  - woit-qtgr/2025-10-20 §9.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `An indecomposable state in the tensor-product state space $\mathcal H_T=\mathcal H_1\otimes\mathcal H_2$ is called an entangled state.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/03-indecomposable-vectors-and-entanglement.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 帕塞瓦尔恒等式(Parseval’s identity)

- UUID: `0e2d5395-c9d1-498d-8e23-b70a76943307` · okf_url: https://limbo137.top/okf/concepts/0e2d5395-c9d1-498d-8e23-b70a76943307.md
- labels: en = Parseval’s identity；zh-CN = 帕塞瓦尔恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#first-f230af882cbb22d24846-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-f230af882cbb22d24846-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 符号差(signature)

- UUID: `0e4afba4-5ba4-439a-b77b-cd49f88c44f3` · okf_url: https://limbo137.top/okf/concepts/0e4afba4-5ba4-439a-b77b-cd49f88c44f3.md
- labels: en = signature；zh-CN = 符号差
- aliases: signature of a quadratic form；二次型的符号差
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §28.2 · `books/woit-qtgr/2025-10-20/en/chapter-28/05-real-cliford-algebras#first-9f35a1c74678f1509bb6-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §28.2 · `books/woit-qtgr/2025-10-20/zh/chapter-28/05-real-cliford-algebras#first-9f35a1c74678f1509bb6-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/05-real-cliford-algebras.md

## Weyl 方程(Weyl equations)

- UUID: `0e844881-34ba-4782-be6c-b34f2c93a208` · okf_url: https://limbo137.top/okf/concepts/0e844881-34ba-4782-be6c-b34f2c93a208.md
- labels: en = Weyl equations；zh-CN = Weyl 方程
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §47.3 · `books/woit-qtgr/2025-10-20/en/chapter-47/05-weyl-spinors#first-acf9070fa374b3225b61-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §47.3 · `books/woit-qtgr/2025-10-20/zh/chapter-47/05-weyl-spinors#first-dcbf76b9f24df73cb24a-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §47.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `二分量旋量的 Weyl 波动方程为 $$ \left(\frac {\partial}{\partial t} + \pmb {\sigma} \cdot \pmb {\nabla}\right) \psi_ {R} = 0\tag{47.16} $$ $$ \left(\frac {\partial}{\partial t} - \boldsymbol {\sigma} \cdot \nabla\right) \psi_ {L} = 0\tag{47.17} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-47/05-weyl-spinors.md

## 李代数自同构作用(Action by Lie algebra automorphisms)

- UUID: `0ee9d2e9-d61c-4c92-90bc-0c305f648a98` · okf_url: https://limbo137.top/okf/concepts/0ee9d2e9-d61c-4c92-90bc-0c305f648a98.md
- labels: en = Action by Lie algebra automorphisms；zh-CN = 李代数自同构作用
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §16.2 · `books/woit-qtgr/2025-10-20/zh/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-2323599dcec4f6d6191c-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §16.2 · `books/woit-qtgr/2025-10-20/en/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-873ca166df5674b0fe14-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `An action of a group $G$ on a Lie algebra $\mathfrak h$, written $$ X\in\mathfrak h\longmapsto\phi_g(X)\in\mathfrak h, $$ is by automorphisms when $$ [\phi_g(X),\phi_g(Y)]=\phi_g([X,Y]) $$ for every $g\in G$ and $X,Y\in\mathfrak h$.`
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `群 $G$ 对李代数 $\mathfrak h$ 的作用记作 $$ X\in\mathfrak h\longmapsto\phi_g(X)\in\mathfrak h $$ 当它满足 $$ [\phi_g(X),\phi_g(Y)]=\phi_g([X,Y]) $$ 对所有 $g\in G$ 及 $X,Y\in\mathfrak h$ 都成立时，称该作用为自同构作用。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group.md

## 离散拓扑(discrete topology)

- UUID: `0effb2c1-c32e-4d63-80be-762ba1393f3b` · okf_url: https://limbo137.top/okf/concepts/0effb2c1-c32e-4d63-80be-762ba1393f3b.md
- labels: en = discrete topology；zh-CN = 离散拓扑
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-4fb6a5c1fc44932c3391-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 上极限(limit superior)

- UUID: `0f3e2bdd-02cb-43d0-b1da-3a225197856f` · okf_url: https://limbo137.top/okf/concepts/0f3e2bdd-02cb-43d0-b1da-3a225197856f.md
- labels: en = limit superior；zh-CN = 上极限
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-9004b3ee00037450534b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 世界线(world-line)

- UUID: `0f98d8d9-386f-44fc-b786-2c81bdb50d6e` · okf_url: https://limbo137.top/okf/concepts/0f98d8d9-386f-44fc-b786-2c81bdb50d6e.md
- labels: en = world-line；zh-CN = 世界线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-5d5ac951d2864fd33092-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-5d5ac951d2864fd33092-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## Pauli不相容原理(Pauli exclusion principle)

- UUID: `0faa133b-6cda-4cf8-a159-9cf3987a7652` · okf_url: https://limbo137.top/okf/concepts/0faa133b-6cda-4cf8-a159-9cf3987a7652.md
- labels: en = Pauli exclusion principle；zh-CN = Pauli不相容原理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §36.1 · `books/woit-qtgr/2025-10-20/zh/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-20a52d97eb40cde8705d-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 自治系统(autonomous system)

- UUID: `0fbe108b-1f27-4fb8-83b1-605f2050e9a7` · okf_url: https://limbo137.top/okf/concepts/0fbe108b-1f27-4fb8-83b1-605f2050e9a7.md
- labels: en = autonomous system；zh-CN = 自治系统
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-autonomous-system` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 复射影空间(Complex projective space)

- UUID: `1002fcb8-4c5d-4471-a9be-ebff4f4e5dd1` · okf_url: https://limbo137.top/okf/concepts/1002fcb8-4c5d-4471-a9be-ebff4f4e5dd1.md
- labels: en = Complex projective space；zh-CN = 复射影空间
- aliases: complex projective space
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-f8b69cced03b8c0c2678-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-f8b69cced03b8c0c2678-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 反同态(anti-homomorphism)

- UUID: `1015d31e-07e0-47cf-8a6d-4020f4408a3f` · okf_url: https://limbo137.top/okf/concepts/1015d31e-07e0-47cf-8a6d-4020f4408a3f.md
- labels: en = anti-homomorphism；zh-CN = 反同态
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-anti-homomorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md

## 标量积(scalar product)

- UUID: `10861f1d-f72d-4e92-808c-bba46d7cd49c` · okf_url: https://limbo137.top/okf/concepts/10861f1d-f72d-4e92-808c-bba46d7cd49c.md
- labels: en = scalar product；zh-CN = 标量积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-scalar-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 立方链(cubical chain)

- UUID: `10ef9d40-5020-4092-b70f-6d91f54ca82c` · okf_url: https://limbo137.top/okf/concepts/10ef9d40-5020-4092-b70f-6d91f54ca82c.md
- labels: en = cubical chain；zh-CN = 立方链
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-0851303442da353fac20-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 边界映射(boundary map)

- UUID: `110a004a-8beb-4140-bb65-e3894be9875d` · okf_url: https://limbo137.top/okf/concepts/110a004a-8beb-4140-bb65-e3894be9875d.md
- labels: en = boundary map；zh-CN = 边界映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-6c96ebf86c51fb558345-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-6c96ebf86c51fb558345-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 上同调的(cohomologous)

- UUID: `114ca9e4-e0ac-482e-9731-3186bbc7dbfa` · okf_url: https://limbo137.top/okf/concepts/114ca9e4-e0ac-482e-9731-3186bbc7dbfa.md
- labels: en = cohomologous；zh-CN = 上同调的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-9f5e2943e091be4de6eb-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-9f5e2943e091be4de6eb-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 等价范数(equivalent norms)

- UUID: `11c7fa5b-d5f8-40eb-ae0b-7e37ff4bebb1` · okf_url: https://limbo137.top/okf/concepts/11c7fa5b-d5f8-40eb-ae0b-7e37ff4bebb1.md
- labels: en = equivalent norms；zh-CN = 等价范数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.9 · `zh/chapter-10/09-topological-vector-spaces#first-9551e9906b8359a8622e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md

## 对偶空间(dual space)

- UUID: `1212910c-fd6e-468e-8315-db95c69ef22e` · okf_url: https://limbo137.top/okf/concepts/1212910c-fd6e-468e-8315-db95c69ef22e.md
- labels: en = dual space；zh-CN = 对偶空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-dual-space` · translation)；woit-qtgr/2025-10-20 (zh-CN · §4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/index#first-e7062a6978544669061d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md

## 真空态(vacuum state)

- UUID: `125009af-e084-4c92-b27e-de031a9fa1fb` · okf_url: https://limbo137.top/okf/concepts/125009af-e084-4c92-b27e-de031a9fa1fb.md
- labels: en = vacuum state；zh-CN = 真空态
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.2 · `books/woit-qtgr/2025-10-20/en/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-04d70685b6bf85740fae-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §36.2 · `books/woit-qtgr/2025-10-20/zh/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-04d70685b6bf85740fae-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism.md

## 对称变换群(symmetry group)

- UUID: `126c82d1-85b0-4175-af2f-9d01a27e0e2e` · okf_url: https://limbo137.top/okf/concepts/126c82d1-85b0-4175-af2f-9d01a27e0e2e.md
- labels: en = symmetry group；zh-CN = 对称变换群
- aliases: 对称群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-symmetry-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.5 · `books/woit-qtgr/2025-10-20/zh/chapter-01/05-groups-and-symmetries#concept-1-symmetry-group` · translation)；woit-qtgr/2025-10-20 (en · §1.5 · `books/woit-qtgr/2025-10-20/en/chapter-01/05-groups-and-symmetries#concept-1-symmetry-group` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §2.7 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `It is a straightforward matter to extend the above theorem to an arbitrary set $\mathcal { F }$ offunctions on $X$. The group oftransformations leaving all functions $F \in { \mathcal { F } }$ invariant will be called the invariance group or symmetry group of $\mathcal { F }$ . The following are some important examples of symmetry groups in mathematical p...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/04-conservation-of-charge-and-u-1-symmetry.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/05-groups-and-symmetries.md

## 群表示(group representation)

- UUID: `1273c312-ea76-4998-85e7-e75280ef3ebb` · okf_url: https://limbo137.top/okf/concepts/1273c312-ea76-4998-85e7-e75280ef3ebb.md
- labels: en = group representation；zh-CN = 群表示
- aliases: Representation；group representations；representation；表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-representation` · translation)；woit-qtgr/2025-10-20 (en · §1.3 · `books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations#first-c96dd35415fecc4ceac3-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-c96dd35415fecc4ceac3-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §1.3 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Representation). A representation $( \pi , V )$ of a group $G$ is a homomorphism $$ \pi : g \in G \to \pi (g) \in GL(V) $$ where $GL(V)$ denotes the [general linear group](/concepts/linear-algebra/general-linear-group), the group of invertible linear maps $V\to V$ for a vector space $V$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 辛结构(symplectic structure)

- UUID: `128dd2b3-9341-4ec8-80d7-756b3c868174` · okf_url: https://limbo137.top/okf/concepts/128dd2b3-9341-4ec8-80d7-756b3c868174.md
- labels: en = symplectic structure；zh-CN = 辛结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-d0cb31504300ef340ba8-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-d0cb31504300ef340ba8-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 类时向量(timelike vector)

- UUID: `132c84d0-902e-4ab2-91b9-d41dd49e754c` · okf_url: https://limbo137.top/okf/concepts/132c84d0-902e-4ab2-91b9-d41dd49e754c.md
- labels: en = timelike vector；zh-CN = 类时向量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.1 · `books/woit-qtgr/2025-10-20/zh/chapter-40/01-minkowski-space#first-08e3f1fa825acf85da13-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 单位分解(partition of unity)

- UUID: `1343270b-b407-4d8c-bb65-0292de19ed56` · okf_url: https://limbo137.top/okf/concepts/1343270b-b407-4d8c-bb65-0292de19ed56.md
- labels: en = partition of unity；zh-CN = 单位分解
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.1 · `zh/chapter-17/01-partitions-of-unity#first-7ce7e75d69be1587295d-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.1 · `en/chapter-17/01-partitions-of-unity#first-7ce7e75d69be1587295d-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Although a different atlas $\{(V_i, \psi_i)\}$ gives different coordinates and a different partition of unity, the integral defined by (5.102) remains the same.`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ f (p) = \sum_ {i} f (p) \varepsilon_ {i} (p) = \sum_ {i} f _ {i} (p) \tag {5.101} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.7. Take an open covering $\{U_i\}$ of $M$ such that each point of $M$ is covered with a finite number of $U_i$ . [If this is always possible, $M$ is called paracompact, which we assume to be the case.] If a family of differentiable functions $\varepsilon_i(p)$ satisfies`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \int_ {M} f \omega \equiv \sum_ {i} \int_ {U _ {i}} f _ {i} \omega . \tag {5.102} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $0\leq \varepsilon_{i}(p)\leq 1$ (ii) $\varepsilon_{i}(p) = 0$ if $p\notin U_i$ and (iii) $\varepsilon_1(p) + \varepsilon_2(p) + \ldots = 1$ for any point $p\in M$`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $f_{i}(p) \equiv f(p)\varepsilon_{i}(p)$ vanishes outside $U_{i}$ by (ii). Hence, given a point $p \in M$ , assumed paracompactness ensures that there are only finite terms in the summation over $i$ in (5.101). For each $f_{i}(p)$ , we may define the integral over $U_{i}$ according to (5.100). Finally the integral of $f$ on $M$ is given by`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `the family $\{\varepsilon(p)\}$ is called a partition of unity subordinate to the covering $\{U_i\}$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.2 (pdf_page=225, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.7：单位分解。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `From condition (iii), it follows that`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/01-partitions-of-unity.md

## 度量拓扑(metric topology)

- UUID: `135f90bc-5ac6-4fc2-a4f1-8642700ae19a` · okf_url: https://limbo137.top/okf/concepts/135f90bc-5ac6-4fc2-a4f1-8642700ae19a.md
- labels: en = metric topology；zh-CN = 度量拓扑
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-c481de73d384a57a12fa-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 核磁共振波谱学(Nuclear magnetic resonance spectroscopy)

- UUID: `13e70843-34ca-4e4e-9bc3-d336dbb4c734` · okf_url: https://limbo137.top/okf/concepts/13e70843-34ca-4e4e-9bc3-d336dbb4c734.md
- labels: en = Nuclear magnetic resonance spectroscopy；zh-CN = 核磁共振波谱学
- aliases: nuclear magnetic resonance spectroscopy
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-nuclear-magnetic-resonance-spectroscopy` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-33f4fa18cf04ef68901a-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 支集(support)

- UUID: `13eb108a-e792-47e5-999f-e7000f71deee` · okf_url: https://limbo137.top/okf/concepts/13eb108a-e792-47e5-999f-e7000f71deee.md
- labels: en = support；zh-CN = 支集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.2 · `en/chapter-17/02-integration-of-n-forms#first-2f2d8bdeefcf3528a381-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-2f2d8bdeefcf3528a381-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/02-integration-of-n-forms.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 核心指标记法(kernel index notation)

- UUID: `1435ed7d-50c4-4a83-adfe-50f270d61d40` · okf_url: https://limbo137.top/okf/concepts/1435ed7d-50c4-4a83-adfe-50f270d61d40.md
- labels: en = kernel index notation；zh-CN = 核心指标记法
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.3 · `en/chapter-07/03-basis-representation-of-tensors#first-9682eeb03d3d57ac469d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.3 · `zh/chapter-07/03-basis-representation-of-tensors#first-9682eeb03d3d57ac469d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/03-basis-representation-of-tensors.md

## 实内积(real inner product)

- UUID: `1438172c-4529-4171-9338-e797f954f1fb` · okf_url: https://limbo137.top/okf/concepts/1438172c-4529-4171-9338-e797f954f1fb.md
- labels: en = real inner product；zh-CN = 实内积
- aliases: inner product；内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-real-inner-product` · translation)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-7fbfc15c27cd8776f6cd-real-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §5.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 实向量空间上的对称、非退化双线性映射V×V→R。Szekeres§5.1在有限维情形定义，明确不要求正定；Woit§4.4也将正定与不定区分。不得自动等同于正定实/复内积的统称，也不包括V*×V自然配对。原文inner product在明确实数域语境可绑定本概念而译为“内积”，不可仅因首选名而在译文擅加“实”；使用积分配对时保留作者的空间及适用条件，不外推有限维结论。 · 原文: `Let $V$ be a real finite dimensional vector space with $\dim$ $V = n$ . A real inner product, often referred to simply as an inner product when there is no danger of confusion, on the vector space $V$ is a map $V \times V \to \mathbb { R }$ that assigns a real number $u \cdot v \in \mathbb { R }$ to every pair of vectors $u$, $v \in V$ satisfying the foll...`
  - szekeres/source-snapshot-bc3404156f6e §5.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 实向量空间上的对称、非退化双线性映射V×V→R。Szekeres§5.1在有限维情形定义，明确不要求正定；Woit§4.4也将正定与不定区分。不得自动等同于正定实/复内积的统称，也不包括V*×V自然配对。原文inner product在明确实数域语境可绑定本概念而译为“内积”，不可仅因首选名而在译文擅加“实”；使用积分配对时保留作者的空间及适用条件，不外推有限维结论。 · 原文: `$$ (a u + b v) \cdot w = w \cdot (a u + b v) = a w \cdot u + b w \cdot v = a u \cdot w + b v \cdot w. $$`
  - szekeres/source-snapshot-bc3404156f6e §5.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 实向量空间上的对称、非退化双线性映射V×V→R。Szekeres§5.1在有限维情形定义，明确不要求正定；Woit§4.4也将正定与不定区分。不得自动等同于正定实/复内积的统称，也不包括V*×V自然配对。原文inner product在明确实数域语境可绑定本概念而译为“内积”，不可仅因首选名而在译文擅加“实”；使用积分配对时保留作者的空间及适用条件，不外推有限维结论。 · 原文: `We often refer to this linearity in both arguments by saying that the inner product is bilinear. As a consequence of property (RIP3) the inner product is said to be non-singular and is often referred to as pseudo-Euclidean. Sometimes (RIP3) is replaced by the stronger condition (RIP3') $u \cdot u > 0$ for all vectors $u \neq 0 .$ In this case the inner pr...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（实内积(real inner product)）。实向量空间 $V$ 上的一个内积是一个对称的 $( ( v , w ) = ( w , v ) )$ 映射 $$ (\cdot , \cdot): V \times V \to \mathbf {R} $$ 即非退化且对两个变量均为线性。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 自发破缺(spontaneously broken)

- UUID: `14428911-958a-420b-b2c4-d079f505c8b7` · okf_url: https://limbo137.top/okf/concepts/14428911-958a-420b-b2c4-d079f505c8b7.md
- labels: en = spontaneously broken；zh-CN = 自发破缺
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33.2 · `books/woit-qtgr/2025-10-20/en/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential#first-e861fec40e0a2930db31-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §33.2 · `books/woit-qtgr/2025-10-20/zh/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential#first-e861fec40e0a2930db31-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential.md

## 绝热过程(adiabatic process)

- UUID: `14623694-1df3-4e83-a965-41f9ba35cef2` · okf_url: https://limbo137.top/okf/concepts/14623694-1df3-4e83-a965-41f9ba35cef2.md
- labels: en = adiabatic process；zh-CN = 绝热过程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-3d772f2db1d521a15f32-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 不变群(invariance group)

- UUID: `14a17663-a83c-4229-aa36-585971c3f84f` · okf_url: https://limbo137.top/okf/concepts/14a17663-a83c-4229-aa36-585971c3f84f.md
- labels: en = invariance group；zh-CN = 不变群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-invariance-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 反自伴矩阵(skew-adjoint matrix)

- UUID: `14b2f149-bd42-4f49-b8b9-6bcd538c6992` · okf_url: https://limbo137.top/okf/concepts/14b2f149-bd42-4f49-b8b9-6bcd538c6992.md
- labels: en = skew-adjoint matrix；zh-CN = 反自伴矩阵
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.2 · `books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices#first-17f0edac1ef6688ad627-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/02-commutation-relations-for-pauli-matrices.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 非奇异矩阵(non-singular matrix)

- UUID: `14cdbba0-4af8-4d23-a2a9-3cfb5c0ad747` · okf_url: https://limbo137.top/okf/concepts/14cdbba0-4af8-4d23-a2a9-3cfb5c0ad747.md
- labels: en = non-singular matrix；zh-CN = 非奇异矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-non-singular-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 自旋角动量(spin angular momentum)

- UUID: `14f530a7-cae3-4199-a129-754aeece7b1f` · okf_url: https://limbo137.top/okf/concepts/14f530a7-cae3-4199-a129-754aeece7b1f.md
- labels: en = spin angular momentum；zh-CN = 自旋角动量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-7d8bfa0700d40e3eaff6-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-7d8bfa0700d40e3eaff6-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §34.2 · `books/woit-qtgr/2025-10-20/zh/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-bbddee0bc3721a136a9a-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §34.2 · `books/woit-qtgr/2025-10-20/en/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-bbddee0bc3721a136a9a-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/05-spatial-symmetries.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3.md

## 对称算符(symmetric operator)

- UUID: `1510ce98-1ea2-4c9a-8847-10f4ea3df1e4` · okf_url: https://limbo137.top/okf/concepts/1510ce98-1ea2-4c9a-8847-10f4ea3df1e4.md
- labels: en = symmetric operator；zh-CN = 对称算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-3610f6d7e6ef544619d8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-symmetric-operator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 等距的(isometric)

- UUID: `151179b3-1206-4b6c-a3b9-d229f1f77314` · okf_url: https://limbo137.top/okf/concepts/151179b3-1206-4b6c-a3b9-d229f1f77314.md
- labels: en = isometric；zh-CN = 等距的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-4e0f8481f19555a6cfd3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-isometric` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 不稳定结点(unstable node)

- UUID: `156a1541-7be0-497b-99b7-df1db1262250` · okf_url: https://limbo137.top/okf/concepts/156a1541-7be0-497b-99b7-df1db1262250.md
- labels: en = unstable node；zh-CN = 不稳定结点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-unstable-node` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 自同构(automorphism)

- UUID: `159b1327-9fd5-45ae-8358-8a668636d673` · okf_url: https://limbo137.top/okf/concepts/159b1327-9fd5-45ae-8358-8a668636d673.md
- labels: en = automorphism；zh-CN = 自同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-automorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 仿射变换(affine transformation)

- UUID: `15b28761-a301-4d39-b113-7b1f58d08987` · okf_url: https://limbo137.top/okf/concepts/15b28761-a301-4d39-b113-7b1f58d08987.md
- labels: en = affine transformation；zh-CN = 仿射变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-affine-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 完备可观测量(complete observable)

- UUID: `15b77a32-b796-4d9e-9b2a-b42524bbd897` · okf_url: https://limbo137.top/okf/concepts/15b77a32-b796-4d9e-9b2a-b42524bbd897.md
- labels: en = complete observable；zh-CN = 完备可观测量
- aliases: complete
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-2d6d7e1b24ad7449faba-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-d4248df8d36a72045755-observable-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 紧致空间(compact space)

- UUID: `15bc6cfe-c68f-43af-aed8-4b5136e30628` · okf_url: https://limbo137.top/okf/concepts/15bc6cfe-c68f-43af-aed8-4b5136e30628.md
- labels: en = compact space；zh-CN = 紧致空间
- aliases: compact；compactness；紧致；紧致性；紧致的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-ab22e837ff7606614c9e-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.5 (pdf_page=103, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.7：紧致空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `In general, if a set is compact in $\mathbb{R}^n$ , it must be bounded. What else is needed? We state the result without the proof.`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.5 (pdf_page=103, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.7：紧致空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.7. Consider a set $X$ and all possible coverings of $X$ . The set $X$ is compact if, for every open covering $\{U_i | i \in I\}$ , there exists a finite subset $J$ of $I$ such that $\{U_j | j \in J\}$ is also a covering of $X$ .`
  - szekeres/source-snapshot-bc3404156f6e §10.6 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书逐字条件保留；不同书路径分别记录。 · 原文: `A topological space $( X , { \mathcal { O } } )$ is said to be compact if every open covering of $X$ contains a finite subcovering. The motivation for this definition lies in the following theorem, the proof of which can be found in standard books on analysis [10–12].`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 李导数(Lie derivative)

- UUID: `16292e18-9c89-4c3a-b3b4-104cec97244e` · okf_url: https://limbo137.top/okf/concepts/16292e18-9c89-4c3a-b3b4-104cec97244e.md
- labels: en = Lie derivative；zh-CN = 李导数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-8579cbe7493608abf999-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-lie-derivative` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md

## 罗素悖论(Russell's paradox)

- UUID: `16a93206-68c6-4611-a639-0f12fe231a59` · okf_url: https://limbo137.top/okf/concepts/16a93206-68c6-4611-a639-0f12fe231a59.md
- labels: en = Russell's paradox；zh-CN = 罗素悖论
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-a1a1dabeea070f314a31-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 拉格朗日定理(Lagrange’s theorem)

- UUID: `16df3086-ee73-47de-8a9c-e0a17709c146` · okf_url: https://limbo137.top/okf/concepts/16df3086-ee73-47de-8a9c-e0a17709c146.md
- labels: en = Lagrange’s theorem；zh-CN = 拉格朗日定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §2.5 · `en/chapter-02/05-normal-subgroups-and-factor-groups#first-e3b29e3a2bdd76ae5eeb-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#first-e3b29e3a2bdd76ae5eeb-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 鬼(ghosts)

- UUID: `16e15429-a004-4dab-a343-1d520139d5ff` · okf_url: https://limbo137.top/okf/concepts/16e15429-a004-4dab-a343-1d520139d5ff.md
- labels: en = ghosts；zh-CN = 鬼
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §31.2 · `books/woit-qtgr/2025-10-20/en/chapter-31/03-the-schr-odinger-representation-for-fermions-ghosts#first-0f2028ef8130e4980332-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §31.2 · `books/woit-qtgr/2025-10-20/zh/chapter-31/03-the-schr-odinger-representation-for-fermions-ghosts#first-0f2028ef8130e4980332-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/03-the-schr-odinger-representation-for-fermions-ghosts.md

## 洛伦兹变换(Lorentz transformation)

- UUID: `16fe5129-802f-4513-bc1c-e3aba06cdc0a` · okf_url: https://limbo137.top/okf/concepts/16fe5129-802f-4513-bc1c-e3aba06cdc0a.md
- labels: en = Lorentz transformation；zh-CN = 洛伦兹变换
- aliases: Lorentz transformations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-lorentz-transformation` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-dac2b73f271fd296dd15-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 带手柄和洞的可定向曲面(orientable surface with handles and holes)

- UUID: `171228e3-696c-4c54-83ee-25852fa9d668` · okf_url: https://limbo137.top/okf/concepts/171228e3-696c-4c54-83ee-25852fa9d668.md
- labels: en = orientable surface with handles and holes；zh-CN = 带手柄和洞的可定向曲面
- aliases: 2-sphere with handles and holes；orientable two-dimensional surface
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 正则配分函数(canonical partition function)

- UUID: `1717a8e7-a0ea-44a0-bc1a-cd3aacaac36d` · okf_url: https://limbo137.top/okf/concepts/1717a8e7-a0ea-44a0-bc1a-cd3aacaac36d.md
- labels: en = canonical partition function；zh-CN = 正则配分函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.4 · `zh/chapter-14/04-quantum-statistical-mechanics#first-275b69d7b37208835d63-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.4 · `en/chapter-14/04-quantum-statistical-mechanics#first-275b69d7b37208835d63-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/04-quantum-statistical-mechanics.md

## 斯特恩–盖拉赫实验(Stern–Gerlach experiment)

- UUID: `17282fd6-0595-40df-a266-9ec044b18a12` · okf_url: https://limbo137.top/okf/concepts/17282fd6-0595-40df-a266-9ec044b18a12.md
- labels: en = Stern–Gerlach experiment；zh-CN = 斯特恩–盖拉赫实验
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-148f5599bd45c2f21453-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-stern-gerlach-experiment` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 外延公理(axiom of extensionality)

- UUID: `173248e5-fde2-41cb-a2da-8c6ae8ce612c` · okf_url: https://limbo137.top/okf/concepts/173248e5-fde2-41cb-a2da-8c6ae8ce612c.md
- labels: en = axiom of extensionality；zh-CN = 外延公理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-d736b244ccd70161a951-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 闵可夫斯基内积(Minkowskian inner product)

- UUID: `17c9f610-9860-4c29-9004-115ae6174af4` · okf_url: https://limbo137.top/okf/concepts/17c9f610-9860-4c29-9004-115ae6174af4.md
- labels: en = Minkowskian inner product；zh-CN = 闵可夫斯基内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-minkowskian-inner-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 协变次数(covariant degree)

- UUID: `17cee705-a2d5-4453-8b19-52af4f65fd59` · okf_url: https://limbo137.top/okf/concepts/17cee705-a2d5-4453-8b19-52af4f65fd59.md
- labels: en = covariant degree；zh-CN = 协变次数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-ca9bb531b7d5655024de-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-ca9bb531b7d5655024de-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 向量(vector)

- UUID: `17eeb36d-36ff-4715-8dd6-5eca459daf1e` · okf_url: https://limbo137.top/okf/concepts/17eeb36d-36ff-4715-8dd6-5eca459daf1e.md
- labels: en = vector；zh-CN = 向量
- aliases: 矢量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-vector` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 四维标量(4-scalar)

- UUID: `184255fe-e4dd-40ca-a6b9-ec43b3eb54f1` · okf_url: https://limbo137.top/okf/concepts/184255fe-e4dd-40ca-a6b9-ec43b3eb54f1.md
- labels: en = 4-scalar；zh-CN = 四维标量
- aliases: 4-scalars
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-54c08880b8b953aeb2b8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-5b2bf690504e62558aa1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 斯通定理(Stone’s theorem)

- UUID: `19388d06-7557-4098-b606-17bd8ea7fcae` · okf_url: https://limbo137.top/okf/concepts/19388d06-7557-4098-b606-17bd8ea7fcae.md
- labels: en = Stone’s theorem；zh-CN = 斯通定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-55a7a9e7bc456db63afe-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#first-55a7a9e7bc456db63afe-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 真空真(vacuous truth)

- UUID: `197008e2-3d14-48ff-ade5-a7008b2596cf` · okf_url: https://limbo137.top/okf/concepts/197008e2-3d14-48ff-ade5-a7008b2596cf.md
- labels: en = vacuous truth；zh-CN = 真空真
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-5d6cb6950b9bdce72947-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 实Clifford代数(real Clifford algebra)

- UUID: `1975f833-3088-44ad-97dd-6f1f7cf97583` · okf_url: https://limbo137.top/okf/concepts/1975f833-3088-44ad-97dd-6f1f7cf97583.md
- labels: en = real Clifford algebra；zh-CN = 实Clifford代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-eb0e4a68b0a47e3ae817-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-clifford-algebra` · translation)；woit-qtgr/2025-10-20 (zh-CN · §28.2 · `books/woit-qtgr/2025-10-20/zh/chapter-28/05-real-cliford-algebras#first-0e4918fc565d97754b74-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §28.2 · `books/woit-qtgr/2025-10-20/en/chapter-28/05-real-cliford-algebras#first-0e4918fc565d97754b74-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/05-real-cliford-algebras.md

## 特征向量(eigenvector)

- UUID: `19c6e3d7-39e1-4c32-8812-3bd9d8a9958e` · okf_url: https://limbo137.top/okf/concepts/19c6e3d7-39e1-4c32-8812-3bd9d8a9958e.md
- labels: en = eigenvector；zh-CN = 特征向量
- aliases: eigenvectors；本征向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-eigenvector` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-6e51c37775e465f0897f-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-b8509e0d59e70e8efc9b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-84bccf053b8b34d182e5-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §13.5 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 非零向量u满足Au=αu；不能把零向量当作特征向量。 · 原文: `As in Chapter 4 a complex number $\alpha$ is an eigenvalue of a bounded linear operator $A$ : $\mathcal { H } \to \mathcal { H }$ if there exists a non-zero vector $u \in \mathcal { H }$ such that $$ A u = \alpha u. $$ $u$ is called the eigenvector of $A$ corresponding to the eigenvalue $\alpha$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 协变(covariant)

- UUID: `19d0d910-333a-4359-9edf-c6a555de1f6d` · okf_url: https://limbo137.top/okf/concepts/19d0d910-333a-4359-9edf-c6a555de1f6d.md
- labels: en = covariant；zh-CN = 协变
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-covariant` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md

## 对易子(commutator)

- UUID: `1a15b2f0-5129-43ae-a79b-291a51fb196e` · okf_url: https://limbo137.top/okf/concepts/1a15b2f0-5129-43ae-a79b-291a51fb196e.md
- labels: en = commutator；zh-CN = 对易子
- aliases: commutators
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-commutator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-3cb8369bd67f180c0676-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §3.2 · `books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices#first-522bb45847f0c2c64d7a-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §3.2 · `books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices#first-c01d6e6a0ebd504fbd78-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/02-commutation-relations-for-pauli-matrices.md

## 谱定理(spectral theorem)

- UUID: `1a675aab-c7ae-434f-b33d-dbac51890000` · okf_url: https://limbo137.top/okf/concepts/1a675aab-c7ae-434f-b33d-dbac51890000.md
- labels: en = spectral theorem；zh-CN = 谱定理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.7 · `books/woit-qtgr/2025-10-20/zh/chapter-04/07-eigenvalues-and-eigenvectors#first-65a47865eb101c002a18-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 克利福德代数(Clifford algebra)

- UUID: `1a91f050-4afa-5797-99a5-a6c742286b15` · okf_url: https://limbo137.top/okf/concepts/1a91f050-4afa-5797-99a5-a6c742286b15.md
- labels: en = Clifford algebra；zh-CN = 克利福德代数
- aliases: Clifford代数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 积分曲线(integral curve)

- UUID: `1a927244-c87d-49c1-9cbe-ea6d8ae7c2d6` · okf_url: https://limbo137.top/okf/concepts/1a927244-c87d-49c1-9cbe-ea6d8ae7c2d6.md
- labels: en = integral curve；zh-CN = 积分曲线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-4b72a7387de67b3e6724-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-integral-curve` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md

## 置换表示(permutation representation)

- UUID: `1ad68c54-876c-4290-88be-64552e9af83e` · okf_url: https://limbo137.top/okf/concepts/1ad68c54-876c-4290-88be-64552e9af83e.md
- labels: en = permutation representation；zh-CN = 置换表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-90759a874fd66575f054-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 复合(composition)

- UUID: `1ad7f304-3bf4-48cb-a56f-44b509bd7bd7` · okf_url: https://limbo137.top/okf/concepts/1ad7f304-3bf4-48cb-a56f-44b509bd7bd7.md
- labels: en = composition；zh-CN = 复合
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-30aee277558d95155bf2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 第二可数(second countable)

- UUID: `1ae3cb73-f44a-4483-b175-f7481d779ac8` · okf_url: https://limbo137.top/okf/concepts/1ae3cb73-f44a-4483-b175-f7481d779ac8.md
- labels: en = second countable；zh-CN = 第二可数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-d3666106efe7fd9144c7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 单位四元数(unit quaternion)

- UUID: `1afbc47e-4026-42fb-a8c1-d2b6a02638c4` · okf_url: https://limbo137.top/okf/concepts/1afbc47e-4026-42fb-a8c1-d2b6a02638c4.md
- labels: en = unit quaternion；zh-CN = 单位四元数
- aliases: unit quaternions
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.3 · `books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary#concept-6-3-quaternion` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.3 · `books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary#concept-6-3-quaternion` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 拉格朗日函数(Lagrangian function)

- UUID: `1b151f83-697f-43f4-b6a1-aa3dfd976f64` · okf_url: https://limbo137.top/okf/concepts/1b151f83-697f-43f4-b6a1-aa3dfd976f64.md
- labels: en = Lagrangian function；zh-CN = 拉格朗日函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-8d838ffb5d6b1f86836a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-8d838ffb5d6b1f86836a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 超对称振子(supersymmetric oscillator)

- UUID: `1b370f94-adf5-40b2-8a15-70d09940d393` · okf_url: https://limbo137.top/okf/concepts/1b370f94-adf5-40b2-8a15-70d09940d393.md
- labels: en = supersymmetric oscillator；zh-CN = 超对称振子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33.1 · `books/woit-qtgr/2025-10-20/en/chapter-33/01-the-supersymmetric-oscillator#first-c2ed14d5226d06f7a497-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §33.1 · `books/woit-qtgr/2025-10-20/zh/chapter-33/01-the-supersymmetric-oscillator#first-c2ed14d5226d06f7a497-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/01-the-supersymmetric-oscillator.md

## 标势(scalar potential)

- UUID: `1bb86a2d-02e3-4010-a168-8df5d51e88e1` · okf_url: https://limbo137.top/okf/concepts/1bb86a2d-02e3-4010-a168-8df5d51e88e1.md
- labels: en = scalar potential；zh-CN = 标势
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-280adbc13f7a2466cd76-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-280adbc13f7a2466cd76-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 束缚指标(bound index)

- UUID: `1bc09023-d978-4710-a687-1c826fbdf172` · okf_url: https://limbo137.top/okf/concepts/1bc09023-d978-4710-a687-1c826fbdf172.md
- labels: en = bound index；zh-CN = 束缚指标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-bound-index` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 同调类(homology class)

- UUID: `1c78e56f-4242-5823-8a95-f2b292733971` · okf_url: https://limbo137.top/okf/concepts/1c78e56f-4242-5823-8a95-f2b292733971.md
- labels: en = homology class；zh-CN = 同调类
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H _ {r} (K) \equiv Z _ {r} (K) / B _ {r} (K). \tag {3.28} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Since $B_r(K)$ is a subgroup of $Z_r(K)$ , $H_r(K)$ is well defined. The group $H_r(K)$ is the set of equivalence classes of $r$ -cycles,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where each equivalence class $[z]$ is called a homology class. Two $r$ -cycles $z$ and $z'$ are in the same equivalence class if and only if $z - z' \in B_r(K)$ , in which case $z$ is said to be homologous to $z'$ and denoted by $z \sim z'$ or $[z] = [z']$ . Geometrically $z - z'$ is a boundary of some space. By definition, any boundary $b \in B_r(K)$ is ...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H _ {r} (K) \equiv \{[ z ] | z \in Z _ {r} (K) \} \tag {3.29} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `[Remarks: If necessary, we define $H_r(K) = 0$ for $r > n$ or $r < 0$ . If we want to stress that the group structure is defined with integer coefficients, we`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=125, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.5. Let $K$ be an $n$ -dimensional simplicial complex. The $r$ th homology group $H_{r}(K)$ , $0 \leq r \leq n$ , associated with $K$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.3 (pdf_page=126, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.5：同调类。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `write $H_{r}(K;\mathbb{Z})$ . We may also define the homology groups with $\mathbb{R}$ -coefficients, $H_{r}(K;\mathbb{R})$ or those with $\mathbb{Z}_2$ -coefficients, $H_{r}(K;\mathbb{Z}_{2})$ .]`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 反酉算子(anti-unitary operator)

- UUID: `1c7f22da-eed4-4732-89e8-1921feebfcf0` · okf_url: https://limbo137.top/okf/concepts/1c7f22da-eed4-4732-89e8-1921feebfcf0.md
- labels: en = anti-unitary operator；zh-CN = 反酉算子
- aliases: anti-unitary
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-8d9d442a5f13efdad443-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-bd359cae2eb1c256e1ad-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 表示的次数(degree of a representation)

- UUID: `1c8384ff-2641-4168-ba63-d81a2dffcea1` · okf_url: https://limbo137.top/okf/concepts/1c8384ff-2641-4168-ba63-d81a2dffcea1.md
- labels: en = degree of a representation；zh-CN = 表示的次数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-degree-of-a-representation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md

## 重心坐标(barycentric coordinate)

- UUID: `1d69ceb7-ab2c-4a69-9c03-48429733f60b` · okf_url: https://limbo137.top/okf/concepts/1d69ceb7-ab2c-4a69-9c03-48429733f60b.md
- labels: en = barycentric coordinate；zh-CN = 重心坐标
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=117, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯形中x=Σc_i p_i、c_i≥0且Σc_i=1的系数；完整假设和公式3.11。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$(c_0, \ldots, c_r)$ is called the barycentric coordinate of $x$ . Since $\sigma_r$ is a bounded and closed subset of $\mathbb{R}^m$ , it is compact.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 广义动量(generalized momenta)

- UUID: `1d84079a-778c-4a1c-bd5d-6518d57af2b6` · okf_url: https://limbo137.top/okf/concepts/1d84079a-778c-4a1c-bd5d-6518d57af2b6.md
- labels: en = generalized momenta；zh-CN = 广义动量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-cab1c4adb17c06967cba-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 积分流形(integral manifold)

- UUID: `1e3de021-b956-44d2-baf0-fb310e41bc05` · okf_url: https://limbo137.top/okf/concepts/1e3de021-b956-44d2-baf0-fb310e41bc05.md
- labels: en = integral manifold；zh-CN = 积分流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.6 · `zh/chapter-15/06-distributions-and-frobenius-theorem#concept-15-integral-manifold` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.6 · `en/chapter-15/06-distributions-and-frobenius-theorem#first-5407589057aa2384f435-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/06-distributions-and-frobenius-theorem.md

## Hirzebruch符号差定理(Hirzebruch signature theorem)

- UUID: `1e67ed27-0d7a-5549-baa9-605e65e4d107` · okf_url: https://limbo137.top/okf/concepts/1e67ed27-0d7a-5549-baa9-605e65e4d107.md
- labels: en = Hirzebruch signature theorem；zh-CN = Hirzebruch符号差定理
- aliases: Hirzebruch 符号差定理；希策布鲁赫符号差定理
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara11.4.3例11.8，在紧致连通可定向四维M条件下，式11.99给同一定理的四维版本；同书多处陈述分别保留。 · 原文: `According to the Hirzebruch signature theorem (see section 12.5), this number is also given in terms of the $L$ -polynomial as`
  - nakahara-geometry-topology-physics/second-edition-local §12.5.2 (pdf_page=486, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.5.2习题12.3的定理陈述式12.68，假设依同节前文紧致无边黎曼流形及中维度数限制；不补造作者Theorem编号，不把引句单独当完整定义。 · 原文: `$$ \tau (M) = \int_ {M} L (T M) | _ {\text {v o l}} \tag {12.68} $$`
  - nakahara-geometry-topology-physics/second-edition-local §12.5.2 (pdf_page=486, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.5.2习题12.3的定理陈述式12.68，假设依同节前文紧致无边黎曼流形及中维度数限制；不补造作者Theorem编号，不把引句单独当完整定义。 · 原文: `where the last equality is true only for the $2l$ -forms in the expansion and $x_{i} = x_{i}(TM^{\mathbb{C}})$ . Now we have obtained the Hirzebruch signature theorem`
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara11.4.3例11.8，在紧致连通可定向四维M条件下，式11.99给同一定理的四维版本；同书多处陈述分别保留。 · 原文: `$$ \tau (M) = \int_ {M} L _ {1} (M) = \frac {1}{3} \int_ {M} p _ {1} (M). \tag {11.99} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 群元素的逆元(inverse of a group element)

- UUID: `1eb6bc94-07d7-48f3-b03b-60b1c2dbb415` · okf_url: https://limbo137.top/okf/concepts/1eb6bc94-07d7-48f3-b03b-60b1c2dbb415.md
- labels: en = inverse of a group element；zh-CN = 群元素的逆元
- aliases: inverse；逆元
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-inverse` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 拓扑(topology)

- UUID: `1ebf8613-2556-44a4-854a-86128769ba52` · okf_url: https://limbo137.top/okf/concepts/1ebf8613-2556-44a4-854a-86128769ba52.md
- labels: en = topology；zh-CN = 拓扑
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-2564e08cd4afded1d225-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 集合上的开集族与Top1–Top3公理；不同于学科。 · 原文: `Given a set $X$, a topology on $X$ consists of a family of subsets $\mathcal{O}$, called open sets, which satisfy the following conditions: (Top1) The empty set $\varnothing$ is open and the entire space $X$ is open, $\{ \varnothing , X \} \subset { \mathcal { O } }$ (Top2) If $U$ and $V$ are open sets then so is their intersection $U \cap V$ $$ U \in \ma...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 佐恩引理(Zorn’s lemma)

- UUID: `1ee2a9ac-e285-44ea-b156-a143a1875246` · okf_url: https://limbo137.top/okf/concepts/1ee2a9ac-e285-44ea-b156-a143a1875246.md
- labels: en = Zorn’s lemma；zh-CN = 佐恩引理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-3096d224960591128aad-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §1.5 · `en/chapter-01/05-infinite-sets#first-3096d224960591128aad-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 单纯复形(simplicial complex)

- UUID: `1ee7bbbb-6d66-4bf4-beb6-68f069aefc76` · okf_url: https://limbo137.top/okf/concepts/1ee7bbbb-6d66-4bf4-beb6-68f069aefc76.md
- labels: en = simplicial complex；zh-CN = 单纯复形
- aliases: 单形复形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.2 (pdf_page=118, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `(i) an arbitrary face of a simplex of $K$ belongs to $K$ , that is, if $\sigma \in K$ and $\sigma' \leq \sigma$ then $\sigma' \in K$ ; and (ii) if $\sigma$ and $\sigma'$ are two simplexes of $K$ , the intersection $\sigma \cap \sigma'$ is either empty or a common face of $\sigma$ and $\sigma'$ , that is, if $\sigma, \sigma' \in K$ then either $\sigma \cap...`
  - nakahara-geometry-topology-physics/second-edition-local §3.2.2 (pdf_page=118, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Let $K$ be a set of finite number of simplexes in $\mathbb{R}^m$ . If these simplexes are nicely fitted together, $K$ is called a simplicial complex. By 'nicely' we mean:`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 可分解向量与不可分解向量(Decomposable and indecomposable vectors)

- UUID: `1eff654d-4586-46ff-b65d-12fcb385b4ae` · okf_url: https://limbo137.top/okf/concepts/1eff654d-4586-46ff-b65d-12fcb385b4ae.md
- labels: en = Decomposable and indecomposable vectors；zh-CN = 可分解向量与不可分解向量
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §9.3 · `books/woit-qtgr/2025-10-20/en/chapter-09/03-indecomposable-vectors-and-entanglement#first-f495c58517fe632c83b6-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §9.3 · `books/woit-qtgr/2025-10-20/zh/chapter-09/03-indecomposable-vectors-and-entanglement#first-f3360697a392c46eea20-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §9.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若$V\otimes W$中的向量可写成$v\otimes w$（其中$v\in V$、$w\in W$），则称其为可分解向量；否则称为不可分解向量。`
  - woit-qtgr/2025-10-20 §9.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A vector in $V\otimes W$ is called decomposable if it has the form $v\otimes w$ for some $v\in V$ and $w\in W$. Otherwise, it is called indecomposable.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/03-indecomposable-vectors-and-entanglement.md

## 海森堡绘景(Heisenberg picture)

- UUID: `1f30c21f-6f29-4be4-8c57-f3cdbc4c3786` · okf_url: https://limbo137.top/okf/concepts/1f30c21f-6f29-4be4-8c57-f3cdbc4c3786.md
- labels: en = Heisenberg picture；zh-CN = 海森堡绘景
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-cf15684370463e5ff631-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-cf15684370463e5ff631-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §7.3 · `books/woit-qtgr/2025-10-20/zh/chapter-07/03-the-heisenberg-picture#first-20e9bdd9ef523a597726-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.3 · `books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture#first-20e9bdd9ef523a597726-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 量子可观测量(quantum observable)

- UUID: `1f30c262-f238-4dfd-acec-55532f758890` · okf_url: https://limbo137.top/okf/concepts/1f30c262-f238-4dfd-acec-55532f758890.md
- labels: en = quantum observable；zh-CN = 量子可观测量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#concept-1-quantum-observable` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 浸入(immersion)

- UUID: `1f605ba7-d55c-4c45-aa84-5df7c7af88e2` · okf_url: https://limbo137.top/okf/concepts/1f605ba7-d55c-4c45-aa84-5df7c7af88e2.md
- labels: en = immersion；zh-CN = 浸入
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.4 · `en/chapter-15/04-tangent-map-and-submanifolds#first-ea170ef86bf2fd24dc34-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.4 · `zh/chapter-15/04-tangent-map-and-submanifolds#concept-15-immersion` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：浸入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) The map $f$ is called an immersion of $M$ into $N$ if $f_{*}: T_{p}M \to T_{f(p)}N$ is an injection (one to one), that is $\operatorname{rank} f_{*} = \dim M$ . (b) The map $f$ is called an **embedding** if $f$ is an injection and an immersion. The image $f(M)$ is called a **submanifold** of $N$ . [In practice, $f(M)$ thus defined is diffeomorphic to ...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：浸入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.3. (Immersion, submanifold, embedding) Let $f: M \to N$ be a smooth map and let $\dim M \leq \dim N$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：浸入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If $f$ is an immersion, $f^{*}$ maps $T_{p}M$ isomorphically to an $m$ -dimensional vector subspace of $T_{f(p)}N$ since $\mathrm{rank}f_{*} = \dim M$ . From theorem 2.1, we also find $\ker f_{*} = \{0\}$ . If $f$ is an embedding, $M$ is diffeomorphic to $f(M)$ . Examples will clarify these rather technical points. Consider a map $f: S^1 \to \mathbb{R}^2$...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/04-tangent-map-and-submanifolds.md

## 度量(metric)

- UUID: `1f8aca83-3403-4584-b621-eed33886f812` · okf_url: https://limbo137.top/okf/concepts/1f8aca83-3403-4584-b621-eed33886f812.md
- labels: en = metric；zh-CN = 度量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-19bd3976430bde51d2c9-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 距离函数意义的度量：集合X上的d:X×X→R，非负、d(x,y)=0当且仅当x=y、对称且满足三角不等式。原始身份来自Szekeres§10.3度量空间，Nakahara§2.3给出同一公理；不指流形上的度规张量g、向量空间上的双线性形式或纤维度规。集合不必预先带光滑结构；该函数可诱导拓扑。 · 原文: `(i) $d(x,y) = d(y,x)$ (ii) $d(x,y)\geq 0$ where the equality holds if and only if $x = y$ (iii) $d(x,y) + d(y,z)\geq d(x,z)$`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 距离函数意义的度量：集合X上的d:X×X→R，非负、d(x,y)=0当且仅当x=y、对称且满足三角不等式。原始身份来自Szekeres§10.3度量空间，Nakahara§2.3给出同一公理；不指流形上的度规张量g、向量空间上的双线性形式或纤维度规。集合不必预先带光滑结构；该函数可诱导拓扑。 · 原文: `and all their possible unions. The topology $\mathcal{T}$ thus defined is called the metric topology determined by $d$ . The topological space $(X, \mathcal{T})$ is called a metric space. [Exercise: Verify that a metric space $(X, \mathcal{T})$ is indeed a topological space.]`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 距离函数意义的度量：集合X上的d:X×X→R，非负、d(x,y)=0当且仅当x=y、对称且满足三角不等式。原始身份来自Szekeres§10.3度量空间，Nakahara§2.3给出同一公理；不指流形上的度规张量g、向量空间上的双线性形式或纤维度规。集合不必预先带光滑结构；该函数可诱导拓扑。 · 原文: `$$ U _ {\varepsilon} (X) = \{y \in X | d (x, y) < \varepsilon \} \tag {2.25} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 距离函数意义的度量：集合X上的d:X×X→R，非负、d(x,y)=0当且仅当x=y、对称且满足三角不等式。原始身份来自Szekeres§10.3度量空间，Nakahara§2.3给出同一公理；不指流形上的度规张量g、向量空间上的双线性形式或纤维度规。集合不必预先带光滑结构；该函数可诱导拓扑。 · 原文: `A metric $d:X\times X\to \mathbb{R}$ is a function that satisfies the conditions:`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 距离函数意义的度量：集合X上的d:X×X→R，非负、d(x,y)=0当且仅当x=y、对称且满足三角不等式。原始身份来自Szekeres§10.3度量空间，Nakahara§2.3给出同一公理；不指流形上的度规张量g、向量空间上的双线性形式或纤维度规。集合不必预先带光滑结构；该函数可诱导拓扑。 · 原文: `for any $x, y, z \in X$ . If $X$ is endowed with a metric $d$ , $X$ is made into a topological space whose open sets are given by 'open discs',`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 宇称变换(parity transformation)

- UUID: `20808338-dff0-4dc9-affd-ba73b8240dae` · okf_url: https://limbo137.top/okf/concepts/20808338-dff0-4dc9-affd-ba73b8240dae.md
- labels: en = parity transformation；zh-CN = 宇称变换
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.2 · `books/woit-qtgr/2025-10-20/zh/chapter-40/02-the-lorentz-group-and-its-lie-algebra#first-0951678b910658242a97-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §40.2 · `books/woit-qtgr/2025-10-20/en/chapter-40/02-the-lorentz-group-and-its-lie-algebra#first-0951678b910658242a97-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/02-the-lorentz-group-and-its-lie-algebra.md

## 可定向流形(orientable manifold)

- UUID: `2093e556-26bf-4b56-9963-d96c911c50ca` · okf_url: https://limbo137.top/okf/concepts/2093e556-26bf-4b56-9963-d96c911c50ca.md
- labels: en = orientable manifold；zh-CN = 可定向流形
- aliases: orientable；可定向的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=223, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.6. Let $M$ be a connected manifold covered by $\{U_i\}$ . The manifold $M$ is orientable if, for any overlapping charts $U_i$ and $U_j$ , there exist local coordinates $\{x^\mu\}$ for $U_i$ and $\{y^\alpha\}$ for $U_j$ such that $J = \det(\partial x^\mu / \partial y^\alpha) > 0$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=223, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If $M$ is non-orientable, $J$ cannot be positive in all intersections of charts. For example, the Möbius strip in figure 5.14(a) is non-orientable since we have to choose $J$ to be negative in the intersection B.`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=224, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The determinant in (5.99) is the Jacobian of the coordinate transformation and must be positive by assumed orientability. If $M$ is non-orientable, $\omega$ with a positive-definite component cannot be defined on $M$ . Let us look at figure 5.14 again. If we circumnavigate the strip along the direction shown in the figure, $\omega = \mathrm{d}x \wedge \ma...`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=224, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `with a positive-definite $h(p)$ on a chart $(U, \varphi)$ whose coordinate is $x = \varphi(p)$ . If $M$ is orientable, we may extend $\omega$ throughout $M$ such that the component $h$ is positive definite on any chart $U_i$ . If $M$ is orientable, this $\omega$ is a volume element. Note that this positivity of $h$ is independent of the choice of coordina...`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=223, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \omega = h (p) \mathrm {d} x ^ {1} \wedge \dots \wedge \mathrm {d} x ^ {m} \tag {5.98} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=223, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If an $m$ -dimensional manifold $M$ is orientable, there exists an $m$ -form $\omega$ which vanishes nowhere. This $m$ -form $\omega$ is called a volume element, which plays the role of a measure when we integrate a function $f \in \mathcal{F}(M)$ over $M$ . Two volume elements $\omega$ and $\omega'$ are said to be equivalent if there exists a strictly po...`
  - nakahara-geometry-topology-physics/second-edition-local §5.5.1 (pdf_page=224, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.6：可定向流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \omega = h (p) \frac {\partial x ^ {1}}{\partial y ^ {\mu_ {1}}} \mathrm {d} y ^ {\mu_ {1}} \wedge \dots \wedge \frac {\partial x ^ {m}}{\partial y ^ {\mu_ {m}}} \mathrm {d} y ^ {\mu_ {m}} = h (p) \det \left(\frac {\partial x ^ {\mu}}{\partial y ^ {\nu}}\right) \mathrm {d} y ^ {1} \wedge \dots \wedge \mathrm {d} y ^ {m}. \tag {5.99} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 康托尔对角论证(Cantor's diagonal argument)

- UUID: `20ad21a3-f998-4cb6-8041-7cd13c0d1119` · okf_url: https://limbo137.top/okf/concepts/20ad21a3-f998-4cb6-8041-7cd13c0d1119.md
- labels: en = Cantor's diagonal argument；zh-CN = 康托尔对角论证
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-9bcb5544386f7b94e549-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 三阶实反对称矩阵(real skew-symmetric 3 by 3 matrix)

- UUID: `20de28a4-da1e-4c03-9e2c-3e13cb0ba553` · okf_url: https://limbo137.top/okf/concepts/20de28a4-da1e-4c03-9e2c-3e13cb0ba553.md
- labels: en = real skew-symmetric 3 by 3 matrix；zh-CN = 三阶实反对称矩阵
- aliases: antisymmetric $3\times3$ matrices
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.2 · `books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-e58d9f3ee96dd36857e2-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md

## 海森堡群，$d=1$(Heisenberg group, $d=1$)

- UUID: `2114b0de-b7cd-4083-b0b5-884f1b96c3e0` · okf_url: https://limbo137.top/okf/concepts/2114b0de-b7cd-4083-b0b5-884f1b96c3e0.md
- labels: en = Heisenberg group, $d=1$；zh-CN = 海森堡群，$d=1$
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.2 · `books/woit-qtgr/2025-10-20/zh/chapter-13/02-the-heisenberg-group#first-15a1a2569da51ffd71da-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.2 · `books/woit-qtgr/2025-10-20/en/chapter-13/02-the-heisenberg-group#first-a3326968b0e583eb7f8e-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Heisenberg group $H_3$ is the space $\mathbf R^3=\mathbf R^2\oplus\mathbf R$ with the group law $$ \left(\binom{x}{y},z\right)\left(\binom{x'}{y'},z'\right)=\left(\binom{x+x'}{y+y'},z+z'+\frac12(xy'-yx')\right)\tag{13.2} $$`
  - woit-qtgr/2025-10-20 §13.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡群 $H_3$ 是空间 $\mathbf R^3=\mathbf R^2\oplus\mathbf R$，其群乘法为 $$ \left(\binom{x}{y},z\right)\left(\binom{x'}{y'},z'\right)=\left(\binom{x+x'}{y+y'},z+z'+\frac12(xy'-yx')\right)\tag{13.2} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/02-the-heisenberg-group.md

## 测地线(geodesic)

- UUID: `211557f7-a3f5-49e5-945e-06820fdf4de7` · okf_url: https://limbo137.top/okf/concepts/211557f7-a3f5-49e5-945e-06820fdf4de7.md
- labels: en = geodesic；zh-CN = 测地线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-6105a376bad25830405f-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-6105a376bad25830405f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 德西特时空(de Sitter space-time)

- UUID: `2166279d-867c-45ec-836c-c0d714e0668e` · okf_url: https://limbo137.top/okf/concepts/2166279d-867c-45ec-836c-c0d714e0668e.md
- labels: en = de Sitter space-time；zh-CN = 德西特时空
- aliases: de Sitter space；德西特空间
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-de-sitter-space-time` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 马约拉纳场算符(Majorana field operator)

- UUID: `217fa41e-b347-4210-af2d-3fbb37069123` · okf_url: https://limbo137.top/okf/concepts/217fa41e-b347-4210-af2d-3fbb37069123.md
- labels: en = Majorana field operator；zh-CN = 马约拉纳场算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §47.2 · `books/woit-qtgr/2025-10-20/en/chapter-47/02-majorana-spinors-and-the-majorana-field#first-2e9d8a043cb2513cd794-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §47.2 · `books/woit-qtgr/2025-10-20/zh/chapter-47/02-majorana-spinors-and-the-majorana-field#first-34bedb23f78eb075f19e-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §47.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `马约拉纳场算符由下式给出 $$ \widehat {\Psi} (\mathbf {x}, t) = \frac {1}{(2 \pi) ^ {3 / 2}} \int_ {\mathbf {R} ^ {3}} \sum_ {s = \pm} (a _ {s} (\mathbf {p}) u _ {s} (\mathbf {p}) e ^ {- i \omega_ {\mathbf {p}} t} e ^ {i \mathbf {p} \cdot \mathbf {x}} + a _ {s} ^ {\dagger} (\mathbf {p}) \overline {{u _ {s} (\mathbf {p})}} e ^ {i \omega_ {\mathbf {p}} t} e ^ {- i \math...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-47/02-majorana-spinors-and-the-majorana-field.md

## 泡利-卢班斯基算符(Pauli-Lubanski operator)

- UUID: `219d8d9a-81e0-4039-91d1-3c87115ef4fa` · okf_url: https://limbo137.top/okf/concepts/219d8d9a-81e0-4039-91d1-3c87115ef4fa.md
- labels: en = Pauli-Lubanski operator；zh-CN = 泡利-卢班斯基算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §42.2 · `books/woit-qtgr/2025-10-20/en/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-05540d234a1bff0a09c7-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §42.2 · `books/woit-qtgr/2025-10-20/zh/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-47d82a4319f55d7e5f40-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §42.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `泡利-卢班斯基算符是四分量算符 $$ W _ {0} = - \mathbf {P} \cdot \mathbf {J}, \quad \mathbf {W} = - P _ {0} \mathbf {J} + \mathbf {P} \times \mathbf {K} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-42/02-irreducible-representations-of-the-poincar-e-group.md

## 聚点(accumulation point)

- UUID: `219d9a3a-91e9-494e-a7b3-9e7b8ac0a94a` · okf_url: https://limbo137.top/okf/concepts/219d9a3a-91e9-494e-a7b3-9e7b8ac0a94a.md
- labels: en = accumulation point；zh-CN = 聚点
- aliases: cluster point
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-ee256cff758d65df9a43-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `A point $x$ is said to be an accumulation point of a set $A$ if every open neighbourhood $U$ of $x$ contains points of $A$ other than $x$ itself, as shown in Fig. 10.3. What this means is that $x$ may or may not lie in $A$, but points of $A$ ‘cluster’ arbitrarily close to it (sometimes it is also called a cluster point of $A$). $A$ related concept is comm...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 逻辑联结词(logical connective)

- UUID: `21a7ab40-1a23-4f76-b330-509920b3fbea` · okf_url: https://limbo137.top/okf/concepts/21a7ab40-1a23-4f76-b330-509920b3fbea.md
- labels: en = logical connective；zh-CN = 逻辑联结词
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-ed32546170d33f79aab6-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 零向量(zero vector)

- UUID: `21e4a707-dc48-420c-8294-547e3a049005` · okf_url: https://limbo137.top/okf/concepts/21e4a707-dc48-420c-8294-547e3a049005.md
- labels: en = zero vector；zh-CN = 零向量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-zero-vector` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 李群的李代数(Lie algebra of a Lie group)

- UUID: `22099e8e-a506-43d9-bc9d-a814a0190816` · okf_url: https://limbo137.top/okf/concepts/22099e8e-a506-43d9-bc9d-a814a0190816.md
- labels: en = Lie algebra of a Lie group；zh-CN = 李群的李代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-lie-algebra-of-a-lie-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 泛函的临界点(critical point of a functional)

- UUID: `224e68d3-89ec-4082-94fa-986f333f6ac4` · okf_url: https://limbo137.top/okf/concepts/224e68d3-89ec-4082-94fa-986f333f6ac4.md
- labels: en = critical point of a functional；zh-CN = 泛函的临界点
- aliases: critical point；临界点
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-b171dfa376a348885515-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-b171dfa376a348885515-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 第二卡西米尔算符(Second Casimir operator)

- UUID: `22bbf2c3-ffe3-4147-9f96-96908e85df9a` · okf_url: https://limbo137.top/okf/concepts/22bbf2c3-ffe3-4147-9f96-96908e85df9a.md
- labels: en = Second Casimir operator；zh-CN = 第二卡西米尔算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §42.2 · `books/woit-qtgr/2025-10-20/en/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-ccd3c35fff3477805158-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §42.2 · `books/woit-qtgr/2025-10-20/zh/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-223fc8dca151b7e5b362-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §42.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `庞加莱李代数的第二卡西米尔算符是 $$ W ^ {2} = - W _ {0} ^ {2} + W _ {1} ^ {2} + W _ {2} ^ {2} + W _ {3} ^ {2} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-42/02-irreducible-representations-of-the-poincar-e-group.md

## 相干态(Coherent states)

- UUID: `22c1f4e6-3a65-4868-8d58-dde99c2cea9a` · okf_url: https://limbo137.top/okf/concepts/22c1f4e6-3a65-4868-8d58-dde99c2cea9a.md
- labels: en = Coherent states；zh-CN = 相干态
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §23.1 · `books/woit-qtgr/2025-10-20/zh/chapter-23/01-coherent-states-and-the-heisenberg-group-action#first-d03ad8d5070c6f987425-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §23.1 · `books/woit-qtgr/2025-10-20/en/chapter-23/01-coherent-states-and-the-heisenberg-group-action#first-3dd1594c89a2d8dae535-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §23.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `$\mathcal H$ 中的相干态定义为以下形式的向量： $$ |\alpha\rangle=D(\alpha)|0\rangle=e^{\alpha a^\dagger-\bar\alpha a}|0\rangle. $$ 其中 $\alpha\in\mathbb C$。`
  - woit-qtgr/2025-10-20 §23.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The coherent states are vectors in $\mathcal H$ of the form $$ |\alpha\rangle=D(\alpha)|0\rangle=e^{\alpha a^\dagger-\bar\alpha a}|0\rangle. $$ Here $\alpha\in\mathbb C$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-23/01-coherent-states-and-the-heisenberg-group-action.md

## Schwinger 玻色子方法(Schwinger boson method)

- UUID: `22d066f1-dd17-450c-8457-1f856104d052` · okf_url: https://limbo137.top/okf/concepts/22d066f1-dd17-450c-8457-1f856104d052.md
- labels: en = Schwinger boson method；zh-CN = Schwinger 玻色子方法
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §25.4 · `books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3#first-20e213766c5b80def17e-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §25.4 · `books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3#first-20e213766c5b80def17e-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/04-examples-in-d-2-and-3.md

## 自反(reflexive)

- UUID: `2396ddfd-f7f2-4e75-8107-4e0be31adab1` · okf_url: https://limbo137.top/okf/concepts/2396ddfd-f7f2-4e75-8107-4e0be31adab1.md
- labels: en = reflexive；zh-CN = 自反
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-d1aad3541c402ead298d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 齐性流形(homogeneous manifold)

- UUID: `23af9bd6-7543-4dde-91ca-b4fc9dd181d4` · okf_url: https://limbo137.top/okf/concepts/23af9bd6-7543-4dde-91ca-b4fc9dd181d4.md
- labels: en = homogeneous manifold；zh-CN = 齐性流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-f700cb4a4a1c75e995d3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.4 · `zh/chapter-19/04-lie-groups-of-transformations#concept-19-homogeneous-manifold` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md

## 局部流(local flow)

- UUID: `240a4900-b14c-4484-a71f-870546a4daca` · okf_url: https://limbo137.top/okf/concepts/240a4900-b14c-4484-a71f-870546a4daca.md
- labels: en = local flow；zh-CN = 局部流
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-5399531bbe0c65157435-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-local-flow` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md

## 多面体(polyhedron)

- UUID: `241c2a28-4e3b-486a-87c5-a0b9766d2b29` · okf_url: https://limbo137.top/okf/concepts/241c2a28-4e3b-486a-87c5-a0b9766d2b29.md
- labels: en = polyhedron；zh-CN = 多面体
- aliases: $|K|$
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.2 (pdf_page=119, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara3.2.2中单纯复形K在R^m内所有单纯形的并|K|；不是仅指三维实体。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `A simplicial complex $K$ is a set whose elements are simplexes. If each simplex is regarded as a subset of $\mathbb{R}^m$ ( $m \geq \dim K$ ), the union of all the simplexes becomes a subset of $\mathbb{R}^m$ . This subset is called the polyhedron $|K|$ of a simplicial complex $K$ . The dimension of $|K|$ as a subset of $\mathbb{R}^m$ is the same as that ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 外微分(exterior derivative)

- UUID: `245d07b1-e2d8-46ce-86c4-13d7dff9755f` · okf_url: https://limbo137.top/okf/concepts/245d07b1-e2d8-46ce-86c4-13d7dff9755f.md
- labels: en = exterior derivative；zh-CN = 外微分
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.1 · `en/chapter-16/01-differential-forms-and-exterior-derivative#first-5a3376d5c2162d6914d8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.1 · `zh/chapter-16/01-differential-forms-and-exterior-derivative#first-5a3376d5c2162d6914d8-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.4.2 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.5：外微分。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \omega = \frac {1}{r !} \omega_ {\mu_ {1} \dots \mu_ {r}} d x ^ {\mu_ {1}} \wedge \dots \wedge d x ^ {\mu_ {r}} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.2 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.5：外微分。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \mathrm {d} _ {r} \omega = \frac {1}{r !} \left(\frac {\partial}{\partial x ^ {v}} \omega_ {\mu_ {1} \dots \mu_ {r}}\right) \mathrm {d} x ^ {v} \wedge \mathrm {d} x ^ {\mu_ {1}} \wedge \dots \wedge \mathrm {d} x ^ {\mu_ {r}}. \tag {5.68} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.2 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.5：外微分。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.2 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.5：外微分。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `It is common to drop the subscript $r$ and write simply d. The wedge product automatically anti-symmetrizes the coefficient.`
  - nakahara-geometry-topology-physics/second-edition-local §5.4.2 (pdf_page=217, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.5：外微分。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.5. The exterior derivative $\mathrm{d}_r$ is a map $\Omega^r (M)\to \Omega^{r + 1}(M)$ whose action on an $r$ -form`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/01-differential-forms-and-exterior-derivative.md

## 有效作用(effective action)

- UUID: `249e0182-e4dd-485a-9d74-9906d1ab1de7` · okf_url: https://limbo137.top/okf/concepts/249e0182-e4dd-485a-9d74-9906d1ab1de7.md
- labels: en = effective action；zh-CN = 有效作用
- aliases: effective；有效的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.4 · `zh/chapter-19/04-lie-groups-of-transformations#concept-19-effective-action` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：有效群作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.15. Let $G$ be a Lie group that acts on a manifold $M$ by $\sigma : G \times M \to M$ . The action $\sigma$ is said to be`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：有效群作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) transitive if, for any $p_1, p_2 \in M$ , there exists an element $g \in G$ such that $\sigma(g, p_1) = p_2$ ; (b) free if every non-trivial element $g \neq e$ of $G$ has no fixed points in $M$ , that is, if there exists an element $p \in M$ such that $\sigma(g, p) = p$ , then $g$ must be the unit element $e$ ; and (c) effective if the unit element $e...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md

## 胞腔(cell)

- UUID: `24c541a9-58f0-4e62-b38e-b7b1496c08ce` · okf_url: https://limbo137.top/okf/concepts/24c541a9-58f0-4e62-b38e-b7b1496c08ce.md
- labels: en = cell；zh-CN = 胞腔
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-31fa70610c94c9d9a446-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-31fa70610c94c9d9a446-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 闭球(closed ball)

- UUID: `24dc4f81-4498-495a-a55e-63506d0e67b4` · okf_url: https://limbo137.top/okf/concepts/24dc4f81-4498-495a-a55e-63506d0e67b4.md
- labels: en = closed ball；zh-CN = 闭球
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-08a9b05ac88b7c29c88e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 保向欧氏群(proper Euclidean group)

- UUID: `24e2efa3-553e-461a-9117-448a9406db0f` · okf_url: https://limbo137.top/okf/concepts/24e2efa3-553e-461a-9117-448a9406db0f.md
- labels: en = proper Euclidean group；zh-CN = 保向欧氏群
- aliases: Euclidean group；欧氏群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §18.1 · `books/woit-qtgr/2025-10-20/zh/chapter-18/01-an-example-the-euclidean-group#first-5845b2591058532eccee-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §18.1 · `books/woit-qtgr/2025-10-20/en/chapter-18/01-an-example-the-euclidean-group#first-ddffe0abff15f1c0a4dd-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §18.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Euclidean group $E ( d )$ (sometimes written $I S O ( d )$ for “inhomogeneous” rotation group) in dimension $d$ is the product of the translation and rotation groups of $\mathbf { R } ^ { d }$ as a set, with multiplication law $$ (\mathbf {a} _ {1}, R _ {1}) (\mathbf {a} _ {2}, R _ {2}) = (\mathbf {a} _ {1} + R _ {1} \mathbf {a} _ {2}, R _ {1} R _ {2}...`
  - woit-qtgr/2025-10-20 §18.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `维数 $d$ 中的欧氏群 $E ( d )$（有时写作 $I S O ( d )$，表示“非齐次”旋转群）作为集合是 $\mathbf { R } ^ { d }$ 的平移群与旋转群的乘积，其乘法法则为 $$ (\mathbf {a} _ {1}, R _ {1}) (\mathbf {a} _ {2}, R _ {2}) = (\mathbf {a} _ {1} + R _ {1} \mathbf {a} _ {2}, R _ {1} R _ {2}) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-18/01-an-example-the-euclidean-group.md

## 欧氏向量空间(Euclidean vector space)

- UUID: `253d9f09-01fb-4357-9e2c-0221063865de` · okf_url: https://limbo137.top/okf/concepts/253d9f09-01fb-4357-9e2c-0221063865de.md
- labels: en = Euclidean vector space；zh-CN = 欧氏向量空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-euclidean-vector-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 狄拉克旋量(Dirac spinor)

- UUID: `26ae89e4-05be-4f48-9a53-8ecdb3861cc9` · okf_url: https://limbo137.top/okf/concepts/26ae89e4-05be-4f48-9a53-8ecdb3861cc9.md
- labels: en = Dirac spinor；zh-CN = 狄拉克旋量
- aliases: Dirac旋量；相对论Dirac旋量；相对论Dirac旋量（至少四分量）
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-6383790d2cefd4edbd3d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-spinor` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §12.6.1 (pdf_page=488, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.6.1例12.4后的Dirac旋量说明；按该章偶数维和Clifford表示约定理解，完整原文保留。此处不是普通spinor的总定义，也不宣称全书首次出现。 · 原文: `A Dirac spinor $\psi \in \Delta (M)$ is an irreducible representation of the Clifford algebra but not that of $\mathrm{SPIN}(2l)$ . Irreducible representations of $\mathrm{SPIN}(2l)$ are obtained by separating $\Delta (M)$ according to the eigenvalues of $\gamma^{m + 1}$ . Since $(\gamma^{m + 1})^2 = I$ , the eigenvalues of $\gamma^{m + 1}$ , called the c...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md

## 置换(permutation)

- UUID: `26cfbcfb-a481-41ee-a74f-cd41f06ecb0f` · okf_url: https://limbo137.top/okf/concepts/26cfbcfb-a481-41ee-a74f-cd41f06ecb0f.md
- labels: en = permutation；zh-CN = 置换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-permutation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 过渡函数(transition function)

- UUID: `26de7693-2eaf-45d5-9d89-9ef22e571302` · okf_url: https://limbo137.top/okf/concepts/26de7693-2eaf-45d5-9d89-9ef22e571302.md
- labels: en = transition function；zh-CN = 过渡函数
- aliases: transition functions
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-transition-function` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-c427f4c3a47a90836b8e-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 同态(homomorphism)

- UUID: `2713e2aa-a3f5-43bc-8e80-739d42457330` · okf_url: https://limbo137.top/okf/concepts/2713e2aa-a3f5-43bc-8e80-739d42457330.md
- labels: en = homomorphism；zh-CN = 同态
- aliases: homomorphisms
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=88, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定义条件与符号保留在引文中；不推断跨书等价。 · 原文: `Suppose certain algebraic structures (product or addition, say) are endowed with the sets $X$ and $Y$ . If $f: X \to Y$ preserves these algebraic structures, then $f$ is called a homomorphism. For example, let $X$ be endowed with a product. If $f$ is a homomorphism, it preserves the product, $f(ab) = f(a)f(b)$ . Note that $ab$ is defined by the product ru...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 双线性形式(Bilinear forms)

- UUID: `2733ca54-c706-432a-b96d-195c3a4e7fb4` · okf_url: https://limbo137.top/okf/concepts/2733ca54-c706-432a-b96d-195c3a4e7fb4.md
- labels: en = Bilinear forms；zh-CN = 双线性形式
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §9.5 · `books/woit-qtgr/2025-10-20/en/chapter-09/08-bilinear-forms-and-tensor-products#first-570976711025e90451ed-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §9.5 · `books/woit-qtgr/2025-10-20/zh/chapter-09/08-bilinear-forms-and-tensor-products#first-db16e626bde79def3d73-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §9.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `域$k$上的向量空间$V$上的双线性形式$B$是一个映射，其中本书取$k=\mathbf R$或$\mathbf C$： $$ B:V\times V\to k $$ 它对两个自变量分别线性，即 $$ \begin{aligned} B(v+v^{\prime},v^{\prime\prime})&=B(v,v^{\prime\prime})+B(v^{\prime},v^{\prime\prime}), & B(cv,v^{\prime})&=cB(v,v^{\prime}),\\ B(v,v^{\prime}+v^{\prime\prime})&=B(v,v^{\prime})+B(v,v^{\prime\prime}), & B(v,cv^{\prime})&=cB(v,v^{\prime}), \...`
  - woit-qtgr/2025-10-20 §9.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A bilinear form $B$ on a vector space $V$ over a field $k$ (for us, $k=\mathbf R$ or $\mathbf C$) is a map $$ B:V\times V\to k, $$ that is linear in each argument; that is, $$ \begin{aligned} B(v+v^{\prime},v^{\prime\prime})&=B(v,v^{\prime\prime})+B(v^{\prime},v^{\prime\prime}), & B(cv,v^{\prime})&=cB(v,v^{\prime}),\\ B(v,v^{\prime}+v^{\prime\prime})&=B(v...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/08-bilinear-forms-and-tensor-products.md

## 动力系统的平衡点(equilibrium point)

- UUID: `278239da-53dc-45e3-8256-1f7bdcdc9037` · okf_url: https://limbo137.top/okf/concepts/278239da-53dc-45e3-8256-1f7bdcdc9037.md
- labels: en = equilibrium point；zh-CN = 动力系统的平衡点
- aliases: critical point；临界点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-critical-point` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 拓扑群(topological group)

- UUID: `278b7ab5-c1c7-44b6-9b34-05f394a2a1cb` · okf_url: https://limbo137.top/okf/concepts/278b7ab5-c1c7-44b6-9b34-05f394a2a1cb.md
- labels: en = topological group；zh-CN = 拓扑群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-2569f1230c5d961fe4f0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 闭包(closure)

- UUID: `2796d975-cc82-4cfa-a082-466a4fb5312e` · okf_url: https://limbo137.top/okf/concepts/2796d975-cc82-4cfa-a082-466a4fb5312e.md
- labels: en = closure；zh-CN = 闭包
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-78396653ed1c67bcb709-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-78396653ed1c67bcb709-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 实射影平面(real projective plane)

- UUID: `27e5661f-6553-4188-ae36-895e9a5bbe86` · okf_url: https://limbo137.top/okf/concepts/27e5661f-6553-4188-ae36-895e9a5bbe86.md
- labels: en = real projective plane；zh-CN = 实射影平面
- aliases: projective plane；射影平面
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-12319e47d0a2daf98c6c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 矩阵李群(matrix Lie group)

- UUID: `27f431a1-3297-467d-9e80-14e2140b1c8e` · okf_url: https://limbo137.top/okf/concepts/27f431a1-3297-467d-9e80-14e2140b1c8e.md
- labels: en = matrix Lie group；zh-CN = 矩阵李群
- aliases: matrix Lie groups
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.3 · `en/chapter-19/03-lie-subgroups#first-826af530ba2784b5fb5e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.3 · `zh/chapter-19/03-lie-subgroups#concept-19-matrix-lie-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-4f1d114ac97cf00e41ba-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/03-lie-subgroups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 升算符(raising operator)

- UUID: `27f67021-0375-4a5d-9cac-fe0b1732d93d` · okf_url: https://limbo137.top/okf/concepts/27f67021-0375-4a5d-9cac-fe0b1732d93d.md
- labels: en = raising operator；zh-CN = 升算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-1a7da8cabae448280d89-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 范数(norm)

- UUID: `27f7027f-1327-4cb7-a562-0f72318ff75c` · okf_url: https://limbo137.top/okf/concepts/27f7027f-1327-4cb7-a562-0f72318ff75c.md
- labels: en = norm；zh-CN = 范数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-norm` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 复Weyl代数(complex Weyl algebra)

- UUID: `280d509b-0425-4ad3-8a7d-aa0b795679ac` · okf_url: https://limbo137.top/okf/concepts/280d509b-0425-4ad3-8a7d-aa0b795679ac.md
- labels: en = complex Weyl algebra；zh-CN = 复Weyl代数
- aliases: Weyl代数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §28.1 · `books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-f5ce9dbff97d7055f7b0-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §28.1 · `books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-f5ce9dbff97d7055f7b0-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/01-the-complex-weyl-and-cliford-algebras.md

## 李代数同态(Lie algebra homomorphism)

- UUID: `28a435be-ede9-40fc-a9b2-f7175d5b3159` · okf_url: https://limbo137.top/okf/concepts/28a435be-ede9-40fc-a9b2-f7175d5b3159.md
- labels: en = Lie algebra homomorphism；zh-CN = 李代数同态
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-2bdd1ef3a458da2c3528-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-2bdd1ef3a458da2c3528-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 张量积(tensor product)

- UUID: `293da78b-f7a4-4572-b27a-2fe3d41262ce` · okf_url: https://limbo137.top/okf/concepts/293da78b-f7a4-4572-b27a-2fe3d41262ce.md
- labels: en = tensor product；zh-CN = 张量积
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-bfce7827433ee3ca115b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-bfce7827433ee3ca115b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 哑指标(dummy index)

- UUID: `295f8e59-9315-4d22-98cc-e13dd1b2f3c8` · okf_url: https://limbo137.top/okf/concepts/295f8e59-9315-4d22-98cc-e13dd1b2f3c8.md
- labels: en = dummy index；zh-CN = 哑指标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-dummy-index` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 卡拉泰奥多里定理(Carathéodory’s theorem)

- UUID: `29a54469-3dea-4ec4-ac3d-3836313569ec` · okf_url: https://limbo137.top/okf/concepts/29a54469-3dea-4ec4-ac3d-3836313569ec.md
- labels: en = Carathéodory’s theorem；zh-CN = 卡拉泰奥多里定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-484658dbed86a4af7e02-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-484658dbed86a4af7e02-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 降算符(lowering operator)

- UUID: `29b6091c-063a-4f0d-9eda-a479679da7a2` · okf_url: https://limbo137.top/okf/concepts/29b6091c-063a-4f0d-9eda-a479679da7a2.md
- labels: en = lowering operator；zh-CN = 降算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-4f2b7426ee5a9462a3dc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 张量积空间(tensor product space)

- UUID: `29ce5567-7fa2-4197-bafd-711866405ac1` · okf_url: https://limbo137.top/okf/concepts/29ce5567-7fa2-4197-bafd-711866405ac1.md
- labels: en = tensor product space；zh-CN = 张量积空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-2d34aa34aabe11722d64-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 位形空间(configuration space)

- UUID: `29e9ae2c-fc05-406f-b27b-8e941535770b` · okf_url: https://limbo137.top/okf/concepts/29e9ae2c-fc05-406f-b27b-8e941535770b.md
- labels: en = configuration space；zh-CN = 位形空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-f74f5b4a5023a774121e-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-f74f5b4a5023a774121e-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 单群(simple group)

- UUID: `29ff0c46-457b-4746-9a48-2753ab5bbc85` · okf_url: https://limbo137.top/okf/concepts/29ff0c46-457b-4746-9a48-2753ab5bbc85.md
- labels: en = simple group；zh-CN = 单群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-simple-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 厄米内积(Hermitian inner product)

- UUID: `2aac3b49-341f-4592-97ad-65a31b630e89` · okf_url: https://limbo137.top/okf/concepts/2aac3b49-341f-4592-97ad-65a31b630e89.md
- labels: en = Hermitian inner product；zh-CN = 厄米内积
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-636eed7b2ce59c34871d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/01-introduction.md

## 约化普朗克常数(reduced Planck constant)

- UUID: `2acc1cb3-ebc4-4dc0-9fae-65f247a55564` · okf_url: https://limbo137.top/okf/concepts/2acc1cb3-ebc4-4dc0-9fae-65f247a55564.md
- labels: en = reduced Planck constant；zh-CN = 约化普朗克常数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-078f03645f2ee1390495-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-reduced-planck-constant` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 正则域(regular domain)

- UUID: `2aff6b0a-acc1-4f76-b66d-c5efc8920a7c` · okf_url: https://limbo137.top/okf/concepts/2aff6b0a-acc1-4f76-b66d-c5efc8920a7c.md
- labels: en = regular domain；zh-CN = 正则域
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-1df947831e84d69d16ef-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-1df947831e84d69d16ef-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 绝热围壁(adiabatic enclosure)

- UUID: `2b09d243-b049-45c7-a004-8b1e128ddafd` · okf_url: https://limbo137.top/okf/concepts/2b09d243-b049-45c7-a004-8b1e128ddafd.md
- labels: en = adiabatic enclosure；zh-CN = 绝热围壁
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-ca5337bafcf16b12ca78-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-ca5337bafcf16b12ca78-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 右平移(right translation)

- UUID: `2b863af3-ea59-4b11-82de-1ef87468e6d4` · okf_url: https://limbo137.top/okf/concepts/2b863af3-ea59-4b11-82de-1ef87468e6d4.md
- labels: en = right translation；zh-CN = 右平移
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-right-translation` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：右平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ R _ {a} g = g a \tag {5.109a} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：右平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.9. Let $a$ and $g$ be elements of a Lie group $G$ . The right-translation $R_{a}: G \to G$ and the left-translation $L_{a}: G \to G$ of $g$ by $a$ are defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：右平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Given a Lie group $G$ , there exists a special class of vector fields characterized by an invariance under group action. [On the usual manifold there is no canonical way of discriminating some vector fields from the others.]`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：右平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `By definition, $R_{a}$ and $L_{a}$ are diffeomorphisms from $G$ to $G$ . Hence, the maps $L_{a}: G \to G$ and $R_{a}: G \to G$ induce $L_{a*}: T_{g}G \to T_{ag}G$ and $R_{a*}: T_{g}G \to T_{ga}G$ ; see section 5.2. Since these translations give equivalent theories, we are concerned mainly with the left-translation in the following. The analysis based on t...`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：右平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ L _ {a} g = a g. \tag {5.109b} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 特殊线性群(special linear group)

- UUID: `2b8f1e0e-bfcc-4391-ace1-5809a76204e3` · okf_url: https://limbo137.top/okf/concepts/2b8f1e0e-bfcc-4391-ace1-5809a76204e3.md
- labels: en = special linear group；zh-CN = 特殊线性群
- aliases: special linear groups
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-special-linear-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.2 · `books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-c49554d0da92dba324b1-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.2 · `books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-2a00b0bb9c6def9040ef-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/03-a-summary.md

## 斯托克斯定理(Stokes’ theorem)

- UUID: `2bac2a32-add3-4717-a57f-9190d1e8fb61` · okf_url: https://limbo137.top/okf/concepts/2bac2a32-add3-4717-a57f-9190d1e8fb61.md
- labels: en = Stokes’ theorem；zh-CN = 斯托克斯定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-198c6a3ed04baee03bfe-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-198c6a3ed04baee03bfe-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 红外截断(infrared cutoff)

- UUID: `2bf16f8e-01b1-462f-807c-fde871a89a6f` · okf_url: https://limbo137.top/okf/concepts/2bf16f8e-01b1-462f-807c-fde871a89a6f.md
- labels: en = infrared cutoff；zh-CN = 红外截断
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.2 · `books/woit-qtgr/2025-10-20/en/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-ee4a811202cac3168b9b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §36.2 · `books/woit-qtgr/2025-10-20/zh/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-ee4a811202cac3168b9b-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism.md

## 哈密顿方程(Hamilton’s equations)

- UUID: `2c564b8b-ea23-4822-b010-583fc86fb905` · okf_url: https://limbo137.top/okf/concepts/2c564b8b-ea23-4822-b010-583fc86fb905.md
- labels: en = Hamilton’s equations；zh-CN = 哈密顿方程
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-5868fd80b3ed210241c9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 外法向规则(outward normal rule)

- UUID: `2c8de24d-b302-4e88-9c0f-db37c70e2f7d` · okf_url: https://limbo137.top/okf/concepts/2c8de24d-b302-4e88-9c0f-db37c70e2f7d.md
- labels: en = outward normal rule；zh-CN = 外法向规则
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-85d5bb855bb031797b63-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-85d5bb855bb031797b63-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 有界的(bounded)

- UUID: `2cc2ae75-e460-4c7d-9b9f-acc867bbc0fb` · okf_url: https://limbo137.top/okf/concepts/2cc2ae75-e460-4c7d-9b9f-acc867bbc0fb.md
- labels: en = bounded；zh-CN = 有界的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-1fe2dd8db55ae82672c4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-bounded` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 勒贝格可测集(Lebesgue measurable set)

- UUID: `2cf3719f-bc6d-437a-96f9-c39c07fde7b9` · okf_url: https://limbo137.top/okf/concepts/2cf3719f-bc6d-437a-96f9-c39c07fde7b9.md
- labels: en = Lebesgue measurable set；zh-CN = 勒贝格可测集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-0c7821d3a69868989200-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 最小作用量原理(principle of least action)

- UUID: `2d1c5e15-26a6-4ca8-816a-91461fc9e654` · okf_url: https://limbo137.top/okf/concepts/2d1c5e15-26a6-4ca8-816a-91461fc9e654.md
- labels: en = principle of least action；zh-CN = 最小作用量原理
- aliases: principle of stationary action；平稳作用量原理；驻定作用量原理
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-3d6fc0d9443abdc40360-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-3d6fc0d9443abdc40360-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 全序(total order)

- UUID: `2d575fbe-b2df-46dc-a490-93876671fcbb` · okf_url: https://limbo137.top/okf/concepts/2d575fbe-b2df-46dc-a490-93876671fcbb.md
- labels: en = total order；zh-CN = 全序
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-66ca8cc6b6a85e6c191b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 多重线性映射(multilinear map)

- UUID: `2d72c1c4-0bf9-45f2-8afb-09fd91d06565` · okf_url: https://limbo137.top/okf/concepts/2d72c1c4-0bf9-45f2-8afb-09fd91d06565.md
- labels: en = multilinear map；zh-CN = 多重线性映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-b525b209eeaa0f496666-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 投影映射(projection)

- UUID: `2e84c121-965c-417d-93ec-e9ac516e3b29` · okf_url: https://limbo137.top/okf/concepts/2e84c121-965c-417d-93ec-e9ac516e3b29.md
- labels: en = projection；zh-CN = 投影映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-f0f80475fcc5c4877e5b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 超对称(supersymmetries)

- UUID: `2e8a69f5-6827-4c61-ad3b-dcc809cf942d` · okf_url: https://limbo137.top/okf/concepts/2e8a69f5-6827-4c61-ad3b-dcc809cf942d.md
- labels: en = supersymmetries；zh-CN = 超对称
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33 · `books/woit-qtgr/2025-10-20/en/chapter-33/index#first-f211e0acc19f155b1f69-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §33 · `books/woit-qtgr/2025-10-20/zh/chapter-33/index#first-f211e0acc19f155b1f69-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/index.md

## 李括号(Lie bracket)

- UUID: `2ec20f01-7d55-4d42-bfe7-cf436d149036` · okf_url: https://limbo137.top/okf/concepts/2ec20f01-7d55-4d42-bfe7-cf436d149036.md
- labels: en = Lie bracket；zh-CN = 李括号
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-114d1afaafbdde84e335-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.5 · `zh/chapter-15/05-commutators-flows-and-lie-derivatives#concept-15-lie-bracket` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.1 · `books/woit-qtgr/2025-10-20/zh/chapter-05/01-lie-algebras#first-51d70695e5ad73ea5e67-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.1 · `books/woit-qtgr/2025-10-20/en/chapter-05/01-lie-algebras#first-51d70695e5ad73ea5e67-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §5.1 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Lie bracket). The Lie bracket operation on $\mathfrak{g}$ is the bilinear antisymmetric map given by the commutator of matrices $$ [ \cdot , \cdot ]: (X, Y) \in \mathfrak {g} \times \mathfrak {g} \rightarrow [ X, Y ] = X Y - Y X \in \mathfrak {g} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/01-lie-algebras.md

## 内积的指数(index of an inner product)

- UUID: `2ed710df-ecf3-492f-9156-7475f79446ce` · okf_url: https://limbo137.top/okf/concepts/2ed710df-ecf3-492f-9156-7475f79446ce.md
- labels: en = index of an inner product；zh-CN = 内积的指数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-index-of-an-inner-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 外直和(external direct sum)

- UUID: `2f193cba-653a-4154-850a-a379f0081576` · okf_url: https://limbo137.top/okf/concepts/2f193cba-653a-4154-850a-a379f0081576.md
- labels: en = external direct sum；zh-CN = 外直和
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-external-direct-sum` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 群(group)

- UUID: `2f3aced7-6c7e-48e2-b0c1-4950a8da9857` · okf_url: https://limbo137.top/okf/concepts/2f3aced7-6c7e-48e2-b0c1-4950a8da9857.md
- labels: en = group；zh-CN = 群
- aliases: Group
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-d266984ecf4cc0979e3c-zh` · translation)；woit-qtgr/2025-10-20 (en · §1.3 · `books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations#first-d266984ecf4cc0979e3c-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §1.3 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Group). A group $G$ is a set with an associative multiplication, such that the set contains an identity element, as well as the multiplicative inverse of each element.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 数学结构(mathematical structure)

- UUID: `2f65813e-7535-4165-b134-2c2b474cf7e7` · okf_url: https://limbo137.top/okf/concepts/2f65813e-7535-4165-b134-2c2b474cf7e7.md
- labels: en = mathematical structure；zh-CN = 数学结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-92d3af98b2074ef659bc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 矩阵指数(matrix exponential)

- UUID: `2f6a696e-9405-429a-a1c7-180a8a780010` · okf_url: https://limbo137.top/okf/concepts/2f6a696e-9405-429a-a1c7-180a8a780010.md
- labels: en = matrix exponential；zh-CN = 矩阵指数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-matrix-exponential` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 定向0-单纯形(oriented 0-simplex)

- UUID: `2f89dbe4-42bd-481d-8c00-3afff9ee25f6` · okf_url: https://limbo137.top/okf/concepts/2f89dbe4-42bd-481d-8c00-3afff9ee25f6.md
- labels: en = oriented 0-simplex；zh-CN = 定向0-单纯形
- aliases: $\sigma_0 = p_0$；定向 0-单形；定向 0-单纯形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书r=0的特殊约定：定向0单纯形是点p0。维数受限版本，不同于任意维定向单纯形。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `For $r = 0$ , we formally define an oriented 0-simplex to be just a point $\sigma_0 = p_0$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 满射(surjection)

- UUID: `2fe19f0b-4db1-4a5b-b7e1-dd0e6281d9a0` · okf_url: https://limbo137.top/okf/concepts/2fe19f0b-4db1-4a5b-b7e1-dd0e6281d9a0.md
- labels: en = surjection；zh-CN = 满射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-7e183040ab4a74c5973c-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：满射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) A map $f: X \to Y$ is called injective (or one to one) if $x \neq x'$ implies $f(x) \neq f(x')$ . (b) A map $f: X \to Y$ is called surjective (or onto) if for each $y \in Y$ there exists at least one element $x \in X$ such that $f(x) = y$ . (c) A map $f: X \to Y$ is called bijective if it is both injective and surjective.`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：满射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.1. If a map satisfies a certain condition it bears a special name.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 测量问题(measurement problem)

- UUID: `3035413b-7cca-4017-a6d2-0152295e6e2a` · okf_url: https://limbo137.top/okf/concepts/3035413b-7cca-4017-a6d2-0152295e6e2a.md
- labels: en = measurement problem；zh-CN = 测量问题
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-34c0038a462ffee601dc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 等价表示(equivalent representations)

- UUID: `30471a7b-45c7-4fed-b604-6391300884fa` · okf_url: https://limbo137.top/okf/concepts/30471a7b-45c7-4fed-b604-6391300884fa.md
- labels: en = equivalent representations；zh-CN = 等价表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-equivalent-representations` · translation)；woit-qtgr/2025-10-20 (zh-CN · §4.3 · `books/woit-qtgr/2025-10-20/zh/chapter-04/03-change-of-basis#first-98dbbb60c1daac63b437-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/03-change-of-basis.md

## 正交反射(orthogonal reflection)

- UUID: `3057bc98-1560-4f68-971f-f741195b6499` · okf_url: https://limbo137.top/okf/concepts/3057bc98-1560-4f68-971f-f741195b6499.md
- labels: en = orthogonal reflection；zh-CN = 正交反射
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §29.2 · `books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry#first-a01df8be599f2af812c2-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §29.2 · `books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry#first-a01df8be599f2af812c2-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md

## 洛伦兹推动(Lorentz boost)

- UUID: `316c2c4b-6cd0-4631-add0-0a3dbbd743fd` · okf_url: https://limbo137.top/okf/concepts/316c2c4b-6cd0-4631-add0-0a3dbbd743fd.md
- labels: en = Lorentz boost；zh-CN = 洛伦兹推动
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.2 · `books/woit-qtgr/2025-10-20/zh/chapter-40/02-the-lorentz-group-and-its-lie-algebra#first-0cc4ab25f30451674969-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/02-the-lorentz-group-and-its-lie-algebra.md

## 完备度量空间(complete metric space)

- UUID: `31e7202c-8861-477b-9c46-0e0f2c97ff2c` · okf_url: https://limbo137.top/okf/concepts/31e7202c-8861-477b-9c46-0e0f2c97ff2c.md
- labels: en = complete metric space；zh-CN = 完备度量空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-75e5ab09c214cd7b12a0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 伽利略变换(Galilean transformation)

- UUID: `32374bbc-32fd-4f94-ad88-af53a211015b` · okf_url: https://limbo137.top/okf/concepts/32374bbc-32fd-4f94-ad88-af53a211015b.md
- labels: en = Galilean transformation；zh-CN = 伽利略变换
- aliases: Galilean transformations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-aed77135dff4bd1f8419-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-galilean-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 群的张量积表示(Tensor product representation of a group)

- UUID: `325a50ce-fadc-478e-9816-feb8f2a6af8d` · okf_url: https://limbo137.top/okf/concepts/325a50ce-fadc-478e-9816-feb8f2a6af8d.md
- labels: en = Tensor product representation of a group；zh-CN = 群的张量积表示
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §9.4 · `books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations#first-cfe253822791be105054-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §9.4 · `books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations#first-9f4db3e7c9c929ad26b4-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §9.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `对群$G$的表示$(\pi_V,V)$和$(\pi_W,W)$，定义其张量积表示$(\pi_{V\otimes W},V\otimes W)$如下： $$ (\pi_{V\otimes W}(g))(v\otimes w)=\pi_V(g)v\otimes\pi_W(g)w $$`
  - woit-qtgr/2025-10-20 §9.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `Given representations $(\pi_V,V)$ and $(\pi_W,W)$ of a group $G$, define their tensor-product representation $(\pi_{V\otimes W},V\otimes W)$ by $$ (\pi_{V\otimes W}(g))(v\otimes w)=\pi_V(g)v\otimes\pi_W(g)w. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/04-tensor-products-of-representations.md

## 完全可约表示(completely reducible representation)

- UUID: `32e00ccc-beaf-4a1d-987e-5b5c58a23d2b` · okf_url: https://limbo137.top/okf/concepts/32e00ccc-beaf-4a1d-987e-5b5c58a23d2b.md
- labels: en = completely reducible representation；zh-CN = 完全可约表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-completely-reducible-representation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md

## 多重形式(multiform)

- UUID: `32f4fedb-4606-4366-ab1e-8542f7d27682` · okf_url: https://limbo137.top/okf/concepts/32f4fedb-4606-4366-ab1e-8542f7d27682.md
- labels: en = multiform；zh-CN = 多重形式
- aliases: multiforms
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.3 · `en/chapter-08/03-exterior-product#first-302e6435bcc1425049d0-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.3 · `zh/chapter-08/03-exterior-product#first-fc962e3bfee5b0d1f998-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md

## 受限辛群(restricted symplectic group)

- UUID: `32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5` · okf_url: https://limbo137.top/okf/concepts/32fecc2c-a1e4-4f9b-b57c-68d2bed32ce5.md
- labels: en = restricted symplectic group；zh-CN = 受限辛群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §39.2 · `books/woit-qtgr/2025-10-20/zh/chapter-39/02-the-restricted-symplectic-group#first-3ad5554d4db186cd1885-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §39.2 · `books/woit-qtgr/2025-10-20/en/chapter-39/02-the-restricted-symplectic-group#first-3ad5554d4db186cd1885-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/02-the-restricted-symplectic-group.md

## 复Clifford代数(complex Clifford algebra)

- UUID: `33849610-9088-454e-a11d-ba3188cc7663` · okf_url: https://limbo137.top/okf/concepts/33849610-9088-454e-a11d-ba3188cc7663.md
- labels: en = complex Clifford algebra；zh-CN = 复Clifford代数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §28.1 · `books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-e703c0ef1873250004e6-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §28.1 · `books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-e703c0ef1873250004e6-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/01-the-complex-weyl-and-cliford-algebras.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md

## 贝塞尔不等式(Bessel’s inequality)

- UUID: `33daafc2-7c8a-412a-b694-353c4330607e` · okf_url: https://limbo137.top/okf/concepts/33daafc2-7c8a-412a-b694-353c4330607e.md
- labels: en = Bessel’s inequality；zh-CN = 贝塞尔不等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-e8eba56b1e007056cf30-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-bessel-s-inequality` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 庞加莱变换(Poincaré transformation)

- UUID: `3449b917-afff-47a2-bbfe-9b907e1af7a4` · okf_url: https://limbo137.top/okf/concepts/3449b917-afff-47a2-bbfe-9b907e1af7a4.md
- labels: en = Poincaré transformation；zh-CN = 庞加莱变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-poincare-transformation` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-defed131d18ea2091359-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 特征函数(characteristic function)

- UUID: `3451c6bf-4a36-41fe-9926-ea47ffdecc9e` · okf_url: https://limbo137.top/okf/concepts/3451c6bf-4a36-41fe-9926-ea47ffdecc9e.md
- labels: en = characteristic function；zh-CN = 特征函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-d9290747478d392f04de-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 法图引理(Fatou’s lemma)

- UUID: `34654eec-cd5e-4045-aa32-cbbe8fd63052` · okf_url: https://limbo137.top/okf/concepts/34654eec-cd5e-4045-aa32-cbbe8fd63052.md
- labels: en = Fatou’s lemma；zh-CN = 法图引理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-2783e36cf95c02d27031-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §11.3 · `en/chapter-11/03-lebesgue-integration#first-2783e36cf95c02d27031-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 向量丛(vector bundle)

- UUID: `346db9a3-44c0-42d4-beda-f834200ff290` · okf_url: https://limbo137.top/okf/concepts/346db9a3-44c0-42d4-beda-f834200ff290.md
- labels: en = vector bundle；zh-CN = 向量丛
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.5 · `books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere#first-997b23b0e8427f44f092-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-997b23b0e8427f44f092-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 希尔伯特空间(Hilbert space)

- UUID: `34c6c0e8-a933-4a51-bbf6-7721075b239c` · okf_url: https://limbo137.top/okf/concepts/34c6c0e8-a933-4a51-bbf6-7721075b239c.md
- labels: en = Hilbert space；zh-CN = 希尔伯特空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.1 · `en/chapter-13/01-definitions-and-examples#concept-13-hilbert-space` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-hilbert-space` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-5efc7defe0f023f15efe-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/01-definitions-and-examples.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 动量算符(momentum operator)

- UUID: `34c7e92b-ccec-4288-a000-265917ff0ffb` · okf_url: https://limbo137.top/okf/concepts/34c7e92b-ccec-4288-a000-265917ff0ffb.md
- labels: en = momentum operator；zh-CN = 动量算符
- aliases: Momentum operators
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-momentum-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-bd2b1c6544bf298b5e00-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §10.2 · `books/woit-qtgr/2025-10-20/zh/chapter-10/02-translations-in-time-and-space#first-142e8da5eb4195114da7-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §10.2 · `books/woit-qtgr/2025-10-20/en/chapter-10/02-translations-in-time-and-space#first-e6f103383e85fa6de992-en` · machine_draft)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §13.6 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `In the Hilbert space $L ^ { 2 } ( \mathbb { R } )$ of equivalence classes of square integrable functions (see Example 13.4), set $D$ to be the vector subspace of elements $\widetilde { \varphi }$ having a representative $\varphi$ from the $C^{\infty}$ functions on $\mathbb R$ of compact support. This is essentially the space of test functions ${ \mathcal ...`
  - woit-qtgr/2025-10-20 §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `For a quantum system with state space $\mathcal { H } = L ^ { 2 } ( \mathbf { R } ^ { 3 } )$ given by complex-valued functions of position variables $q _ { 1 } , q _ { 2 } , q _ { 3 }$ momentum operators $P _ { 1 } , P _ { 2 } , P _ { 3 }$ are defined by $$ P _ {1} = - i \hbar \frac {\partial}{\partial q _ {1}}, P _ {2} = - i \hbar \frac {\partial}{\parti...`
  - woit-qtgr/2025-10-20 §10.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `对于状态空间为 $\mathcal { H } = L ^ { 2 } ( \mathbf { R } ^ { 3 } )$、由位置变量 $q _ { 1 } , q _ { 2 } , q _ { 3 }$ 的复值函数给出的量子系统，动量算符 $P _ { 1 } , P _ { 2 } , P _ { 3 }$ 定义为 $$ P _ {1} = - i \hbar \frac {\partial}{\partial q _ {1}}, P _ {2} = - i \hbar \frac {\partial}{\partial q _ {2}}, P _ {3} = - i \hbar \frac {\partial}{\partial q _ {3}} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-10/02-translations-in-time-and-space.md

## 商群(factor group)

- UUID: `34d7ee62-cceb-462f-a9c9-8307ed5c5244` · okf_url: https://limbo137.top/okf/concepts/34d7ee62-cceb-462f-a9c9-8307ed5c5244.md
- labels: en = factor group；zh-CN = 商群
- aliases: factor groups；quotient group；quotient groups
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-factor-group` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §— (pdf_page=113, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 本章每个子群均正规，不能省去一般群商构造中的正规性条件。 · 原文: `Let $H$ be a subgroup of $G$ . We say $x, y \in G$ are equivalent if`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Szekeres 要求 $N$ 正规；不是任意子群都可形成群商。
  - szekeres/source-snapshot-bc3404156f6e §2.5 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: N是正规子群；G/N的乘法由代表元相乘诱导。 · 原文: `Furthermore, the coset $e N = N$ plays the role of an identity element, while every coset has an inverse $( g N ) ^ { - 1 } = g ^ { - 1 } N$ . Hence the cosets of a normal subgroup $N$ form a group called the factor group of $G$ by $N$, denoted $G / N$`
- 显式依赖组(dependency_groups): nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Proof. Both sides are groups according to lemma 3.1. Define a map $\varphi : G_1 / \ker f \to \operatorname{im} f$ by $\varphi([x]) = f(x)$ . This map is well d...`；nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Example 3.2. Let $f: \mathbb{Z} \to \mathbb{Z}_2$ be defined by $f(2n) = 0$ and $f(2n + 1) = 1$ . Then $\ker f = 2\mathbb{Z}$ and $\operatorname{im} f = \mathbb...`
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 黎曼联络(Riemannian connection)

- UUID: `34f22db1-cc59-44be-9cd8-005c9cadb6ad` · okf_url: https://limbo137.top/okf/concepts/34f22db1-cc59-44be-9cd8-005c9cadb6ad.md
- labels: en = Riemannian connection；zh-CN = 黎曼联络
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-0dad657edfc9067c98b1-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-0dad657edfc9067c98b1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 半直积群(semi-direct product group)

- UUID: `3518e444-803d-459b-bd0c-654aa486613e` · okf_url: https://limbo137.top/okf/concepts/3518e444-803d-459b-bd0c-654aa486613e.md
- labels: en = semi-direct product group；zh-CN = 半直积群
- aliases: Semi-direct product group
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.6 · `books/woit-qtgr/2025-10-20/zh/chapter-04/06-orthogonal-and-unitary-transformations#first-40cd3161ae8073520fc2-zh` · translation)；woit-qtgr/2025-10-20 (en · §18.2 · `books/woit-qtgr/2025-10-20/en/chapter-18/02-semi-direct-product-groups#woit-semi-direct-product-group` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-18/02-semi-direct-product-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/06-orthogonal-and-unitary-transformations.md

## 正则对易关系(Canonical commutation relations)

- UUID: `3547e89d-df9b-4a7a-9f78-22fe3f20a50b` · okf_url: https://limbo137.top/okf/concepts/3547e89d-df9b-4a7a-9f78-22fe3f20a50b.md
- labels: en = Canonical commutation relations；zh-CN = 正则对易关系
- aliases: canonical commutation relations
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §25.1 · `books/woit-qtgr/2025-10-20/en/chapter-25/01-multiple-degrees-of-freedom#first-f2557acc058ab1a8ecb7-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §25.1 · `books/woit-qtgr/2025-10-20/zh/chapter-25/01-multiple-degrees-of-freedom#first-f2557acc058ab1a8ecb7-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §25.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `正则对易关系（常缩写为 CCR）为 $$ [a_j,a_k^\dagger]=\delta_{jk}\mathbf 1,\qquad [a_j,a_k]=[a_j^\dagger,a_k^\dagger]=0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/01-multiple-degrees-of-freedom.md

## 轨道(orbit)

- UUID: `357477c3-6241-452b-834b-833bbcf1ea71` · okf_url: https://limbo137.top/okf/concepts/357477c3-6241-452b-834b-833bbcf1ea71.md
- labels: en = orbit；zh-CN = 轨道
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.5 · `en/chapter-15/05-commutators-flows-and-lie-derivatives#first-be5c0d4a8e58f2dac890-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-orbit` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/05-commutators-flows-and-lie-derivatives.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md

## 圆群(circle group)

- UUID: `35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9` · okf_url: https://limbo137.top/okf/concepts/35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9.md
- labels: en = circle group；zh-CN = 圆群
- aliases: The group $U(1)$；unit circle group
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §2 · `books/woit-qtgr/2025-10-20/zh/chapter-02/index#first-c205b531a9556fbe5a3d-zh` · translation)；woit-qtgr/2025-10-20 (en · §2.2 · `books/woit-qtgr/2025-10-20/en/chapter-02/02-the-group-u-1-and-its-representations#first-a6eca3bcad136ed9baa5-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §2.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `One might think that the simplest Lie group is the one dimensional additive group $\mathbf{R}$, a group that we will study together with its representations beginning in chapter 10. It turns out that one gets a much easier to analyze Lie group by adding a periodicity condition (which removes the problem of what happens as you go to $\pm\infty$), getting t...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/02-the-group-u-1-and-its-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/index.md

## 张量(tensor)

- UUID: `36315ed7-174d-4460-b840-89d2dc130e21` · okf_url: https://limbo137.top/okf/concepts/36315ed7-174d-4460-b840-89d2dc130e21.md
- labels: en = tensor；zh-CN = 张量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-bbf3a8c07408c18d2b7e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-bbf3a8c07408c18d2b7e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 同时性的相对性(relativity of simultaneity)

- UUID: `36553f3d-3063-49ab-861c-47fd4eec7cfd` · okf_url: https://limbo137.top/okf/concepts/36553f3d-3063-49ab-861c-47fd4eec7cfd.md
- labels: en = relativity of simultaneity；zh-CN = 同时性的相对性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.2 · `zh/chapter-09/02-relativistic-kinematics#first-26267762dabbe9a4ba43-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.2 · `en/chapter-09/02-relativistic-kinematics#first-26267762dabbe9a4ba43-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/02-relativistic-kinematics.md

## 接触变换(contact transformation)

- UUID: `368c43a6-d587-4e9d-96c7-99da4d648aaa` · okf_url: https://limbo137.top/okf/concepts/368c43a6-d587-4e9d-96c7-99da4d648aaa.md
- labels: en = contact transformation；zh-CN = 接触变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-050ffff04cc9cd79f7f8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-050ffff04cc9cd79f7f8-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 对称福克空间(symmetric Fock space)

- UUID: `3774fca5-2e21-4a5d-af64-923eb14eb423` · okf_url: https://limbo137.top/okf/concepts/3774fca5-2e21-4a5d-af64-923eb14eb423.md
- labels: en = symmetric Fock space；zh-CN = 对称福克空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-e3dae41920d742085441-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-symmetric-fock-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 傅里叶逆变换(inverse Fourier transform)

- UUID: `379e1338-0ea5-4619-b007-835a58c6e36c` · okf_url: https://limbo137.top/okf/concepts/379e1338-0ea5-4619-b007-835a58c6e36c.md
- labels: en = inverse Fourier transform；zh-CN = 傅里叶逆变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.3 · `zh/chapter-12/03-fourier-transforms#first-05b5d995e33f21f31b15-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/03-fourier-transforms.md

## 海涅–博雷尔定理(Heine–Borel theorem)

- UUID: `381a967a-011c-46ad-9fd8-208068cc9d29` · okf_url: https://limbo137.top/okf/concepts/381a967a-011c-46ad-9fd8-208068cc9d29.md
- labels: en = Heine–Borel theorem；zh-CN = 海涅–博雷尔定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §10.6 · `en/chapter-10/06-compact-spaces#first-e98f4d1c40f2c1f13f7c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-e98f4d1c40f2c1f13f7c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 线性泛函(linear functional)

- UUID: `38526a73-c4a9-4cef-bdee-8d961da25fda` · okf_url: https://limbo137.top/okf/concepts/38526a73-c4a9-4cef-bdee-8d961da25fda.md
- labels: en = linear functional；zh-CN = 线性泛函
- aliases: covector；余向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-linear-functional` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §3.7 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书n元关系与二元关系不同；covector与linear functional明确同义。 · 原文: `With respect to these operations the set oflinear functionals on $V$ forms a vector space over $\mathbb{K}$ called the dual space of $V _ { ; }$ , usually denoted $V ^ { * }$ . In keeping with earlier conventions, other possible notations for this space are $L ( V , \mathbb { K } )$ or $\operatorname{Hom}(V,K)$. Frequently, linear functionals on $V$ will ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md

## 共轭类(conjugacy class)

- UUID: `38649abc-85e1-4f7f-af1d-9f644a37a91b` · okf_url: https://limbo137.top/okf/concepts/38649abc-85e1-4f7f-af1d-9f644a37a91b.md
- labels: en = conjugacy class；zh-CN = 共轭类
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-conjugacy-class` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 阶(order)

- UUID: `38710033-3794-4234-8bbd-5f060b249b28` · okf_url: https://limbo137.top/okf/concepts/38710033-3794-4234-8bbd-5f060b249b28.md
- labels: en = order；zh-CN = 阶
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-order` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 简单函数(simple function)

- UUID: `38b6973b-8b4f-4cd3-97ae-bb7e46697725` · okf_url: https://limbo137.top/okf/concepts/38b6973b-8b4f-4cd3-97ae-bb7e46697725.md
- labels: en = simple function；zh-CN = 简单函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-9759b3f1d335692b30dd-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## Weyl代数(Weyl algebra)

- UUID: `38dc6df0-5043-4a95-9110-147b40ca7e51` · okf_url: https://limbo137.top/okf/concepts/38dc6df0-5043-4a95-9110-147b40ca7e51.md
- labels: en = Weyl algebra；zh-CN = Weyl代数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §32 · `books/woit-qtgr/2025-10-20/en/chapter-32/index#first-5ad217a6b870b4e0aa0b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §32 · `books/woit-qtgr/2025-10-20/zh/chapter-32/index#first-5ad217a6b870b4e0aa0b-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md

## 射影表示(projective representation)

- UUID: `38ecd601-9467-4080-844c-8b76816d3b95` · okf_url: https://limbo137.top/okf/concepts/38ecd601-9467-4080-844c-8b76816d3b95.md
- labels: en = projective representation；zh-CN = 射影表示
- aliases: projective；projective representations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-cc1844bb005888d81b42-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-de8e2adf86ce24a42446-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §7.1 · `books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation#concept-7-1-projective-representation` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.1 · `books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation#concept-7-1-projective-representation` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/03-the-anomaly-and-the-schwinger-term.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 线性算符(linear operator)

- UUID: `39428ebb-bc71-4f72-9ebe-c940f88d9879` · okf_url: https://limbo137.top/okf/concepts/39428ebb-bc71-4f72-9ebe-c940f88d9879.md
- labels: en = linear operator；zh-CN = 线性算符
- aliases: linear operators
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.3 · `zh/chapter-03/03-vector-space-homomorphisms#concept-3-linear-operator` · translation)；woit-qtgr/2025-10-20 (en · §4.1 · `books/woit-qtgr/2025-10-20/en/chapter-04/01-vector-spaces-and-linear-maps#concept-4-linear-map` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.1 · `books/woit-qtgr/2025-10-20/zh/chapter-04/01-vector-spaces-and-linear-maps#concept-4-linear-map` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/01-vector-spaces-and-linear-maps.md

## 线性化(linearize)

- UUID: `39c83087-8f6e-4028-bca4-e15886984c77` · okf_url: https://limbo137.top/okf/concepts/39c83087-8f6e-4028-bca4-e15886984c77.md
- labels: en = linearize；zh-CN = 线性化
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-add16b5bc60391c7391b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 元胞自动机(cellular automaton)

- UUID: `39c9ede5-d966-4d1a-a6f5-6118869c3c2c` · okf_url: https://limbo137.top/okf/concepts/39c9ede5-d966-4d1a-a6f5-6118869c3c2c.md
- labels: en = cellular automaton；zh-CN = 元胞自动机
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-d96d067d8bfcf0a33f4c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 海森堡不确定性定理(Heisenberg uncertainty theorem)

- UUID: `3a23f505-a2e2-4be4-92e0-ccc73f9a600a` · okf_url: https://limbo137.top/okf/concepts/3a23f505-a2e2-4be4-92e0-ccc73f9a600a.md
- labels: en = Heisenberg uncertainty theorem；zh-CN = 海森堡不确定性定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-6bb255744d1d4ebd815a-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-6bb255744d1d4ebd815a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 逻辑命题(proposition)

- UUID: `3a308f27-dd99-4ab6-a7ae-09666829b159` · okf_url: https://limbo137.top/okf/concepts/3a308f27-dd99-4ab6-a7ae-09666829b159.md
- labels: en = proposition；zh-CN = 逻辑命题
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-61caf15f825f6b36fa43-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 手性(Chirality)

- UUID: `3a39f940-8a51-4881-8f68-5fc85f295341` · okf_url: https://limbo137.top/okf/concepts/3a39f940-8a51-4881-8f68-5fc85f295341.md
- labels: en = Chirality；zh-CN = 手性
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §47.3 · `books/woit-qtgr/2025-10-20/zh/chapter-47/05-weyl-spinors#first-6d52c23daf6d5ca5b499-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §47.3 · `books/woit-qtgr/2025-10-20/en/chapter-47/05-weyl-spinors#first-6d38f617f31be57d1d94-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §47.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `该算符 $$ \gamma_ {5} = i \gamma_ {0} \gamma_ {1} \gamma_ {2} \gamma_ {3} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-47/05-weyl-spinors.md

## 对称张量(symmetric tensor)

- UUID: `3a5b55ef-0814-48af-910a-8879b536855b` · okf_url: https://limbo137.top/okf/concepts/3a5b55ef-0814-48af-910a-8879b536855b.md
- labels: en = symmetric tensor；zh-CN = 对称张量
- aliases: symmetric
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-760eb7e9150caa9c7ef3-tensor-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-f75f84548821dcd41bf9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 基数(cardinality)

- UUID: `3ac29f1f-bcfb-44ee-a0ae-8ca8187902cb` · okf_url: https://limbo137.top/okf/concepts/3ac29f1f-bcfb-44ee-a0ae-8ca8187902cb.md
- labels: en = cardinality；zh-CN = 基数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-9765ed468cf20b453f9e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 相对拓扑(relative topology)

- UUID: `3b04747e-23e9-4c1b-86a6-e07fc32de2e1` · okf_url: https://limbo137.top/okf/concepts/3b04747e-23e9-4c1b-86a6-e07fc32de2e1.md
- labels: en = relative topology；zh-CN = 相对拓扑
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-589585a61281f0346f8f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 李超代数(Lie superalgebra)

- UUID: `3b4db342-d792-4a76-bd33-37c7ca43c704` · okf_url: https://limbo137.top/okf/concepts/3b4db342-d792-4a76-bd33-37c7ca43c704.md
- labels: en = Lie superalgebra；zh-CN = 李超代数
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §30.2 · `books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-8a89b3a46e2cfbec8006-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §30.2 · `books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-77567078e09f32fab57f-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §30.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A [Lie superalgebra](/concepts/lie-theory/lie-superalgebra) structure on a real or complex vector space $V$ is given by a [Lie superbracket](/concepts/lie-theory/lie-superbracket) $[\cdot,\cdot]_\pm$, a bilinear map satisfying, on usual or fermionic generators $X,Y,Z$, $$ [X,Y]_\pm=-(-1)^{|X||Y|}[Y,X]_\pm $$ and the [super-Jaco...`
  - woit-qtgr/2025-10-20 §30.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `实或复向量空间 $V$ 上的[李超代数(Lie superalgebra)](/concepts/lie-theory/lie-superalgebra)结构由双线性映射[李超括号(Lie superbracket)](/concepts/lie-theory/lie-superbracket) $[\cdot,\cdot]_\pm$ 给出。它在通常的或费米子的生成元 $X,Y,Z$ 上满足 $$ [X,Y]_\pm=-(-1)^{|X||Y|}[Y,X]_\pm $$ 以及[超雅可比恒等式(super-Jacobi identity)](../../../../../concepts/lie-theory/super-jacobi-identity...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket.md

## 正则值(regular value)

- UUID: `3c19d901-1c08-4e84-a9c5-1658b4447b88` · okf_url: https://limbo137.top/okf/concepts/3c19d901-1c08-4e84-a9c5-1658b4447b88.md
- labels: en = regular value；zh-CN = 正则值
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-c9c9a931a1650d24a179-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#concept-13-regular-value` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 量子达尔文主义(quantum Darwinism)

- UUID: `3c247d87-47d1-44a1-8de6-205f195cc69e` · okf_url: https://limbo137.top/okf/concepts/3c247d87-47d1-44a1-8de6-205f195cc69e.md
- labels: en = quantum Darwinism；zh-CN = 量子达尔文主义
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.6 · `books/woit-qtgr/2025-10-20/zh/chapter-01/06-for-further-reading#concept-1-quantum-darwinism` · translation)；woit-qtgr/2025-10-20 (en · §1.6 · `books/woit-qtgr/2025-10-20/en/chapter-01/06-for-further-reading#concept-1-quantum-darwinism` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/06-for-further-reading.md

## 相似变换(similarity transformation)

- UUID: `3c375c9d-6401-49d6-bf50-aae52921d4a5` · okf_url: https://limbo137.top/okf/concepts/3c375c9d-6401-49d6-bf50-aae52921d4a5.md
- labels: en = similarity transformation；zh-CN = 相似变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-similarity-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 经典哈密顿函数(classical Hamiltonian function)

- UUID: `3c76b2f0-1ccb-43f9-b00a-b8ea4bc88a8d` · okf_url: https://limbo137.top/okf/concepts/3c76b2f0-1ccb-43f9-b00a-b8ea4bc88a8d.md
- labels: en = classical Hamiltonian function；zh-CN = 经典哈密顿函数
- aliases: Hamiltonian；哈密顿函数；哈密顿量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-14c43d8a9215edd8df7d-classical-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## Clebsch–Gordan 分解(Clebsch–Gordan decomposition)

- UUID: `3ca9ee3c-aa64-49a8-8eac-e4e9fdde1f33` · okf_url: https://limbo137.top/okf/concepts/3ca9ee3c-aa64-49a8-8eac-e4e9fdde1f33.md
- labels: en = Clebsch–Gordan decomposition；zh-CN = Clebsch–Gordan 分解
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §9.4 · `books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations#first-243cb6d1e32d4cc7f484-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §9.4 · `books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations#first-bee1103babcc35ba870b-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/04-tensor-products-of-representations.md

## 波算子(wave operator)

- UUID: `3d3ba215-ed6c-45ba-a293-b2ab86c51d69` · okf_url: https://limbo137.top/okf/concepts/3d3ba215-ed6c-45ba-a293-b2ab86c51d69.md
- labels: en = wave operator；zh-CN = 波算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-a385e9d66f55d77eab52-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-a385e9d66f55d77eab52-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 自旋(spin)

- UUID: `3e11a671-c004-44f2-8d6e-bf7933e08bbb` · okf_url: https://limbo137.top/okf/concepts/3e11a671-c004-44f2-8d6e-bf7933e08bbb.md
- labels: en = spin；zh-CN = 自旋
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-ca4672a1bebbc3fe7b05-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-spin` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/05-spatial-symmetries.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 洛伦兹规范(Lorenz gauge)

- UUID: `3e455468-be72-4afa-aba9-58c37c036e75` · okf_url: https://limbo137.top/okf/concepts/3e455468-be72-4afa-aba9-58c37c036e75.md
- labels: en = Lorenz gauge；zh-CN = 洛伦兹规范
- aliases: Lorentz gauge
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-d9326ccde36fac5ec14f-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-d9326ccde36fac5ec14f-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §46.6 · `books/woit-qtgr/2025-10-20/en/chapter-46/09-covariant-gauge-quantization#first-4077dbcfd8201c21a572-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §46.6 · `books/woit-qtgr/2025-10-20/zh/chapter-46/09-covariant-gauge-quantization#first-be10a701ba730b37e7f5-zh` · machine_draft)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §9.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Ignoring the tilde over $A$, any choice of 4-potential $A ^ { \mu }$ that satisfies $$ A _ {, \mu} ^ {\mu} = \nabla \cdot \mathbf {A} + \frac {1}{c} \frac {\partial \phi}{\partial t} = 0\tag{9.51} $$ is called a Lorentz gauge.`
  - woit-qtgr/2025-10-20 §46.9 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `$A$ vector potential $A _ { \mu }$ is said to be in Lorenz gauge if $$ \chi (A) \equiv - \frac {\partial A _ {0}}{\partial t} + \pmb {\nabla} \cdot \mathbf {A} = 0 $$ Besides Lorentz invariance`
  - woit-qtgr/2025-10-20 §46.6 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若矢势 $A _ { \mu }$ 满足 $$ \chi (A) \equiv - \frac {\partial A _ {0}}{\partial t} + \pmb {\nabla} \cdot \mathbf {A} = 0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-46/09-covariant-gauge-quantization.md

## 二次量子化(second quantization)

- UUID: `3e50e9ae-4157-4eea-a762-f943f292b9b2` · okf_url: https://limbo137.top/okf/concepts/3e50e9ae-4157-4eea-a762-f943f292b9b2.md
- labels: en = second quantization；zh-CN = 二次量子化
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §36.1 · `books/woit-qtgr/2025-10-20/zh/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-71b2f278d512fe650237-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §36.1 · `books/woit-qtgr/2025-10-20/en/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-71b2f278d512fe650237-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 向量子空间(vector subspace)

- UUID: `3e59dd1f-a3b0-4189-8bc8-363be7807161` · okf_url: https://limbo137.top/okf/concepts/3e59dd1f-a3b0-4189-8bc8-363be7807161.md
- labels: en = vector subspace；zh-CN = 向量子空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-vector-subspace` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 群同态(group homomorphism)

- UUID: `3e749de6-9d6f-43ea-a6de-ddff7c549a00` · okf_url: https://limbo137.top/okf/concepts/3e749de6-9d6f-43ea-a6de-ddff7c549a00.md
- labels: en = group homomorphism；zh-CN = 群同态
- aliases: group homomorphisms；homomorphism；同态
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-homomorphism` · translation)；woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-2468346c80db9d72875d-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-da110d60b99705240778-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §— (pdf_page=112, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 · 原文: `Let $G_{1}$ and $G_{2}$ be Abelian groups. A map $f: G_{1} \to G_{2}$ is said to be a homomorphism if`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
- 显式依赖组(dependency_groups): nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Proof. Both sides are groups according to lemma 3.1. Define a map $\varphi : G_1 / \ker f \to \operatorname{im} f$ by $\varphi([x]) = f(x)$ . This map is well d...`；nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , w...`；nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Example 3.2. Let $f: \mathbb{Z} \to \mathbb{Z}_2$ be defined by $f(2n) = 0$ and $f(2n + 1) = 1$ . Then $\ker f = 2\mathbb{Z}$ and $\operatorname{im} f = \mathbb...`
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 分布导数(distributional derivative)

- UUID: `3e9c67a0-81ad-468d-a98c-68d3af104351` · okf_url: https://limbo137.top/okf/concepts/3e9c67a0-81ad-468d-a98c-68d3af104351.md
- labels: en = distributional derivative；zh-CN = 分布导数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.2 · `zh/chapter-12/02-operations-on-distributions#first-3c942356e4619bea14af-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/02-operations-on-distributions.md

## 角动量算子(angular momentum operator)

- UUID: `3eb9597a-ad5c-4bc3-96cd-60280ad79388` · okf_url: https://limbo137.top/okf/concepts/3eb9597a-ad5c-4bc3-96cd-60280ad79388.md
- labels: en = angular momentum operator；zh-CN = 角动量算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-65644d184cd68c46bbe1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md

## 无限维(infinite dimensional)

- UUID: `3ef9fce1-922f-464f-b973-2d43ecaddccd` · okf_url: https://limbo137.top/okf/concepts/3ef9fce1-922f-464f-b973-2d43ecaddccd.md
- labels: en = infinite dimensional；zh-CN = 无限维
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-infinite-dimensional` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 布洛赫球(Bloch sphere)

- UUID: `3f43164d-2e69-4cc9-98b5-60fdc5e6f434` · okf_url: https://limbo137.top/okf/concepts/3f43164d-2e69-4cc9-98b5-60fdc5e6f434.md
- labels: en = Bloch sphere；zh-CN = 布洛赫球
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.5 · `books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere#first-8c3f0eb4496516550659-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-8c3f0eb4496516550659-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 经典极限(classical limit)

- UUID: `3fa7eb7e-bb45-4bd1-a7a4-a7c40cf9f2f6` · okf_url: https://limbo137.top/okf/concepts/3fa7eb7e-bb45-4bd1-a7a4-a7c40cf9f2f6.md
- labels: en = classical limit；zh-CN = 经典极限
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-e4da13267a1621d28b79-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 光锥(light cone)

- UUID: `3fafde9b-d338-4ed8-8089-0bfa0ee48b30` · okf_url: https://limbo137.top/okf/concepts/3fafde9b-d338-4ed8-8089-0bfa0ee48b30.md
- labels: en = light cone；zh-CN = 光锥
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-e85b302cea3ef1070173-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-e85b302cea3ef1070173-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §40.1 · `books/woit-qtgr/2025-10-20/en/chapter-40/01-minkowski-space#first-1f36e4e2df075fdc2203-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §40.1 · `books/woit-qtgr/2025-10-20/zh/chapter-40/01-minkowski-space#first-1f36e4e2df075fdc2203-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 拓扑等价(topologically equivalent)

- UUID: `3fbea852-b7db-4f32-9aa8-190e797cd8b0` · okf_url: https://limbo137.top/okf/concepts/3fbea852-b7db-4f32-9aa8-190e797cd8b0.md
- labels: en = topologically equivalent；zh-CN = 拓扑等价
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-0dcf34357d50cbdbee63-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 粒子视界(particle horizon)

- UUID: `3fe76a4c-b5f0-4bb9-b384-5f7f83b07128` · okf_url: https://limbo137.top/okf/concepts/3fe76a4c-b5f0-4bb9-b384-5f7f83b07128.md
- labels: en = particle horizon；zh-CN = 粒子视界
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-633d035df38d22e19e62-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-633d035df38d22e19e62-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 线性无关(linearly independent)

- UUID: `400f1610-1eba-4c3c-830b-e21ea6965065` · okf_url: https://limbo137.top/okf/concepts/400f1610-1eba-4c3c-830b-e21ea6965065.md
- labels: en = linearly independent；zh-CN = 线性无关
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-linearly-independent` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 协变张量(covariant tensor)

- UUID: `40115c4b-1875-476b-b2eb-8f9f8b71e6bb` · okf_url: https://limbo137.top/okf/concepts/40115c4b-1875-476b-b2eb-8f9f8b71e6bb.md
- labels: en = covariant tensor；zh-CN = 协变张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-6a893244d4bc6c0a6fa7-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-6a893244d4bc6c0a6fa7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 子覆盖(subcovering)

- UUID: `40652117-ec64-4eda-8fa8-bdfb858b1aa2` · okf_url: https://limbo137.top/okf/concepts/40652117-ec64-4eda-8fa8-bdfb858b1aa2.md
- labels: en = subcovering；zh-CN = 子覆盖
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-76c097697b1f74e8b7ad-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 连续统假设(continuum hypothesis)

- UUID: `41a13648-f379-4328-a0b6-0429f53c6e80` · okf_url: https://limbo137.top/okf/concepts/41a13648-f379-4328-a0b6-0429f53c6e80.md
- labels: en = continuum hypothesis；zh-CN = 连续统假设
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-33d1a1e2bec0d724a4a4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 四维速度(4-velocity)

- UUID: `41bf9209-0d1e-44a8-ad75-8dfeff9e0527` · okf_url: https://limbo137.top/okf/concepts/41bf9209-0d1e-44a8-ad75-8dfeff9e0527.md
- labels: en = 4-velocity；zh-CN = 四维速度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-0981752dd75d35e370cf-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-0981752dd75d35e370cf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 无穷小生成元(infinitesimal generator)

- UUID: `41c5a15f-f943-4381-b7e2-f4e457e84fd1` · okf_url: https://limbo137.top/okf/concepts/41c5a15f-f943-4381-b7e2-f4e457e84fd1.md
- labels: en = infinitesimal generator；zh-CN = 无穷小生成元
- aliases: infinitesimal generators
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-21037cd3830efca48e55-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-infinitesimal-generator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 投影算符(projection operator)

- UUID: `424805ab-e734-4973-9608-07892429420f` · okf_url: https://limbo137.top/okf/concepts/424805ab-e734-4973-9608-07892429420f.md
- labels: en = projection operator；zh-CN = 投影算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-19186d7c1ed96c63f981-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-projection-operator` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-c29f2a07feb2a0bf8fa4-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-c29f2a07feb2a0bf8fa4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 表示的直和(direct sum of representations)

- UUID: `43002c12-43d7-4d07-9293-49f3074b8953` · okf_url: https://limbo137.top/okf/concepts/43002c12-43d7-4d07-9293-49f3074b8953.md
- labels: en = direct sum of representations；zh-CN = 表示的直和
- aliases: direct sum；直和
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §2.1 · `books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory#first-a385c1c9bbbd755de28f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md

## 单调收敛定理(Monotone convergence theorem)

- UUID: `43141104-142c-4e97-b472-61595388c16c` · okf_url: https://limbo137.top/okf/concepts/43141104-142c-4e97-b472-61595388c16c.md
- labels: en = Monotone convergence theorem；zh-CN = 单调收敛定理
- aliases: Monotone convergence theorem (Beppo Levi)；单调收敛定理（贝波·列维） (Monotone convergence theorem (Beppo Levi))
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-54dfd3bed4f5767748da-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §11.3 · `en/chapter-11/03-lebesgue-integration#first-86fea91872b45bc21da6-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 正规序(normal ordering)

- UUID: `43457138-2721-4162-870f-85006ba3e6be` · okf_url: https://limbo137.top/okf/concepts/43457138-2721-4162-870f-85006ba3e6be.md
- labels: en = normal ordering；zh-CN = 正规序
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §25 · `books/woit-qtgr/2025-10-20/zh/chapter-25/index#first-b89fd35829a573405729-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §25 · `books/woit-qtgr/2025-10-20/en/chapter-25/index#first-b89fd35829a573405729-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/01-the-complex-weyl-and-cliford-algebras.md

## 同构(isomorphism)

- UUID: `4353d45e-8bf1-44c5-831b-4abf7a7182bc` · okf_url: https://limbo137.top/okf/concepts/4353d45e-8bf1-44c5-831b-4abf7a7182bc.md
- labels: en = isomorphism；zh-CN = 同构
- aliases: isomorphic；isomorphisms；同构的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=88, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §2.1.1：给定代数结构之间的双射结构保持映射。适用结构条件分别记录；不把任意双射或只有单向连续的映射都叫同构。 · 原文: `Suppose certain algebraic structures (product or addition, say) are endowed with the sets $X$ and $Y$ . If $f: X \to Y$ preserves these algebraic structures, then $f$ is called a homomorphism. For example, let $X$ be endowed with a product. If $f$ is a homomorphism, it preserves the product, $f(ab) = f(a)f(b)$ . Note that $ab$ is defined by the product ru...`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=88, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 给定代数结构之间存在双射结构保持映射 · 原文: `Suppose certain algebraic structures (product or addition, say) are endowed with the sets $X$ and $Y$ . If $f: X \to Y$ preserves these algebraic structures, then $f$ is called a homomorphism. For example, let $X$ be endowed with a product. If $f$ is a homomorphism, it preserves the product, $f(ab) = f(a)f(b)$ . Note that $ab$ is defined by the product ru...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 可数无限集(countably infinite set)

- UUID: `438baf25-c35b-4694-a032-72cff4c8c469` · okf_url: https://limbo137.top/okf/concepts/438baf25-c35b-4694-a032-72cff4c8c469.md
- labels: en = countably infinite set；zh-CN = 可数无限集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-0975a42b100b06860b9d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 克莱因-戈尔登方程(Klein-Gordon equation)

- UUID: `438ec001-b2be-4352-a008-bba8f8a1f3f1` · okf_url: https://limbo137.top/okf/concepts/438ec001-b2be-4352-a008-bba8f8a1f3f1.md
- labels: en = Klein-Gordon equation；zh-CN = 克莱因-戈尔登方程
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §43.1 · `books/woit-qtgr/2025-10-20/en/chapter-43/01-the-klein-gordon-equation-and-its-solutions#first-e92b7ab2feabb4687b62-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §43.1 · `books/woit-qtgr/2025-10-20/zh/chapter-43/01-the-klein-gordon-equation-and-its-solutions#first-dd4cc999609d536dcee9-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §43.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `克莱因-戈尔登方程是二阶偏微分方程 $$ \left(- \frac {\partial^ {2}}{\partial t ^ {2}} + \frac {\partial^ {2}}{\partial x _ {1} ^ {2}} + \frac {\partial^ {2}}{\partial x _ {2} ^ {2}} + \frac {\partial^ {2}}{\partial x _ {3} ^ {2}}\right) \phi = m ^ {2} \phi $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-43/01-the-klein-gordon-equation-and-its-solutions.md

## 海森堡群(Heisenberg group)

- UUID: `443f3561-4bb5-4d5b-b021-e674d7a47664` · okf_url: https://limbo137.top/okf/concepts/443f3561-4bb5-4d5b-b021-e674d7a47664.md
- labels: en = Heisenberg group；zh-CN = 海森堡群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.2 · `books/woit-qtgr/2025-10-20/zh/chapter-13/02-the-heisenberg-group#first-f3d51f414b7a22726f66-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.2 · `books/woit-qtgr/2025-10-20/en/chapter-13/02-the-heisenberg-group#first-a33e9908cbadee10bb61-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡群 $H_{2d+1}$ 是空间 $\mathbf R^{2d+1}$，其群乘法为 $$ \left(\binom{\mathbf x}{\mathbf y},z\right)\left(\binom{\mathbf x'}{\mathbf y'},z'\right)=\left(\binom{\mathbf x+\mathbf x'}{\mathbf y+\mathbf y'},z+z'+\frac12(\mathbf x\cdot\mathbf y'-\mathbf y\cdot\mathbf x')\right) $$ 其中 $\mathbf x,\mathbf x',\mathbf y,\mathbf y'\in\mathbf R^d$。'`
  - woit-qtgr/2025-10-20 §13.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Heisenberg group $H_{2d+1}$ is the space $\mathbf R^{2d+1}$ with the group law $$ \left(\binom{\mathbf x}{\mathbf y},z\right)\left(\binom{\mathbf x'}{\mathbf y'},z'\right)=\left(\binom{\mathbf x+\mathbf x'}{\mathbf y+\mathbf y'},z+z'+\frac12(\mathbf x\cdot\mathbf y'-\mathbf y\cdot\mathbf x')\right) $$ where $\mathbf x,\mathbf x',\mathbf y,\mathbf y'\i...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/02-the-heisenberg-group.md

## 测地坐标(geodesic coordinates)

- UUID: `4447f41d-cf47-4eba-83d4-fbd2afd7d87c` · okf_url: https://limbo137.top/okf/concepts/4447f41d-cf47-4eba-83d4-fbd2afd7d87c.md
- labels: en = geodesic coordinates；zh-CN = 测地坐标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-7fe6a2dd32207ef1f9ef-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-7fe6a2dd32207ef1f9ef-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 列维–奇维塔符号(Levi-Civita symbol)

- UUID: `4473ec98-3b0f-46b4-93de-60629332ba89` · okf_url: https://limbo137.top/okf/concepts/4473ec98-3b0f-46b4-93de-60629332ba89.md
- labels: en = Levi-Civita symbol；zh-CN = 列维–奇维塔符号
- aliases: Levi–Civita symbols
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.5 · `zh/chapter-08/05-oriented-vector-spaces#first-89b88f44acb83211bca9-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.5 · `en/chapter-08/05-oriented-vector-spaces#first-6e7ba517a10d0d34ccd8-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/05-oriented-vector-spaces.md

## 线性自同构(linear automorphism)

- UUID: `44a389d2-b094-42fc-b6bb-a92cd24cdadc` · okf_url: https://limbo137.top/okf/concepts/44a389d2-b094-42fc-b6bb-a92cd24cdadc.md
- labels: en = linear automorphism；zh-CN = 线性自同构
- aliases: linear transformation；线性变换；线性自同构（Szekeres约定）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-linear-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md

## 西格玛代数(sigma-algebra)

- UUID: `44bd4ec6-ac9e-4294-b54f-0decd732aeb5` · okf_url: https://limbo137.top/okf/concepts/44bd4ec6-ac9e-4294-b54f-0decd732aeb5.md
- labels: en = sigma-algebra；zh-CN = 西格玛代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-867c13c9c2d14aeab825-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## n 元关系(n-ary relation)

- UUID: `44c77505-82a8-434d-a726-351f2d95732f` · okf_url: https://limbo137.top/okf/concepts/44c77505-82a8-434d-a726-351f2d95732f.md
- labels: en = n-ary relation；zh-CN = n 元关系
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-0b128aeeb219318b6863-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 生成的拓扑(generated topology)

- UUID: `45298e19-ae4b-4100-9b74-56f108e83258` · okf_url: https://limbo137.top/okf/concepts/45298e19-ae4b-4100-9b74-56f108e83258.md
- labels: en = generated topology；zh-CN = 生成的拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-692f8f21ad12538ff904-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 自伴算符(self-adjoint operator with domain conditions)

- UUID: `454e5037-ea6c-46dc-a2b9-d9b27c051ab1` · okf_url: https://limbo137.top/okf/concepts/454e5037-ea6c-46dc-a2b9-d9b27c051ab1.md
- labels: en = self-adjoint operator with domain conditions；zh-CN = 自伴算符
- aliases: self-adjoint；自伴的；自伴算符（含定义域条件）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-self-adjoint` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-8af32d2d33df8cd4d08f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 实辛群(real symplectic group)

- UUID: `459d2b29-b1f3-4298-b1f9-2df232c37b62` · okf_url: https://limbo137.top/okf/concepts/459d2b29-b1f3-4298-b1f9-2df232c37b62.md
- labels: en = real symplectic group；zh-CN = 实辛群
- aliases: symplectic group；辛群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-symplectic-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.3 · `books/woit-qtgr/2025-10-20/zh/chapter-05/03-a-summary#first-72b4367d38b0209600ee-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.3 · `books/woit-qtgr/2025-10-20/en/chapter-05/03-a-summary#first-72b4367d38b0209600ee-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §16.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [symplectic group](/concepts/lie-theory/symplectic-group) $\mathrm{Sp}(2d,\mathbf R)$ is the subgroup of linear transformations $g$ of $\mathcal M=\mathbf R^{2d}$ satisfying $$ \Omega(gv_1,gv_2)=\Omega(v_1,v_2) $$ for all $v_1,v_2\in\mathcal M$.`
  - woit-qtgr/2025-10-20 §16.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[辛群(symplectic group)](/concepts/lie-theory/symplectic-group) $\mathrm{Sp}(2d,\mathbf R)$ 是 $\mathcal M=\mathbf R^{2d}$ 上保持 $\Omega$ 的线性变换 $g$ 所构成的子群，即满足 $$ \Omega(gv_1,gv_2)=\Omega(v_1,v_2) $$ 对所有 $v_1,v_2\in\mathcal M$ 都成立。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/03-a-summary.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/01-the-symplectic-group.md

## 拓扑空间(topological space)

- UUID: `45a8647a-b930-455d-8207-7edbb089a132` · okf_url: https://limbo137.top/okf/concepts/45a8647a-b930-455d-8207-7edbb089a132.md
- labels: en = topological space；zh-CN = 拓扑空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-dea640c66ed6d34a8246-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.3：拓扑空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $\emptyset, X \in \mathcal{T}$ . (ii) If $\mathcal{T}$ is any (maybe infinite) subcollection of $I$ , the family $\{U_j|j\in J\}$ satisfies $\cup_{j\in J}U_j\in \mathcal{T}$ . (iii) If $K$ is any finite subcollection of $I$ , the family $\{U_k|k\in K\}$ satisfies $\cap_{k\in K}U_k\in \mathcal{T}$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.3：拓扑空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.3. Let $X$ be any set and $\mathcal{T} = \{U_i | i \in I\}$ denote a certain collection of subsets of $X$ . The pair $(X, \mathcal{T})$ is a topological space if $\mathcal{T}$ satisfies the following requirements.`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.1 (pdf_page=100, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.3：拓扑空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$X$ alone is sometimes called a topological space. The $U_{i}$ are called the open sets and $\mathcal{T}$ is said to give a topology to $X$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 伴随矩阵(adjoint matrix)

- UUID: `45d0c13b-cde3-424f-8e70-a8c56a307519` · okf_url: https://limbo137.top/okf/concepts/45d0c13b-cde3-424f-8e70-a8c56a307519.md
- labels: en = adjoint matrix；zh-CN = 伴随矩阵
- aliases: adjoint
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-adjoint` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 指标集(index set)

- UUID: `45ddac14-ce30-4d20-a175-bbc4d85f45d4` · okf_url: https://limbo137.top/okf/concepts/45ddac14-ce30-4d20-a175-bbc4d85f45d4.md
- labels: en = index set；zh-CN = 指标集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-0409e5766920bf4d98fa-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 推动变换(boost)

- UUID: `45e9d7e5-ed93-488b-af37-b939eda83929` · okf_url: https://limbo137.top/okf/concepts/45e9d7e5-ed93-488b-af37-b939eda83929.md
- labels: en = boost；zh-CN = 推动变换
- aliases: boosts
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.2 · `en/chapter-09/02-relativistic-kinematics#first-453657fe974e14aaf56c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.2 · `zh/chapter-09/02-relativistic-kinematics#first-49129302d1f493db81ef-zh` · translation)；woit-qtgr/2025-10-20 (en · §40.2 · `books/woit-qtgr/2025-10-20/en/chapter-40/02-the-lorentz-group-and-its-lie-algebra#first-4fabf34d602bcab2dc76-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/02-relativistic-kinematics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/02-the-lorentz-group-and-its-lie-algebra.md

## 第一比安基恒等式(first Bianchi identity)

- UUID: `45ffeccd-00c3-4397-a705-466728615531` · okf_url: https://limbo137.top/okf/concepts/45ffeccd-00c3-4397-a705-466728615531.md
- labels: en = first Bianchi identity；zh-CN = 第一比安基恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-5827ca9ddfcc26c5cd43-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-5827ca9ddfcc26c5cd43-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 狄拉克弦(Dirac string)

- UUID: `461a79e2-9a12-4460-a940-0b2732f8b00b` · okf_url: https://limbo137.top/okf/concepts/461a79e2-9a12-4460-a940-0b2732f8b00b.md
- labels: en = Dirac string；zh-CN = 狄拉克弦
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.5 · `en/chapter-17/05-the-poincar-e-lemma#first-cee219554a6c2d13e1b7-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.5 · `zh/chapter-17/05-the-poincar-e-lemma#first-cee219554a6c2d13e1b7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/05-the-poincar-e-lemma.md

## 斜伴随矩阵(skew-adjoint matrices)

- UUID: `462a2e50-08b0-4eca-b51b-5b957df943b1` · okf_url: https://limbo137.top/okf/concepts/462a2e50-08b0-4eca-b51b-5b957df943b1.md
- labels: en = skew-adjoint matrices；zh-CN = 斜伴随矩阵
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §3.2 · `books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices#first-94c5c4bc84fb270c7415-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-94c5c4bc84fb270c7415-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/02-commutation-relations-for-pauli-matrices.md

## 收敛(convergence)

- UUID: `4630b373-c711-4f78-89ac-9d391ebb15dc` · okf_url: https://limbo137.top/okf/concepts/4630b373-c711-4f78-89ac-9d391ebb15dc.md
- labels: en = convergence；zh-CN = 收敛
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.1 · `zh/chapter-10/01-euclidean-topology#first-db5608de19c5f800f7f2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## 边缘(boundary)

- UUID: `464db988-1584-4cad-b2c3-a93d61fae5ad` · okf_url: https://limbo137.top/okf/concepts/464db988-1584-4cad-b2c3-a93d61fae5ad.md
- labels: en = boundary；zh-CN = 边缘
- aliases: p-boundary；r-boundary；$p$-边界；$p$-边缘；$r$-边界；$r$-边缘；边界
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-ccd810e4a97cf523defc-homology-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-ccd810e4a97cf523defc-homology-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \operatorname{im} \partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c = \partial_ {r + 1} d \tag {3.25} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \in C_r(K)$ . If there exists an element $d \in C_{r + 1}(K)$ such that`
  - szekeres/source-snapshot-bc3404156f6e §17.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 链群语境的边缘；各构造按原书保留，不与点集边界合并。 · 原文: `A $p$-chain $C$ is said to be a cycle if it has no boundary, $\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\prime}$ such that $C = \partial C ^ { \prime }$ . Clearly every boundary is a $p$-cycle since $\partial C = \partial ^ { 2 } C ^ { \prime } = 0$ , but the converse need not be true.`
  - szekeres/source-snapshot-bc3404156f6e §17.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 链群语境的边缘；各构造按原书保留，不与点集边界合并。 · 原文: `A $p$-boundary $B$ is a singular $p$-cycle on $M$ that is the boundary of a $(p+1)$-chain, $B = \partial _ { p + 1 } C . \mathrm { ~ A ~ }$ $p$-cycle $C$ is a singular $p$-chain on $M$ whose boundary vanishes, $\partial _ { p } C = 0$ Since $\partial ^ { 2 } = 0$ it is clear that every $p$-boundary is a $p$-cycle.`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=librarian-source-reviewed; editorial-sense-evidence, dependency_status=not-recorded) · 原文: `A $p$-boundary $B$ is a singular $p$-cycle on $M$ that is the boundary of a $(p+1)$-chain, $B = \partial _ { p + 1 } C . \mathrm { ~ A ~ }$ $p$-cycle $C$ is a singular $p$-chain on $M$ whose boundary vanishes, $\partial _ { p } C = 0$ Since $\partial ^ { 2 } = 0$ it is clear that every $p$-boundary is a $p$-cycle.`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=librarian-source-reviewed; editorial-sense-evidence, dependency_status=not-recorded) · 原文: `and the boundary of a $p$-simplex $\Delta$ is defined as the $( p - 1 )$ -chain $$ \partial \Delta = \sum_ {i = 0} ^ {p} \Delta_ {i} = \sum_ {i = 0} ^ {p} (- 1) ^ {i} \langle \mathbf {x} _ {0}, \ldots , \widehat {\mathbf {x}} _ {i}, \ldots , \mathbf {x} _ {p} \rangle , $$ ![Figure 17.5 Standard low dimensional $p$-simplexes](../../assets/chapter-17/part-0...`
  - szekeres/source-snapshot-bc3404156f6e §17.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 链群语境的边缘；各构造按原书保留，不与点集边界合并。 · 原文: `and the boundary of a $p$-simplex $\Delta$ is defined as the $( p - 1 )$ -chain $$ \partial \Delta = \sum_ {i = 0} ^ {p} \Delta_ {i} = \sum_ {i = 0} ^ {p} (- 1) ^ {i} \langle \mathbf {x} _ {0}, \ldots , \widehat {\mathbf {x}} _ {i}, \ldots , \mathbf {x} _ {p} \rangle , $$ ![Figure 17.5 Standard low dimensional $p$-simplexes](../../assets/chapter-17/part-0...`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=librarian-source-reviewed; editorial-sense-evidence, dependency_status=not-recorded) · 原文: `A $p$-chain $C$ is said to be a cycle if it has no boundary, $\partial C = 0 .$ . It is said to be a boundary if there exists a $(p+1)$-chain $C^{\prime}$ such that $C = \partial C ^ { \prime }$ . Clearly every boundary is a $p$-cycle since $\partial C = \partial ^ { 2 } C ^ { \prime } = 0$ , but the converse need not be true.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 可分的(separable)

- UUID: `46781123-db78-4492-b540-4ea28f03f02b` · okf_url: https://limbo137.top/okf/concepts/46781123-db78-4492-b540-4ea28f03f02b.md
- labels: en = separable；zh-CN = 可分的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-2c7109926df0c442b1b2-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-separable` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 链群(chain group)

- UUID: `46e6428d-bc3d-49a3-b86a-2e4d5b648ae1` · okf_url: https://limbo137.top/okf/concepts/46e6428d-bc3d-49a3-b86a-2e4d5b648ae1.md
- labels: en = chain group；zh-CN = 链群
- aliases: $C_r(K)$；$r$-chain group
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The reader might wonder about the appearance of a minus sign in front of $p_0$ . This is again related to the orientation. The following examples will clarify this point. In figure 3.7(a), an oriented 1-simplex $(p_0p_2)$ is divided into two, $(p_0p_1)$ and $(p_1p_2)$ . We agree that the boundary of $(p_0p_2)$ is $\{p_0\} \cup \{p_2\}$ and so should be th...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r}: C _ {r} (K) \rightarrow C _ {r - 1} (K). \tag {3.22} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let $K$ be an $n$ -dimensional simplicial complex. There exists a sequence of free Abelian groups and homomorphisms,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {0} p _ {0} = 0. \tag {3.18} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let $\sigma_r(p_0\ldots p_r)$ ( $r > 0$ ) be an oriented $r$ -simplex. The boundary $\partial_r\sigma_r$ of $\sigma_r$ is an $(r - 1)$ -chain defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ C _ {r} (K) \cong \underbrace {\mathbb {Z} \oplus \mathbb {Z} \oplus \cdots \oplus \mathbb {Z}} _ {I _ {r}}. \tag {3.17} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \begin{array}{l} \partial_ {2} \left(p _ {0} p _ {1} p _ {2}\right) = \left(p _ {1} p _ {2}\right) - \left(p _ {0} p _ {2}\right) + \left(p _ {0} p _ {1}\right) \\ \partial_ {3} \left(p _ {0} p _ {1} p _ {2} p _ {3}\right) = \left(p _ {1} p _ {2} p _ {3}\right) - \left(p _ {0} p _ {2} p _ {3}\right) + \left(p _ {0} p _ {1} p _ {3}\right) - \left(p _ {0...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {0} + p _ {1} + p _ {1} + p _ {2} + p _ {2} + p _ {0} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c = \sum_ {i = 1} ^ {I _ {r}} c _ {i} \sigma_ {r, i} \quad c _ {i} \in \mathbb {Z}. \tag {3.15} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The operator $\partial_r$ acts linearly on an element $c = \sum_{i} c_i \sigma_{r,i}$ of $C_r(K)$ ,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} \sigma_ {r} \equiv \sum_ {i = 0} ^ {r} (- 1) ^ {i} \left(p _ {0} p _ {1} \dots \hat {p} _ {i} \dots p _ {r}\right) \tag {3.20} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Before we define the cycle group and the boundary group, we need to introduce the boundary operator. Let us denote the boundary of an $r$ -simplex $\sigma_r$ by $\partial_r\sigma_r$ . $\partial_r$ should be understood as an operator acting on $\sigma_r$ to produce its boundary. This point of view will be elaborated later. Let us look at the boundaries of ...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {1} - p _ {0} + p _ {2} - p _ {1} + p _ {0} - p _ {2} = 0 $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} c = \sum_ {i} c _ {i} \partial_ {r} \sigma_ {r, i}. \tag {3.21} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `We formally define $\partial_0\sigma_0 = 0$ for $r = 0$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `which contradicts our intuition. If, on the other hand, we take $\partial_1(p_0p_1) = p_1 - p_0$ , we have`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$\partial_r$ is called the boundary operator. It is easy to see that the boundary operator is a homomorphism.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where the point $p_i$ under $\hat{\mathbf{\mu}}$ is omitted. For example,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$(p_0p_1) + (p_1p_2) + (p_2p_0)$ . We agree that it has no boundary. If we insisted on the rule $\partial_{1}(p_{0}p_{1}) = p_{0} + p_{1}$ , we would have`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c + c ^ {\prime} = \sum_ {i} \left(c _ {i} + c _ {i} ^ {\prime}\right) \sigma_ {r, i}. \tag {3.16} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.2. The $r$ -chain group $C_r(K)$ of a simplicial complex $K$ is a free Abelian group generated by the oriented $r$ -simplexes of $K$ . If $r > \dim K$ , $C_r(K)$ is defined to be 0. An element of $C_r(K)$ is called an $r$ -chain.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The unit element is $0 = \sum_{i} 0 \cdot \sigma_{r,i}$ , while the inverse element of $c$ is $-c = \sum_{i} (-c_i) \sigma_{r,i}$ . [Remark: An oppositely oriented $r$ -simplex $-\sigma_r$ is identified with $(-1) \sigma_r \in C_r(K)$ .] Thus, $C_r(K)$ is a free Abelian group of rank $I_r$ ,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ 0 \xrightarrow {i} C _ {n} (K) \xrightarrow {\partial_ {n}} C _ {n - 1} (K) \xrightarrow {\partial_ {n - 1}} \dots \xrightarrow {\partial_ {2}} C _ {1} (K) \xrightarrow {\partial_ {1}} C _ {0} (K) \xrightarrow {\partial_ {0}} 0 \tag {3.23} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\partial_r$ defines a map`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The integers $c_{i}$ are called the coefficients of $c$ . The group structure is given as follows. The addition of two $r$ -chains, $c = \sum_{i} c_{i} \sigma_{r,i}$ and $c' = \sum_{i} c_{i}' \sigma_{r,i}$ , is`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For a 1-simplex $(p_0p_1)$ , we define`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `as expected. Hence, we put a plus sign if the first vertex is omitted and a minus sign if the second is omitted. We employ this fact to define the boundary of a general $r$ -simplex.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) = p _ {1} - p _ {0}. \tag {3.19} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $i: 0 \hookrightarrow C_n(K)$ is an inclusion map (0 is regarded as the unit element of $C_n(K)$ ). This sequence is called the chain complex associated with $K$ and is denoted by $C(K)$ . It is interesting to study the image and kernel of the homomorphisms $\partial_r$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let there be $I_r$ $r$ -simplexes in $K$ . We denote each of them by $\sigma_{r,i}$ ( $1 \leq i \leq I_r$ ). Then $c \in C_r(K)$ is expressed as`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 线性映射(linear map)

- UUID: `4702877e-2ad4-4875-8a27-ef99339c8553` · okf_url: https://limbo137.top/okf/concepts/4702877e-2ad4-4875-8a27-ef99339c8553.md
- labels: en = linear map；zh-CN = 线性映射
- aliases: linear mapping；linear maps
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-linear-mapping` · translation)；woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-4c01999493383d81b50c-zh` · translation)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-4c01999493383d81b50c-en` · machine_draft)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §3.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `If $V$ and $W$ are two vector spaces over the same field $\mathbb{K}$, a map $T : V \to W$ is called linear, or a vector space homomorphism from $V$ into $W _ { ; }$ , if $$ T (a u + b v) = a T u + b T v\tag{3.1} $$ for all $a , b \in \mathbb { K }$ and all $u , v \in V$ . The notation $T u$ on the right-hand side is commonly used in place of $T ( u )$ . ...`
  - szekeres/source-snapshot-bc3404156f6e §2.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `A mapping $A : \mathbb { R } ^ { n } \to \mathbb { R } ^ { n }$ is said to be linear if $$ A (a \mathbf {x} + b \mathbf {y}) = a A (\mathbf {x}) + b A (\mathbf {y}) $$ for all vectors $x$, $\mathbf { y } \in \mathbb { R } ^ { n }$ and all real numbers $a , b \in \mathbb { R }$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group.md

## 自由向量空间(free vector space)

- UUID: `47072eab-eb7b-4766-b00a-0bf3fe867c5e` · okf_url: https://limbo137.top/okf/concepts/47072eab-eb7b-4766-b00a-0bf3fe867c5e.md
- labels: en = free vector space；zh-CN = 自由向量空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-7ea590de5c999a6681ac-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-7ea590de5c999a6681ac-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 紧辛群(compact symplectic group)

- UUID: `475077c6-8a0d-4954-9883-ba29330925d3` · okf_url: https://limbo137.top/okf/concepts/475077c6-8a0d-4954-9883-ba29330925d3.md
- labels: en = compact symplectic group；zh-CN = 紧辛群
- aliases: symplectic groups；紧辛群（四元数酉群）；辛群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#first-bc61ca187a358bb1fe74-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#first-0f032f052560f8bd866d-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 薛定谔绘景(Schrödinger picture)

- UUID: `4794ccb8-5d3f-4284-bb16-afe34bfa3d67` · okf_url: https://limbo137.top/okf/concepts/4794ccb8-5d3f-4284-bb16-afe34bfa3d67.md
- labels: en = Schrödinger picture；zh-CN = 薛定谔绘景
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；schwartz-qft/local-experiment = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-de2e21074d229fa07052-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-de2e21074d229fa07052-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §7.3 · `books/woit-qtgr/2025-10-20/zh/chapter-07/03-the-heisenberg-picture#concept-7-3-schrodinger-picture` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.3 · `books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture#concept-7-3-schrodinger-picture` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 逆映射(inverse mapping)

- UUID: `47c1c428-f32a-4c45-8750-2dc68caad65e` · okf_url: https://limbo137.top/okf/concepts/47c1c428-f32a-4c45-8750-2dc68caad65e.md
- labels: en = inverse mapping；zh-CN = 逆映射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-62330ca021eddcddfc36-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 联络1-形式(connection 1-form)

- UUID: `47cb9b8a-f841-4b73-85bc-a777473aa609` · okf_url: https://limbo137.top/okf/concepts/47cb9b8a-f841-4b73-85bc-a777473aa609.md
- labels: en = connection 1-form；zh-CN = 联络1-形式
- aliases: connection 1-forms；联络一形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.7 · `zh/chapter-18/07-cartan-formalism#first-d612eb4de01c68cd76e7-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.7 · `en/chapter-18/07-cartan-formalism#first-c6b3fcebb0af1cf3e6c4-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/07-cartan-formalism.md

## 特征多项式(characteristic polynomial)

- UUID: `482ee8fb-4b0d-42dc-a313-c214cae1ff1e` · okf_url: https://limbo137.top/okf/concepts/482ee8fb-4b0d-42dc-a313-c214cae1ff1e.md
- labels: en = characteristic polynomial；zh-CN = 特征多项式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-characteristic-polynomial` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 调和规范(harmonic gauge)

- UUID: `4838c3a1-746c-4112-b3de-526aab1035f0` · okf_url: https://limbo137.top/okf/concepts/4838c3a1-746c-4112-b3de-526aab1035f0.md
- labels: en = harmonic gauge；zh-CN = 调和规范
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-d9a2910f651fd35cad88-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-d9a2910f651fd35cad88-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 自由结合代数(free associative algebra)

- UUID: `48420078-611a-40d6-b7c5-02935856b181` · okf_url: https://limbo137.top/okf/concepts/48420078-611a-40d6-b7c5-02935856b181.md
- labels: en = free associative algebra；zh-CN = 自由结合代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-67ed67400db27e00e05e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-67ed67400db27e00e05e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 正交(orthogonal)

- UUID: `4843528b-7c62-4895-8f6c-b663cf7ccb91` · okf_url: https://limbo137.top/okf/concepts/4843528b-7c62-4895-8f6c-b663cf7ccb91.md
- labels: en = orthogonal；zh-CN = 正交
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.3 · `en/chapter-13/03-linear-functionals#first-7baa35007a91219ffbd0-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-orthogonal` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/03-linear-functionals.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 限制(restriction)

- UUID: `488b6a4d-4c6a-499a-8696-328fc48e46e8` · okf_url: https://limbo137.top/okf/concepts/488b6a4d-4c6a-499a-8696-328fc48e46e8.md
- labels: en = restriction；zh-CN = 限制
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-7da983e944068568c49d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 受限辛群与受限正交群(Restricted symplectic and orthogonal groups)

- UUID: `489e7a75-33e7-46e1-b794-5c268cca2a23` · okf_url: https://limbo137.top/okf/concepts/489e7a75-33e7-46e1-b794-5c268cca2a23.md
- labels: en = Restricted symplectic and orthogonal groups；zh-CN = 受限辛群与受限正交群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §39.2 · `books/woit-qtgr/2025-10-20/zh/chapter-39/02-the-restricted-symplectic-group#first-f963f2be39fb932857a5-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §39.2 · `books/woit-qtgr/2025-10-20/en/chapter-39/02-the-restricted-symplectic-group#first-4323d7c01dcec4577d14-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §39.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[受限辛群(restricted symplectic group)](/concepts/symplectic-geometry/restricted-symplectic-group) $Sp_{\mathrm{res}}$ 由保持无限维向量空间上的辛结构、并满足下述条件的线性变换 $g$ 组成： $$ \operatorname{tr}(A^\dagger A) $$ [受限正交群(restricted orthogonal group)](/concepts/linear-algebra/restricted-orthogonal-group) $SO_{\mathrm{res}}$ 则由保持无限维空间上的内积、且对 $[J_g,J_0]$ ...`
  - woit-qtgr/2025-10-20 §39.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [restricted symplectic group](/concepts/symplectic-geometry/restricted-symplectic-group) $Sp_{\mathrm{res}}$ consists of symplectic linear transformations $g$ on the infinite-dimensional symplectic space satisfying $$ \operatorname{tr}(A^\dagger A) $$ The [restricted orthogonal group](../../../../../concepts/linear-algebra/restricted-ort...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/02-the-restricted-symplectic-group.md

## 图像(graph)

- UUID: `48bd14ac-fca7-46be-b50e-f729bd475980` · okf_url: https://limbo137.top/okf/concepts/48bd14ac-fca7-46be-b50e-f729bd475980.md
- labels: en = graph；zh-CN = 图像
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-df4287765e747e65cfc8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-graph` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 内测度(inner measure)

- UUID: `496d1b66-73ef-4e6a-965e-6e67183f7c50` · okf_url: https://limbo137.top/okf/concepts/496d1b66-73ef-4e6a-965e-6e67183f7c50.md
- labels: en = inner measure；zh-CN = 内测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-cda9152c54ea73af72d4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 仿射参数(affine parameter)

- UUID: `49a9d4dc-b1df-4c2f-aa1c-041435108e07` · okf_url: https://limbo137.top/okf/concepts/49a9d4dc-b1df-4c2f-aa1c-041435108e07.md
- labels: en = affine parameter；zh-CN = 仿射参数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-d1335b23c7468b19892f-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-d1335b23c7468b19892f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 自伴算符的谱定理(Spectral theorem for self-adjoint operators)

- UUID: `49df86f1-e435-4072-ad8f-2cf200b89a68` · okf_url: https://limbo137.top/okf/concepts/49df86f1-e435-4072-ad8f-2cf200b89a68.md
- labels: en = Spectral theorem for self-adjoint operators；zh-CN = 自伴算符的谱定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-0c7c88a6b2004b097883-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#first-0c7c88a6b2004b097883-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 几何结构(geometric structure)

- UUID: `4a4cc83b-b7d3-45bb-aab2-a4918310d48a` · okf_url: https://limbo137.top/okf/concepts/4a4cc83b-b7d3-45bb-aab2-a4918310d48a.md
- labels: en = geometric structure；zh-CN = 几何结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-3726814eb6fa6dd04cda-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 对偶表示或逆变表示(Dual or contragredient representation)

- UUID: `4ae0743e-27f5-4211-ba73-7671ebc0738a` · okf_url: https://limbo137.top/okf/concepts/4ae0743e-27f5-4211-ba73-7671ebc0738a.md
- labels: en = Dual or contragredient representation；zh-CN = 对偶表示或逆变表示
- aliases: dual or contragredient representation
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.2 · `books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces#first-5854b76d4f5dde5ced68-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-5854b76d4f5dde5ced68-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Dual or contragredient representation). The dual or contragredient representation on $V ^ { * }$ is given by taking as linear operators $$ (\pi^ {- 1}) ^ {t} (g): V ^ {*} \to V ^ {*}\tag{4.2} $$`
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（[对偶表示或逆变表示(dual or contragredient representation)](/concepts/representation-theory/dual-representation)）。$V ^ { * }$ 上的对偶表示或反步表示由取作线性算符给出 $$ (\pi^ {- 1}) ^ {t} (g): V ^ {*} \to V ^ {*}\tag{4.2} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md

## 有效的(effective)

- UUID: `4aea9cc6-3e2f-45a6-bb3d-92a8c19224ae` · okf_url: https://limbo137.top/okf/concepts/4aea9cc6-3e2f-45a6-bb3d-92a8c19224ae.md
- labels: en = effective；zh-CN = 有效的
- aliases: 有效
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-4b12786769396862bf40-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md

## 正则化(regularization)

- UUID: `4b06a6ef-00b0-41c0-b826-3ac3059efaa8` · okf_url: https://limbo137.top/okf/concepts/4b06a6ef-00b0-41c0-b826-3ac3059efaa8.md
- labels: en = regularization；zh-CN = 正则化
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §39.5 · `books/woit-qtgr/2025-10-20/en/chapter-39/05-higher-order-operators-and-renormalization#first-b248617a299cb7252c3e-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §39.5 · `books/woit-qtgr/2025-10-20/zh/chapter-39/05-higher-order-operators-and-renormalization#first-b248617a299cb7252c3e-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/05-higher-order-operators-and-renormalization.md

## 勒贝格控制收敛定理(Lebesgue dominated convergence theorem)

- UUID: `4b34c7bc-ab68-4690-952f-eec2de9e865c` · okf_url: https://limbo137.top/okf/concepts/4b34c7bc-ab68-4690-952f-eec2de9e865c.md
- labels: en = Lebesgue dominated convergence theorem；zh-CN = 勒贝格控制收敛定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §11.3 · `en/chapter-11/03-lebesgue-integration#first-12eea7dbfea39eb77dc3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-12eea7dbfea39eb77dc3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 矩映射(Moment map)

- UUID: `4b3c4f61-3c9d-4308-b4d4-ed0cb6e26fa4` · okf_url: https://limbo137.top/okf/concepts/4b3c4f61-3c9d-4308-b4d4-ed0cb6e26fa4.md
- labels: en = Moment map；zh-CN = 矩映射
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §15.3 · `books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map#first-7afaeb94340449c20f59-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §15.3 · `books/woit-qtgr/2025-10-20/zh/chapter-15/03-group-actions-on-m-and-the-moment-map#first-f57a1cbc45867461ee61-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §15.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `给定 $G$ 对相空间 $M$ 的作用，若有李代数同态 $$ L\longmapsto\mu_L $$ 将 $\mathfrak g$ 映到 $M$ 上的函数，并满足 $$ X_L=X_{\mu_L}, $$ 则称该映射为矩映射。等价地，对于 $M$ 上的函数 $F$，$\mu_L$ 满足 $$ \{\mu_L,F\}(m)=-X_LF(m)=\left.\frac{d}{dt}F(e^{-tL}\cdot m)\right|_{t=0}.\tag{15.11} $$`
  - woit-qtgr/2025-10-20 §15.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `Given an action of $G$ on phase space $M$, a Lie algebra homomorphism $$ L\longmapsto\mu_L $$ from $\mathfrak g$ to functions on $M$ is called a moment map if $$ X_L=X_{\mu_L}. $$ Equivalently, for functions $F$ on $M$, $\mu_L$ satisfies $$ \{\mu_L,F\}(m)=-X_LF(m)=\left.\frac{d}{dt}F(e^{-tL}\cdot m)\right|_{t=0}.\tag{15.11} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-15/03-group-actions-on-m-and-the-moment-map.md

## 张量的分量(tensor components)

- UUID: `4b81a76b-55e7-4097-a345-89d8b778f08e` · okf_url: https://limbo137.top/okf/concepts/4b81a76b-55e7-4097-a345-89d8b778f08e.md
- labels: en = tensor components；zh-CN = 张量的分量
- aliases: components；张量分量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-18ad13985e23c54f76ee-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-d109fe7c9c0f0518f204-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 施瓦西解(Schwarzschild solution)

- UUID: `4b93cc33-a176-4805-bd45-238798451e0c` · okf_url: https://limbo137.top/okf/concepts/4b93cc33-a176-4805-bd45-238798451e0c.md
- labels: en = Schwarzschild solution；zh-CN = 施瓦西解
- aliases: 史瓦西解；施瓦氏解
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-4580eec697fdf3c32123-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-4580eec697fdf3c32123-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 参数曲线(parametrized curve)

- UUID: `4be5393c-e464-4ede-ac10-bacd25e3b1e5` · okf_url: https://limbo137.top/okf/concepts/4be5393c-e464-4ede-ac10-bacd25e3b1e5.md
- labels: en = parametrized curve；zh-CN = 参数曲线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.2 · `zh/chapter-15/02-differentiable-maps-and-curves#concept-15-parametrized-curve` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.2 · `en/chapter-15/02-differentiable-maps-and-curves#first-c6d3928615da6b50e111-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/02-differentiable-maps-and-curves.md

## 自伴矩阵的谱定理(Spectral theorem for self-adjoint matrices)

- UUID: `4c5b32a9-4d9f-4f2b-804a-12c487399d39` · okf_url: https://limbo137.top/okf/concepts/4c5b32a9-4d9f-4f2b-804a-12c487399d39.md
- labels: en = Spectral theorem for self-adjoint matrices；zh-CN = 自伴矩阵的谱定理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.7 · `books/woit-qtgr/2025-10-20/zh/chapter-04/07-eigenvalues-and-eigenvectors#first-95a38030b29b01d5927d-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.7 · `books/woit-qtgr/2025-10-20/en/chapter-04/07-eigenvalues-and-eigenvectors#first-6c94bd48e99862283c91-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 规范群(Gauge group)

- UUID: `4c74fea1-7d81-4015-bdee-4561d67bcd4d` · okf_url: https://limbo137.top/okf/concepts/4c74fea1-7d81-4015-bdee-4561d67bcd4d.md
- labels: en = Gauge group；zh-CN = 规范群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.1 · `books/woit-qtgr/2025-10-20/zh/chapter-45/01-u-1-gauge-symmetry#first-93f4e0d9093d61909885-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.1 · `books/woit-qtgr/2025-10-20/en/chapter-45/01-u-1-gauge-symmetry#first-3bfc8e9d948f62d50be7-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `定义在 $M ^ { 4 }$ 上、取值于单位圆 $U ( 1 )$ 的函数构成的群 $\mathcal { G }$，其群法则由逐点乘法给出 $$ e ^ {i e \varphi_ {1} (x)} \cdot e ^ {i e \varphi_ {2} (x)} = e ^ {i e (\varphi_ {1} (x) + \varphi_ {2} (x))} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/01-u-1-gauge-symmetry.md

## 对称联络(symmetric connection)

- UUID: `4cbbafec-2379-4b3d-b21a-4cc767822af7` · okf_url: https://limbo137.top/okf/concepts/4cbbafec-2379-4b3d-b21a-4cc767822af7.md
- labels: en = symmetric connection；zh-CN = 对称联络
- aliases: symmetric；对称联络（无挠联络）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-7014997d70e6559f6fc8-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-760eb7e9150caa9c7ef3-connection-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 闭链(cycle)

- UUID: `4ce8076e-d99e-40dd-b263-2ba130402d2c` · okf_url: https://limbo137.top/okf/concepts/4ce8076e-d99e-40dd-b263-2ba130402d2c.md
- labels: en = cycle；zh-CN = 闭链
- aliases: homology cycle；p-cycle；r-cycle；$p$-闭链；$r$-闭链
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-20322b3f8009b543e4b2-homology-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-20322b3f8009b543e4b2-homology-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \ker \partial_{r}$ . [Remark: If $r = 0$ , $\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.3. If $c \in C_r(K)$ satisfies`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} c = 0 \tag {3.24} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 伽利略空间(Galilean space)

- UUID: `4ce8148b-d4ae-4583-8172-277abeeb26cf` · okf_url: https://limbo137.top/okf/concepts/4ce8148b-d4ae-4583-8172-277abeeb26cf.md
- labels: en = Galilean space；zh-CN = 伽利略空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-galilean-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 厄米多项式(Hermite polynomials)

- UUID: `4d0a6e4e-434f-447d-a9fd-6747ab984a87` · okf_url: https://limbo137.top/okf/concepts/4d0a6e4e-434f-447d-a9fd-6747ab984a87.md
- labels: en = Hermite polynomials；zh-CN = 厄米多项式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-e68940f41eb517ea3f07-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-hermite-polynomials` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 维数(dimension)

- UUID: `4d8daf81-914a-426c-9cc7-534fc6eed006` · okf_url: https://limbo137.top/okf/concepts/4d8daf81-914a-426c-9cc7-534fc6eed006.md
- labels: en = dimension；zh-CN = 维数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-dimension` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 微积分(calculus)

- UUID: `4ddbcd07-562c-582f-a6e8-7e5974c5cc02` · okf_url: https://limbo137.top/okf/concepts/4ddbcd07-562c-582f-a6e8-7e5974c5cc02.md
- labels: en = calculus；zh-CN = 微积分
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 试验函数空间(test function space)

- UUID: `4e808969-ade5-405c-b05b-2e0bbaaadafa` · okf_url: https://limbo137.top/okf/concepts/4e808969-ade5-405c-b05b-2e0bbaaadafa.md
- labels: en = test function space；zh-CN = 试验函数空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-879ee36cb3f8b9f9881f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 不可约表示(irreducible representation)

- UUID: `4e9e2c89-0422-42cf-8762-6b5e59837ef0` · okf_url: https://limbo137.top/okf/concepts/4e9e2c89-0422-42cf-8762-6b5e59837ef0.md
- labels: en = irreducible representation；zh-CN = 不可约表示
- aliases: Irreducible representation；irreducible representations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-irreducible-representation` · translation)；woit-qtgr/2025-10-20 (zh-CN · §2.1 · `books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory#first-2f3541c4170f75fb114f-zh` · translation)；woit-qtgr/2025-10-20 (en · §2.1 · `books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory#first-2f3541c4170f75fb114f-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §2.1 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Irreducible representation). A representation $\pi$ is called irreducible if it has no subrepresentations, meaning non-zero proper subspaces $W \subset V$ such that $( \pi _ { | W } , W )$ is a representation. A representation that does have such a subrepresentation is called reducible.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 哈密顿函数(Hamiltonian function)

- UUID: `4eb4e3f1-c861-4ad3-aa28-6972dd6ea552` · okf_url: https://limbo137.top/okf/concepts/4eb4e3f1-c861-4ad3-aa28-6972dd6ea552.md
- labels: en = Hamiltonian function；zh-CN = 哈密顿函数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.5 · `books/woit-qtgr/2025-10-20/zh/chapter-01/05-groups-and-symmetries#first-1a5f4280c9dab3c7c162-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/05-groups-and-symmetries.md

## 离散群(discrete group)

- UUID: `4ed166c8-a90c-4c52-9bd7-db726835b380` · okf_url: https://limbo137.top/okf/concepts/4ed166c8-a90c-4c52-9bd7-db726835b380.md
- labels: en = discrete group；zh-CN = 离散群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-ab1d9acc2a37ba9ec0cf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 四维力(4-force)

- UUID: `4f13b233-3060-4c00-a769-c558429c3aa7` · okf_url: https://limbo137.top/okf/concepts/4f13b233-3060-4c00-a769-c558429c3aa7.md
- labels: en = 4-force；zh-CN = 四维力
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-bb7eaa1854dc57fc7706-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-bb7eaa1854dc57fc7706-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## Wiener测度(Wiener measure)

- UUID: `4f2153cc-eb69-4a7d-ba5c-eabaf46627e6` · okf_url: https://limbo137.top/okf/concepts/4f2153cc-eb69-4a7d-ba5c-eabaf46627e6.md
- labels: en = Wiener measure；zh-CN = Wiener测度
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.3 · `books/woit-qtgr/2025-10-20/zh/chapter-35/03-quantization-and-path-integrals#first-bf33b2fb096561412dd4-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.3 · `books/woit-qtgr/2025-10-20/en/chapter-35/03-quantization-and-path-integrals#first-bf33b2fb096561412dd4-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/03-quantization-and-path-integrals.md

## 动力学变量(dynamical variable)

- UUID: `4f329e32-44d0-4b99-86ea-6081a59ce73d` · okf_url: https://limbo137.top/okf/concepts/4f329e32-44d0-4b99-86ea-6081a59ce73d.md
- labels: en = dynamical variable；zh-CN = 动力学变量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-ec9af8e84a0c2190a4c3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-ec9af8e84a0c2190a4c3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 恒等算符(identity operator)

- UUID: `4f5f10c1-2f69-4e89-8fab-84a7f7a3061b` · okf_url: https://limbo137.top/okf/concepts/4f5f10c1-2f69-4e89-8fab-84a7f7a3061b.md
- labels: en = identity operator；zh-CN = 恒等算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-b4aeaf0008ec9304a949-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-b4aeaf0008ec9304a949-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 闭形式(closed form)

- UUID: `4f8194d3-f4b1-47a6-8101-aa8cd016f6e1` · okf_url: https://limbo137.top/okf/concepts/4f8194d3-f4b1-47a6-8101-aa8cd016f6e1.md
- labels: en = closed form；zh-CN = 闭形式
- aliases: closed；闭的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-0bbe12462b1c9456320e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 泡利方程(Pauli equation)

- UUID: `4f857975-df69-4cc2-a60b-7a498719299e` · okf_url: https://limbo137.top/okf/concepts/4f857975-df69-4cc2-a60b-7a498719299e.md
- labels: en = Pauli equation；zh-CN = 泡利方程
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §34.1 · `books/woit-qtgr/2025-10-20/en/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3#first-c70019d0222353c7cd02-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §34.1 · `books/woit-qtgr/2025-10-20/zh/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3#first-c70019d0222353c7cd02-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3.md

## 斜厄米矩阵(skew-Hermitian matrix)

- UUID: `4f876e03-2ba8-494f-b0f2-afd46ab6b934` · okf_url: https://limbo137.top/okf/concepts/4f876e03-2ba8-494f-b0f2-afd46ab6b934.md
- labels: en = skew-Hermitian matrix；zh-CN = 斜厄米矩阵
- aliases: skew-Hermitian；skew-Hermitian matrices
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.2 · `books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-584d72d499a6f41c5867-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.2 · `books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-f5be45f798ddcc61a933-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md

## 局部平坦的(locally flat)

- UUID: `4fdcca86-5235-4133-887e-17b54bf1da24` · okf_url: https://limbo137.top/okf/concepts/4fdcca86-5235-4133-887e-17b54bf1da24.md
- labels: en = locally flat；zh-CN = 局部平坦的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.7 · `zh/chapter-18/07-cartan-formalism#first-b4d5c28c90a66b775f5a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.7 · `en/chapter-18/07-cartan-formalism#first-b4d5c28c90a66b775f5a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/07-cartan-formalism.md

## 拉回(pullback)

- UUID: `4ff6041e-629b-4166-89b6-aba165d8c15f` · okf_url: https://limbo137.top/okf/concepts/4ff6041e-629b-4166-89b6-aba165d8c15f.md
- labels: en = pullback；zh-CN = 拉回
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.4 · `en/chapter-15/04-tangent-map-and-submanifolds#first-a46b28249c38ae049c51-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-a46b28249c38ae049c51-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/04-tangent-map-and-submanifolds.md

## 幂零算符(nilpotent operator)

- UUID: `503e7b48-5ef7-4bf8-b4ab-31d04aaa1559` · okf_url: https://limbo137.top/okf/concepts/503e7b48-5ef7-4bf8-b4ab-31d04aaa1559.md
- labels: en = nilpotent operator；zh-CN = 幂零算符
- aliases: nilpotent；幂零的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.2 · `zh/chapter-04/02-jordan-canonical-form#concept-4-nilpotent-operator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/02-jordan-canonical-form.md

## 量子计算(Quantum computing)

- UUID: `5045f4bb-0f7d-4132-988c-c49d734a1587` · okf_url: https://limbo137.top/okf/concepts/5045f4bb-0f7d-4132-988c-c49d734a1587.md
- labels: en = Quantum computing；zh-CN = 量子计算
- aliases: quantum computing
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-quantum-computing` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-777ef88a4a804344b8b5-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 微分外代数(differential exterior algebra)

- UUID: `5064fac9-6eab-41ae-8c95-5afae9141140` · okf_url: https://limbo137.top/okf/concepts/5064fac9-6eab-41ae-8c95-5afae9141140.md
- labels: en = differential exterior algebra；zh-CN = 微分外代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.1 · `en/chapter-16/01-differential-forms-and-exterior-derivative#first-d5afcc7bdb65e60eec5a-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.1 · `zh/chapter-16/01-differential-forms-and-exterior-derivative#first-d5afcc7bdb65e60eec5a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/01-differential-forms-and-exterior-derivative.md

## 线性联络(linear connection)

- UUID: `50a5c655-e7e1-418e-9486-178a70f24069` · okf_url: https://limbo137.top/okf/concepts/50a5c655-e7e1-418e-9486-178a70f24069.md
- labels: en = linear connection；zh-CN = 线性联络
- aliases: linear
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-38e57951e7b90c74fd78-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-d58ea303f6f67900c798-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 爱因斯坦–德西特宇宙(Einstein–de Sitter universe)

- UUID: `50efd903-fe4f-4a4d-9c73-4e7cd5e825eb` · okf_url: https://limbo137.top/okf/concepts/50efd903-fe4f-4a4d-9c73-4e7cd5e825eb.md
- labels: en = Einstein–de Sitter universe；zh-CN = 爱因斯坦–德西特宇宙
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-8065053991dc0373c7e4-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-8065053991dc0373c7e4-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 引力波(gravitational wave)

- UUID: `5118e748-bc6f-482a-ad9a-7a158a9252a9` · okf_url: https://limbo137.top/okf/concepts/5118e748-bc6f-482a-ad9a-7a158a9252a9.md
- labels: en = gravitational wave；zh-CN = 引力波
- aliases: gravitational waves
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-02b75935ddb66fb28ef9-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-f02f8026ddd0613c5fb4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 左矢(bra)

- UUID: `511e56a9-12c7-411e-989a-1061fc3c2882` · okf_url: https://limbo137.top/okf/concepts/511e56a9-12c7-411e-989a-1061fc3c2882.md
- labels: en = bra；zh-CN = 左矢
- aliases: bra vector
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-3935bac23857d81829af-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-3935bac23857d81829af-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-290e5917b695e1e13d33-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-290e5917b695e1e13d33-en` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §14.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `In this chapter we will adopt Dirac’s bra-ket notation which, though slightly quirky, has largely become the convention of choice among physicists. Vectors $\psi \in { \mathcal { H } }$ are written as kets $|\psi\rangle$ and one makes the identification $| \lambda \psi \rangle = \lambda | \psi \rangle$ . By the Riesz representation theorem 13.10, to each ...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `An element of the dual vector space $V ^ { * }$ is written as a “bra vector” $$ \langle \alpha | $$ with the labeling in terms of $\alpha$ determined by the isomorphism 4.3, i.e., $$ \langle \alpha | = l _ {| \alpha \rangle} $$ Evaluating $\langle \alpha | \in V ^ { * }$ on $| \beta \rangle \in V$ gives an element of $\mathbf{C}$, written $$ \langle \alph...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 里奇张量(Ricci tensor)

- UUID: `51590a3c-c536-46cc-82a2-1a213099de88` · okf_url: https://limbo137.top/okf/concepts/51590a3c-c536-46cc-82a2-1a213099de88.md
- labels: en = Ricci tensor；zh-CN = 里奇张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.6 · `zh/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-6a5d3659e0f77b31409b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.6 · `en/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-6a5d3659e0f77b31409b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/06-the-riemann-tensor-and-its-symmetries.md

## 静止能量(rest-energy)

- UUID: `5164558d-cd81-470d-a435-0da418a7d137` · okf_url: https://limbo137.top/okf/concepts/5164558d-cd81-470d-a435-0da418a7d137.md
- labels: en = rest-energy；zh-CN = 静止能量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-79a6fc5d3d120961afa6-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-79a6fc5d3d120961afa6-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 非相对论场论传播子(Nonrelativistic propagator)

- UUID: `516fb662-512d-42cf-b030-756189a7065e` · okf_url: https://limbo137.top/okf/concepts/516fb662-512d-42cf-b030-756189a7065e.md
- labels: en = Nonrelativistic propagator；zh-CN = 非相对论场论传播子
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §37.3 · `books/woit-qtgr/2025-10-20/en/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-104e66a9a3c499a2299e-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §37.3 · `books/woit-qtgr/2025-10-20/zh/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-a6f7889de0a9ef156552-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §37.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `当 $t_2>t_1$ 时，非相对论量子场论的[传播子(propagator)](/concepts/quantum-field-theory/nonrelativistic-propagator)是振幅 $$ U(x_2,t_2,x_1,t_1)=\langle0|\widehat\Psi(x_2,t_2)\widehat\Psi^\dagger(x_1,t_1)|0\rangle. $$`
  - woit-qtgr/2025-10-20 §37.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [propagator](/concepts/quantum-field-theory/nonrelativistic-propagator) is the amplitude, for $t_2>t_1$, $$ U(x_2,t_2,x_1,t_1)=\langle0|\widehat\Psi(x_2,t_2)\widehat\Psi^\dagger(x_1,t_1)|0\rangle. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory.md

## 连续谱(continuous spectrum)

- UUID: `517e06c6-f907-4d3f-92ca-d3b6f06a4dba` · okf_url: https://limbo137.top/okf/concepts/517e06c6-f907-4d3f-92ca-d3b6f06a4dba.md
- labels: en = continuous spectrum；zh-CN = 连续谱
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#concept-13-continuous-spectrum` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-426d32ca4365ebacef33-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 李群同态(Lie group homomorphism)

- UUID: `517f0a9d-f274-4f99-8731-9dc921f8c7ac` · okf_url: https://limbo137.top/okf/concepts/517f0a9d-f274-4f99-8731-9dc921f8c7ac.md
- labels: en = Lie group homomorphism；zh-CN = 李群同态
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.2 · `zh/chapter-19/02-the-exponential-map#concept-19-lie-group-homomorphism` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.2 · `en/chapter-19/02-the-exponential-map#first-8d142d52dd94eda1ef5f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/02-the-exponential-map.md

## 同调空间(homology space)

- UUID: `51b40e2c-5b4d-4919-bc65-d426422923ef` · okf_url: https://limbo137.top/okf/concepts/51b40e2c-5b4d-4919-bc65-d426422923ef.md
- labels: en = homology space；zh-CN = 同调空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-68cbc20b1eb8a1e723e8-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-68cbc20b1eb8a1e723e8-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 更细的拓扑(finer topology)

- UUID: `51ded9e4-c533-4884-81e6-0b62a8516850` · okf_url: https://limbo137.top/okf/concepts/51ded9e4-c533-4884-81e6-0b62a8516850.md
- labels: en = finer topology；zh-CN = 更细的拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-743c72677d36d2d5a234-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 重数(multiplicity)

- UUID: `520edeaf-4bda-4c66-87bb-d6ba7a65f4ef` · okf_url: https://limbo137.top/okf/concepts/520edeaf-4bda-4c66-87bb-d6ba7a65f4ef.md
- labels: en = multiplicity；zh-CN = 重数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-multiplicity` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 规范变换(gauge transformation)

- UUID: `522df93e-8d78-4aa8-8ca3-e9c88f110c80` · okf_url: https://limbo137.top/okf/concepts/522df93e-8d78-4aa8-8ca3-e9c88f110c80.md
- labels: en = gauge transformation；zh-CN = 规范变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-08d155e8b5f0fa0dd883-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-08d155e8b5f0fa0dd883-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 奇异单纯形(singular simplex)

- UUID: `523d476d-e383-44e0-8cab-50d9f2859647` · okf_url: https://limbo137.top/okf/concepts/523d476d-e383-44e0-8cab-50d9f2859647.md
- labels: en = singular simplex；zh-CN = 奇异单纯形
- aliases: singular
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-c5d4764a83c86eddf2d3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-8696ec0b50f8c5bcd420-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 正部与负部(positive and negative parts)

- UUID: `526b3bdf-ba92-439a-b9d5-23fa4279be45` · okf_url: https://limbo137.top/okf/concepts/526b3bdf-ba92-439a-b9d5-23fa4279be45.md
- labels: en = positive and negative parts；zh-CN = 正部与负部
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-bf0ebedaa2b9c211b427-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 极大图册(maximal atlas)

- UUID: `5297658f-b601-4564-8c96-d8298c08cad7` · okf_url: https://limbo137.top/okf/concepts/5297658f-b601-4564-8c96-d8298c08cad7.md
- labels: en = maximal atlas；zh-CN = 极大图册
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-9d5ce9406e1e672e08ae-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-maximal-atlas` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 福克空间(Fock space)

- UUID: `52a06444-312f-4b90-9174-02d19553722f` · okf_url: https://limbo137.top/okf/concepts/52a06444-312f-4b90-9174-02d19553722f.md
- labels: en = Fock space；zh-CN = 福克空间
- aliases: Fock空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.1 · `books/woit-qtgr/2025-10-20/en/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-457607b5c7969744a740-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 电荷算符(charge operator)

- UUID: `52a09bf2-6b64-4f64-8d17-cb50a2c294d5` · okf_url: https://limbo137.top/okf/concepts/52a09bf2-6b64-4f64-8d17-cb50a2c294d5.md
- labels: en = charge operator；zh-CN = 电荷算符
- aliases: Charge operator
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §2.3 · `books/woit-qtgr/2025-10-20/en/chapter-02/03-the-charge-operator#first-bbf4d016dc21069094bf-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §2 · `books/woit-qtgr/2025-10-20/zh/chapter-02/index#first-bbf4d016dc21069094bf-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §2.3 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition. The charge operator $Q$ for the $U ( 1 )$ representation $( \pi , \mathcal { H } )$ is the self-adjoint linear operator on $\mathcal{H}$ that acts by multiplication by $q _ { j }$ on the irreducible sub-representation $\mathcal { H } _ { q _ { j } }$ . Taking basis elements in $\mathcal { H } _ { q _ { j } }$ it acts on $\mathcal{H}$ as the ma...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/03-the-charge-operator.md

## 开覆盖(open covering)

- UUID: `52e98d74-140a-45ae-9670-fcdb38942d97` · okf_url: https://limbo137.top/okf/concepts/52e98d74-140a-45ae-9670-fcdb38942d97.md
- labels: en = open covering；zh-CN = 开覆盖
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-96a4c3059f1e03dc322d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 准希尔伯特空间(pre-Hilbert space)

- UUID: `533f0fe5-106d-4d9c-8fbb-d18f91263336` · okf_url: https://limbo137.top/okf/concepts/533f0fe5-106d-4d9c-8fbb-d18f91263336.md
- labels: en = pre-Hilbert space；zh-CN = 准希尔伯特空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.1 · `zh/chapter-13/01-definitions-and-examples#concept-13-pre-hilbert-space` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.1 · `en/chapter-13/01-definitions-and-examples#first-7823004a613ae5dd9c35-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/01-definitions-and-examples.md

## 李群同构(Lie group isomorphism)

- UUID: `537f5597-841d-4de6-85fe-976eb10c7db7` · okf_url: https://limbo137.top/okf/concepts/537f5597-841d-4de6-85fe-976eb10c7db7.md
- labels: en = Lie group isomorphism；zh-CN = 李群同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.2 · `en/chapter-19/02-the-exponential-map#first-ff53abc2bdcb7a599733-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.2 · `zh/chapter-19/02-the-exponential-map#concept-19-lie-group-isomorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/02-the-exponential-map.md

## 狄拉克γ矩阵(Dirac gamma matrices)

- UUID: `539bde69-fb14-4a93-ad65-5a676746e932` · okf_url: https://limbo137.top/okf/concepts/539bde69-fb14-4a93-ad65-5a676746e932.md
- labels: en = Dirac gamma matrices；zh-CN = 狄拉克γ矩阵
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §29.2 · `books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry#first-9109fa0b3825bf2af7c6-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md

## Bargmann 变换(Bargmann transform)

- UUID: `5412c8ef-53d3-42d8-b6f9-d39220aff942` · okf_url: https://limbo137.top/okf/concepts/5412c8ef-53d3-42d8-b6f9-d39220aff942.md
- labels: en = Bargmann transform；zh-CN = Bargmann 变换
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §23.5 · `books/woit-qtgr/2025-10-20/zh/chapter-23/08-the-bargmann-transform#first-2fd21d4a1851d9db8701-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §23.5 · `books/woit-qtgr/2025-10-20/en/chapter-23/08-the-bargmann-transform#first-3f6675e1116e564402df-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §23.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `There is a unitary map called the Bargmann transform $$ \mathcal B:\mathcal H_S\to\mathcal F, $$ which intertwines the Schrödinger and Bargmann–Fock representations. That is, for $X\in\mathfrak h_3$, $$ \Gamma_S^{\prime}(X)=\mathcal B^{-1}\Gamma_{BF}^{\prime}(X)\mathcal B. $$`
  - woit-qtgr/2025-10-20 §23.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `存在一个称为 Bargmann 变换的酉映射 $$ \mathcal B:\mathcal H_S\to\mathcal F, $$ 它交织薛定谔表示与 Bargmann–Fock 表示。也就是说，对 $X\in\mathfrak h_3$，有 $$ \Gamma_S^{\prime}(X)=\mathcal B^{-1}\Gamma_{BF}^{\prime}(X)\mathcal B。 $$$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-23/08-the-bargmann-transform.md

## 收缩核(retract)

- UUID: `545035c3-cbeb-55ac-be37-b055f1b481c2` · okf_url: https://limbo137.top/okf/concepts/545035c3-cbeb-55ac-be37-b055f1b481c2.md
- labels: en = retract；zh-CN = 收缩核
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=148, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.7：收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.7. Let $R$ ( $\neq \emptyset$ ) be a subspace of $X$ . If there exists a continuous map $f: X \to R$ such that $f|_{R} = \mathrm{id}_{R}$ , $R$ is called a retract of $X$ and $f$ a retraction.`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=148, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.7：收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Note that the whole of $X$ is mapped onto $R$ keeping points in $R$ fixed. Figure 4.7 is an example of a retract and retraction.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 求和约定(summation convention)

- UUID: `546a74e5-bfed-4ce4-9b19-041d5fdfb388` · okf_url: https://limbo137.top/okf/concepts/546a74e5-bfed-4ce4-9b19-041d5fdfb388.md
- labels: en = summation convention；zh-CN = 求和约定
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-summation-convention` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 群自同构作用(Action by group automorphisms)

- UUID: `54afd32d-e7ca-47b4-b23f-1277bda066e2` · okf_url: https://limbo137.top/okf/concepts/54afd32d-e7ca-47b4-b23f-1277bda066e2.md
- labels: en = Action by group automorphisms；zh-CN = 群自同构作用
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §16.2 · `books/woit-qtgr/2025-10-20/zh/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-2bc452a5b7283a730bd6-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §16.2 · `books/woit-qtgr/2025-10-20/en/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-9a54646d1c2bb19c1110-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `An action of a group $G$ on a group $H$, written $$ h\in H\longmapsto\Phi_g(h)\in H, $$ is an action by automorphisms when $$ \Phi_g(h_1)\Phi_g(h_2)=\Phi_g(h_1h_2) $$ for every $g\in G$ and $h_1,h_2\in H$. In this case, each $\Phi_g$ is an automorphism of $H$. The action law also gives $\Phi_{g_1g_2}=\Phi_{g_1}\circ\Phi_{g_2}$.`
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `群 $G$ 对群 $H$ 的作用记作 $$ h\in H\longmapsto\Phi_g(h)\in H $$ 当它满足 $$ \Phi_g(h_1)\Phi_g(h_2)=\Phi_g(h_1h_2) $$ 对所有 $g\in G$ 及 $h_1,h_2\in H$ 都成立时，称该作用为自同构作用。此时，每个 $\Phi_g$ 都是 $H$ 的自同构；由群作用定律还有 $\Phi_{g_1g_2}=\Phi_{g_1}\circ\Phi_{g_2}$。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group.md

## 有界厄米算符的谱定理(Spectral theorem for bounded Hermitian operators)

- UUID: `54d35d78-da75-48b1-bccb-144eba0804a0` · okf_url: https://limbo137.top/okf/concepts/54d35d78-da75-48b1-bccb-144eba0804a0.md
- labels: en = Spectral theorem for bounded Hermitian operators；zh-CN = 有界厄米算符的谱定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#first-05666e28e8c41b62774c-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-05666e28e8c41b62774c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 向量空间(vector space)

- UUID: `54e19cc9-1e38-4128-acaa-80499dfd908c` · okf_url: https://limbo137.top/okf/concepts/54e19cc9-1e38-4128-acaa-80499dfd908c.md
- labels: en = vector space；zh-CN = 向量空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-vector-space` · translation)；woit-qtgr/2025-10-20 (en · §4.1 · `books/woit-qtgr/2025-10-20/en/chapter-04/01-vector-spaces-and-linear-maps#concept-4-vector-space` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/index#first-7fdcbbb0d58dcf2329fb-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/01-vector-spaces-and-linear-maps.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md

## 时空(spacetime)

- UUID: `550d1c68-68fa-4441-8e34-69825f02d844` · okf_url: https://limbo137.top/okf/concepts/550d1c68-68fa-4441-8e34-69825f02d844.md
- labels: en = spacetime；zh-CN = 时空
- aliases: space-time
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-28deb1980edfcd305655-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-91a79563d627d98979c5-en` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §18.8 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `The basic proposition of general relativity (Einstein, 1916) is the following: the world is a four-dimensional Minkowskian manifold (pseudo-Riemannian of index +2) called spacetime. Its points are called events. The world-line, or space-time history of a materia particle, is a parametrized curve $\gamma : \mathbb { R } M$ whose tangent $\dot { \gamma }$ i...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 满态射(epimorphism)

- UUID: `55363d96-9a7f-484d-be57-a1e93ae4df55` · okf_url: https://limbo137.top/okf/concepts/55363d96-9a7f-484d-be57-a1e93ae4df55.md
- labels: en = epimorphism；zh-CN = 满态射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-db50bfe42e319116dbd4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 弗罗贝尼乌斯定理(Frobenius theorem)

- UUID: `554bac8e-2cd1-477e-b88f-0bfc5f46da87` · okf_url: https://limbo137.top/okf/concepts/554bac8e-2cd1-477e-b88f-0bfc5f46da87.md
- labels: en = Frobenius theorem；zh-CN = 弗罗贝尼乌斯定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.6 · `zh/chapter-15/06-distributions-and-frobenius-theorem#first-e3a51828446c9626f9a5-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.6 · `en/chapter-15/06-distributions-and-frobenius-theorem#first-e3a51828446c9626f9a5-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/06-distributions-and-frobenius-theorem.md

## 狄拉克算符(Dirac operator)

- UUID: `555c6cfc-d07b-463a-9c7d-5fc1fecfd3e4` · okf_url: https://limbo137.top/okf/concepts/555c6cfc-d07b-463a-9c7d-5fc1fecfd3e4.md
- labels: en = Dirac operator；zh-CN = 狄拉克算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §34.1 · `books/woit-qtgr/2025-10-20/en/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3#first-a27e3488ba73108cbb4c-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §34.1 · `books/woit-qtgr/2025-10-20/zh/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3#first-a27e3488ba73108cbb4c-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §34.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[Dirac operator](/concepts/quantum-mechanics/dirac-operator) is the operator $$ \not\partial=\sum_{j=1}^d\gamma_j\frac\partial{\partial q_j}. $$`
  - woit-qtgr/2025-10-20 §34.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[狄拉克算符(Dirac operator)](/concepts/quantum-mechanics/dirac-operator)定义为 $$ \not\partial=\sum_{j=1}^d\gamma_j\frac\partial{\partial q_j}. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/04-the-dirac-operator.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/01-the-pauli-schr-odinger-equation-and-free-spin-1-2-particles-in-d-3.md

## 内积空间(inner product space)

- UUID: `556c76cb-7433-4f01-86f1-cbbf0bc6fe95` · okf_url: https://limbo137.top/okf/concepts/556c76cb-7433-4f01-86f1-cbbf0bc6fe95.md
- labels: en = inner product space；zh-CN = 内积空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-inner-product-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 线性相关(linearly dependent)

- UUID: `55c8e28c-473e-4094-89d6-2a1ea16879e3` · okf_url: https://limbo137.top/okf/concepts/55c8e28c-473e-4094-89d6-2a1ea16879e3.md
- labels: en = linearly dependent；zh-CN = 线性相关
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-linearly-dependent` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 薛定谔表示，李代数版本(Schrödinger representation, Lie algebra version)

- UUID: `561e07f8-1c49-4700-b0af-4df4cf5ee94c` · okf_url: https://limbo137.top/okf/concepts/561e07f8-1c49-4700-b0af-4df4cf5ee94c.md
- labels: en = Schrödinger representation, Lie algebra version；zh-CN = 薛定谔表示，李代数版本
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.3 · `books/woit-qtgr/2025-10-20/zh/chapter-13/03-the-schr-odinger-representation#first-d58b4ee264851626a4f2-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.3 · `books/woit-qtgr/2025-10-20/en/chapter-13/03-the-schr-odinger-representation#first-57afec92c23b30ac7e4d-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡李代数 $\mathfrak h_3$ 的薛定谔表示是 $(\Gamma'_S,L^2(\mathbf R))$，由下列关系定义： $$ \begin{aligned} \Gamma'_S(X)\psi(q)&=-iQ\psi(q)=-iq\psi(q), & \Gamma'_S(Y)\psi(q)&=-iP\psi(q)=-\frac{d}{dq}\psi(q),\\[0.45em] \Gamma'_S(Z)\psi(q)&=-i\psi(q). \end{aligned} $$`
  - woit-qtgr/2025-10-20 §13.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Schrödinger representation of the Heisenberg Lie algebra $\mathfrak h_3$ is the representation $(\Gamma'_S,L^2(\mathbf R))$ defined by $$ \begin{aligned} \Gamma'_S(X)\psi(q)&=-iQ\psi(q)=-iq\psi(q), & \Gamma'_S(Y)\psi(q)&=-iP\psi(q)=-\frac{d}{dq}\psi(q),\\[0.45em] \Gamma'_S(Z)\psi(q)&=-i\psi(q). \end{aligned} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/03-the-schr-odinger-representation.md

## 有限维酉算符(finite-dimensional unitary operator)

- UUID: `56523306-74f3-4753-8214-d0bfa68bc5d5` · okf_url: https://limbo137.top/okf/concepts/56523306-74f3-4753-8214-d0bfa68bc5d5.md
- labels: en = finite-dimensional unitary operator；zh-CN = 有限维酉算符
- aliases: unitary operator；unitary operators；幺正算符；酉算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-unitary-operator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 凯莱–哈密顿定理(Cayley–Hamilton theorem)

- UUID: `56b0284d-ea52-4730-baef-596780d61dcd` · okf_url: https://limbo137.top/okf/concepts/56b0284d-ea52-4730-baef-596780d61dcd.md
- labels: en = Cayley–Hamilton theorem；zh-CN = 凯莱–哈密顿定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#first-f18e90dc0ef0caed99a3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §4.1 · `en/chapter-04/01-eigenspaces-and-characteristic-equations#first-f18e90dc0ef0caed99a3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 四维张量(4-tensor)

- UUID: `56b3a74d-7720-4bf7-b3bd-93993f94ebf3` · okf_url: https://limbo137.top/okf/concepts/56b3a74d-7720-4bf7-b3bd-93993f94ebf3.md
- labels: en = 4-tensor；zh-CN = 四维张量
- aliases: 4-tensors
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-4452bee419902606361b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-c60c66d6dd2d054d0ad2-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 基本链(fundamental chain)

- UUID: `56bd26c9-7d45-4391-ba3e-fb198c908665` · okf_url: https://limbo137.top/okf/concepts/56bd26c9-7d45-4391-ba3e-fb198c908665.md
- labels: en = fundamental chain；zh-CN = 基本链
- aliases: fundamental
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-f6310c1ab2d9fe02018b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-49bab764fcc603fedeab-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 稳定结点(stable node)

- UUID: `57790a83-4023-48b5-9e26-ff4c19323c91` · okf_url: https://limbo137.top/okf/concepts/57790a83-4023-48b5-9e26-ff4c19323c91.md
- labels: en = stable node；zh-CN = 稳定结点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-stable-node` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 图册(atlas)

- UUID: `57951517-a204-4ee3-8fb0-861a36731660` · okf_url: https://limbo137.top/okf/concepts/57951517-a204-4ee3-8fb0-861a36731660.md
- labels: en = atlas；zh-CN = 图册
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-0cad84b788fdf0e94996-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-atlas` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 幂等算符(idempotent operator)

- UUID: `57d0ee24-a7ce-4e13-a210-4cc1e3b00b56` · okf_url: https://limbo137.top/okf/concepts/57d0ee24-a7ce-4e13-a210-4cc1e3b00b56.md
- labels: en = idempotent operator；zh-CN = 幂等算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-idempotent-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-4b64d8ade3f383ecc1b9-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 期望值(expectation value)

- UUID: `57eeacec-5920-4133-83e9-8b4cd48bf759` · okf_url: https://limbo137.top/okf/concepts/57eeacec-5920-4133-83e9-8b4cd48bf759.md
- labels: en = expectation value；zh-CN = 期望值
- aliases: expectation values；expected value
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-expectation-value` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-expectation-value` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 拓扑不变量(topological invariant)

- UUID: `5838084a-ffaf-428a-80a9-26e5da5f38c4` · okf_url: https://limbo137.top/okf/concepts/5838084a-ffaf-428a-80a9-26e5da5f38c4.md
- labels: en = topological invariant；zh-CN = 拓扑不变量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-5b61d32a5b05b27d3d37-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 楔积(wedge product)

- UUID: `586481c1-ffe5-4581-a75e-5a8521c2e3fb` · okf_url: https://limbo137.top/okf/concepts/586481c1-ffe5-4581-a75e-5a8521c2e3fb.md
- labels: en = wedge product；zh-CN = 楔积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.3 · `zh/chapter-08/03-exterior-product#first-e8b41d80cf673cd165ba-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.3 · `en/chapter-08/03-exterior-product#first-e8b41d80cf673cd165ba-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md

## 互补可观测量(complementary observables)

- UUID: `58ad6734-94de-4363-a2b2-559a9a10aebb` · okf_url: https://limbo137.top/okf/concepts/58ad6734-94de-4363-a2b2-559a9a10aebb.md
- labels: en = complementary observables；zh-CN = 互补可观测量
- aliases: complementary
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-5d7e3e3238db3e40f868-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 整体相位因子(overall phase factor)

- UUID: `58e62b67-5cb1-4463-a967-c506506fb472` · okf_url: https://limbo137.top/okf/concepts/58e62b67-5cb1-4463-a967-c506506fb472.md
- labels: en = overall phase factor；zh-CN = 整体相位因子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-c363a4bca209f00248fe-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §3.3 · `books/woit-qtgr/2025-10-20/zh/chapter-03/03-dynamics-of-a-two-state-system#first-c363a4bca209f00248fe-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md

## 广义坐标(generalized coordinate)

- UUID: `58e807c5-1ecd-41cb-b5cb-d975822e7d10` · okf_url: https://limbo137.top/okf/concepts/58e807c5-1ecd-41cb-b5cb-d975822e7d10.md
- labels: en = generalized coordinate；zh-CN = 广义坐标
- aliases: generalized coordinates
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-d75b07f69fb525ba0ca3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-ec2c3e7e75ab34f57f10-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## Weyl 特征标公式(Weyl character formula)

- UUID: `58f1c3d5-9aa5-45b5-b01b-8f998de65e40` · okf_url: https://limbo137.top/okf/concepts/58f1c3d5-9aa5-45b5-b01b-8f998de65e40.md
- labels: en = Weyl character formula；zh-CN = Weyl 特征标公式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §9.4 · `books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations#first-3031bbfcb3e29807b3b5-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §9.4 · `books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations#first-2aeebe1b55db59905bc6-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/04-tensor-products-of-representations.md

## 无穷次可微(infinitely differentiable)

- UUID: `593cdd72-5635-4c79-a9c7-b99346d7c2e1` · okf_url: https://limbo137.top/okf/concepts/593cdd72-5635-4c79-a9c7-b99346d7c2e1.md
- labels: en = infinitely differentiable；zh-CN = 无穷次可微
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-9555df478b16ceb3fe98-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 线性映射的像(image of a linear map)

- UUID: `593dc0f6-cda0-4a27-a905-4d9a84968af1` · okf_url: https://limbo137.top/okf/concepts/593dc0f6-cda0-4a27-a905-4d9a84968af1.md
- labels: en = image of a linear map；zh-CN = 线性映射的像
- aliases: image；image of a linear mapping；image of a linear transformation；像
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-image` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 若尔当标准形(Jordan canonical form)

- UUID: `59e47f44-f852-42f8-ab47-64d212447691` · okf_url: https://limbo137.top/okf/concepts/59e47f44-f852-42f8-ab47-64d212447691.md
- labels: en = Jordan canonical form；zh-CN = 若尔当标准形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.2 · `zh/chapter-04/02-jordan-canonical-form#concept-4-jordan-canonical-form` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/02-jordan-canonical-form.md

## 指数映射(exponential map)

- UUID: `5a2f0d15-d50a-45f3-a76f-87e2968a6aed` · okf_url: https://limbo137.top/okf/concepts/5a2f0d15-d50a-45f3-a76f-87e2968a6aed.md
- labels: en = exponential map；zh-CN = 指数映射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.2 · `en/chapter-19/02-the-exponential-map#first-dc4a9df00eb1ff4f58fd-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.2 · `zh/chapter-19/02-the-exponential-map#concept-19-exponential-map` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.3 (pdf_page=233, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.13：指数映射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $\phi_V$ is a one-parameter subgroup of $G$ generated by the left-invariant vector field $X_V|_g = L_{g^*}V$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.3 (pdf_page=232, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.13：指数映射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.13. Let $G$ be a Lie group and $V \in T_{e}G$ . The exponential map $\exp : T_{e}G \to G$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.3 (pdf_page=232, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.13：指数映射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \exp V \equiv \phi_ {V} (1) \tag {5.127} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/02-the-exponential-map.md

## 伪正交群(pseudo-orthogonal group)

- UUID: `5a7c372b-6b23-4115-8129-84b5fad3a591` · okf_url: https://limbo137.top/okf/concepts/5a7c372b-6b23-4115-8129-84b5fad3a591.md
- labels: en = pseudo-orthogonal group；zh-CN = 伪正交群
- aliases: pseudo-orthogonal groups
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-pseudo-orthogonal-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.3 · `books/woit-qtgr/2025-10-20/zh/chapter-05/03-a-summary#first-fbac5a9241c08a68d2e9-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.3 · `books/woit-qtgr/2025-10-20/en/chapter-05/03-a-summary#first-0574d9cae79869b5978c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/03-a-summary.md

## 仿射空间(affine space)

- UUID: `5a945993-0bbc-4b93-a027-8a9c65eadde0` · okf_url: https://limbo137.top/okf/concepts/5a945993-0bbc-4b93-a027-8a9c65eadde0.md
- labels: en = affine space；zh-CN = 仿射空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-618cb40fe4a5b27a39a1-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-618cb40fe4a5b27a39a1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 左不变的(left-invariant)

- UUID: `5b63b7cd-d37a-4f98-9472-ac67fe1c2dae` · okf_url: https://limbo137.top/okf/concepts/5b63b7cd-d37a-4f98-9472-ac67fe1c2dae.md
- labels: en = left-invariant；zh-CN = 左不变的
- aliases: 左不变
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-861f00be2d1f56c10a36-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 李代数表示(Lie algebra representation)

- UUID: `5bad1df2-281a-4c11-8362-128d54fb523a` · okf_url: https://limbo137.top/okf/concepts/5bad1df2-281a-4c11-8362-128d54fb523a.md
- labels: en = Lie algebra representation；zh-CN = 李代数表示
- aliases: Lie algebra representations
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-7fce2cf1162463ce211a-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-7fce2cf1162463ce211a-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（李代数表示）。李代数 $\mathfrak{g}$ 在 $n$ 维复向量空间 $V$ 上的一个（复）李代数表示(Lie algebra representation) $(\phi,V)$ 由一个实线性映射给出 $$ \phi:X\in\mathfrak{g}\mapsto\phi(X)\in\mathfrak{gl}(n,\mathbf{C})=M(n,\mathbf{C}) $$ 满足 $$ \phi([X,Y])=[\phi(X),\phi(Y)] $$ 如果这样的表示的像包含在 $\mathfrak{u}(n)$ 中，并满足 $$ \phi(X)^\dagger=-\phi(X) $$`
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Lie algebra representation). A (complex) Lie algebra representation $(\phi,V)$ of a Lie algebra $\mathfrak{g}$ on an $n$ dimensional complex vector space $V$ is given by a real-linear map $$ \phi:X\in\mathfrak{g}\mapsto\phi(X)\in\mathfrak{gl}(n,\mathbf{C})=M(n,\mathbf{C}) $$ satisfying $$ \phi([X,Y])=[\phi(X),\phi(Y)] $$ Such a representation ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md

## 李代数导子作用(Action by Lie algebra derivations)

- UUID: `5bd96f47-f41c-4ba5-86a0-0d8cb192c94b` · okf_url: https://limbo137.top/okf/concepts/5bd96f47-f41c-4ba5-86a0-0d8cb192c94b.md
- labels: en = Action by Lie algebra derivations；zh-CN = 李代数导子作用
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §16.2 · `books/woit-qtgr/2025-10-20/en/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-e7b6952e155aac6e5ae1-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §16.2 · `books/woit-qtgr/2025-10-20/zh/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group#first-a479d0dc9f062583980d-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `An action of a Lie algebra $\mathfrak g$ on a Lie algebra $\mathfrak h$ by linear maps $$ X\in\mathfrak h\longmapsto Z\cdot X\in\mathfrak h $$ is an action by derivations when $$ [Z\cdot X,Y]+[X,Z\cdot Y]=Z\cdot[X,Y].\tag{16.17} $$ This condition is required for every $Z\in\mathfrak g$ and $X,Y\in\mathfrak h$. Each $Z$ acts on $\mathfrak h$ as a derivation.`
  - woit-qtgr/2025-10-20 §16.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `李代数 $\mathfrak g$ 通过线性映射作用于李代数 $\mathfrak h$，记作 $$ X\in\mathfrak h\longmapsto Z\cdot X\in\mathfrak h $$ 若满足 $$ [Z\cdot X,Y]+[X,Z\cdot Y]=Z\cdot[X,Y],\tag{16.17} $$ 对所有 $Z\in\mathfrak g$ 及 $X,Y\in\mathfrak h$ 都成立，则称 $\mathfrak g$ 通过导子作用于 $\mathfrak h$。每个 $Z$ 在 $\mathfrak h$ 上的作用都是一个导子。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/04-the-symplectic-group-and-automorphisms-of-the-heisenberg-group.md

## 单位立方体(unit cube)

- UUID: `5bf1070c-8b77-468e-8316-4cfb10732c62` · okf_url: https://limbo137.top/okf/concepts/5bf1070c-8b77-468e-8316-4cfb10732c62.md
- labels: en = unit cube；zh-CN = 单位立方体
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.3 · `zh/chapter-17/03-stokes-theorem#first-b962923d98d2c1269b79-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 李超括号(Lie superbracket)

- UUID: `5c0010c3-e567-45b1-926e-91dd0363f528` · okf_url: https://limbo137.top/okf/concepts/5c0010c3-e567-45b1-926e-91dd0363f528.md
- labels: en = Lie superbracket；zh-CN = 李超括号
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §30.2 · `books/woit-qtgr/2025-10-20/zh/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-7c533cddbd9d2f52b76c-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §30.2 · `books/woit-qtgr/2025-10-20/en/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket#first-7c533cddbd9d2f52b76c-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/02-pseudo-classical-mechanics-and-the-fermionic-poisson-bracket.md

## 密度算子(density operator)

- UUID: `5c04d308-9b9c-437f-b36f-ac6d05359b7c` · okf_url: https://limbo137.top/okf/concepts/5c04d308-9b9c-437f-b36f-ac6d05359b7c.md
- labels: en = density operator；zh-CN = 密度算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.4 · `zh/chapter-14/04-quantum-statistical-mechanics#first-a7bd670b7c98b52f9130-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.4 · `en/chapter-14/04-quantum-statistical-mechanics#first-a7bd670b7c98b52f9130-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/04-quantum-statistical-mechanics.md

## 正规序乘积(Normal ordered product)

- UUID: `5c130474-2e54-4ec8-9a3e-2355428ac1d1` · okf_url: https://limbo137.top/okf/concepts/5c130474-2e54-4ec8-9a3e-2355428ac1d1.md
- labels: en = Normal ordered product；zh-CN = 正规序乘积
- aliases: normal ordered product
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §24.3 · `books/woit-qtgr/2025-10-20/en/chapter-24/03-implications-of-the-choice-of-z-z#first-3fce279692026b05358b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §24.3 · `books/woit-qtgr/2025-10-20/zh/chapter-24/03-implications-of-the-choice-of-z-z#first-3fce279692026b05358b-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-24/03-implications-of-the-choice-of-z-z.md

## 功形式(work form)

- UUID: `5c46b2ed-4fca-4687-ba8b-563256d5f2f4` · okf_url: https://limbo137.top/okf/concepts/5c46b2ed-4fca-4687-ba8b-563256d5f2f4.md
- labels: en = work form；zh-CN = 功形式
- aliases: workform
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-a89c8f244ce4c9186d29-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 可微结构(differentiable structure)

- UUID: `5c84e88a-89e3-4262-b2c7-9fe2ef55d012` · okf_url: https://limbo137.top/okf/concepts/5c84e88a-89e3-4262-b2c7-9fe2ef55d012.md
- labels: en = differentiable structure；zh-CN = 可微结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-361378f15f49e6a6b5e4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-differentiable-structure` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 规范协变导数(gauge covariant derivative)

- UUID: `5c8953d5-b002-431f-8ba7-4b8ce7dd42ab` · okf_url: https://limbo137.top/okf/concepts/5c8953d5-b002-431f-8ba7-4b8ce7dd42ab.md
- labels: en = gauge covariant derivative；zh-CN = 规范协变导数
- aliases: Covariant derivative；协变导数
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.1 · `books/woit-qtgr/2025-10-20/zh/chapter-45/01-u-1-gauge-symmetry#first-18680e9ff64aa9830f24-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.1 · `books/woit-qtgr/2025-10-20/en/chapter-45/01-u-1-gauge-symmetry#first-26422999ff3e64c97f86-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `给定一个联络 $A$，沿 $\mu$ 方向的相伴协变导数是算符 $$ D _ {\mu} ^ {A} = \frac {\partial}{\partial x _ {\mu}} - i e A _ {\mu} (x) $$ 有了这个定义，规范变换的效应是`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/01-u-1-gauge-symmetry.md

## 单射(injection)

- UUID: `5c98a7ca-5c42-4f2b-9236-86c453eddf68` · okf_url: https://limbo137.top/okf/concepts/5c98a7ca-5c42-4f2b-9236-86c453eddf68.md
- labels: en = injection；zh-CN = 单射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-5ec0092aba5eb3211c3a-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：单射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.1. If a map satisfies a certain condition it bears a special name.`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：单射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) A map $f: X \to Y$ is called injective (or one to one) if $x \neq x'$ implies $f(x) \neq f(x')$ . (b) A map $f: X \to Y$ is called surjective (or onto) if for each $y \in Y$ there exists at least one element $x \in X$ such that $f(x) = y$ . (c) A map $f: X \to Y$ is called bijective if it is both injective and surjective.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 线性组合(linear combination)

- UUID: `5c9ca977-04c0-42d0-8b1e-5d85c3fd905c` · okf_url: https://limbo137.top/okf/concepts/5c9ca977-04c0-42d0-8b1e-5d85c3fd905c.md
- labels: en = linear combination；zh-CN = 线性组合
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-c777ebc2168b0a5726f5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 希尔伯特拉格朗日量(Hilbert Lagrangian)

- UUID: `5cac9313-3d4f-4510-8bac-b257c148accb` · okf_url: https://limbo137.top/okf/concepts/5cac9313-3d4f-4510-8bac-b257c148accb.md
- labels: en = Hilbert Lagrangian；zh-CN = 希尔伯特拉格朗日量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.10 · `zh/chapter-18/10-variation-principles-in-space-time#first-585ec3f84b40b0c0f6c5-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.10 · `en/chapter-18/10-variation-principles-in-space-time#first-585ec3f84b40b0c0f6c5-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/10-variation-principles-in-space-time.md

## 实数加法群(additive group of reals)

- UUID: `5cb523fb-06b4-4068-b362-7aeed13d453e` · okf_url: https://limbo137.top/okf/concepts/5cb523fb-06b4-4068-b362-7aeed13d453e.md
- labels: en = additive group of reals；zh-CN = 实数加法群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-additive-group-of-reals` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 封闭性(closure property)

- UUID: `5cbf52ee-08a7-4a19-bb31-d97ab8d074f7` · okf_url: https://limbo137.top/okf/concepts/5cbf52ee-08a7-4a19-bb31-d97ab8d074f7.md
- labels: en = closure property；zh-CN = 封闭性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-closure-property` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 绝对熵(absolute entropy)

- UUID: `5d16fb8c-58f4-4c1b-9d25-40666fc144bd` · okf_url: https://limbo137.top/okf/concepts/5d16fb8c-58f4-4c1b-9d25-40666fc144bd.md
- labels: en = absolute entropy；zh-CN = 绝对熵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-7a41c7fe273382aa349c-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-7a41c7fe273382aa349c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 对偶表示(dual representation)

- UUID: `5d7e6d90-8fc4-4b2f-a97b-04bab6ff6e19` · okf_url: https://limbo137.top/okf/concepts/5d7e6d90-8fc4-4b2f-a97b-04bab6ff6e19.md
- labels: en = dual representation；zh-CN = 对偶表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-9fcf711b7793bf89e6bc-zh` · translation)；woit-qtgr/2025-10-20 (en · §16.1 · `books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group#first-9fcf711b7793bf89e6bc-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/06-orthogonal-and-unitary-transformations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/03-change-of-basis.md

## 表示的复化(complexification of a representation)

- UUID: `5de2ca26-d059-42fc-9ea3-2beff8196917` · okf_url: https://limbo137.top/okf/concepts/5de2ca26-d059-42fc-9ea3-2beff8196917.md
- labels: en = complexification of a representation；zh-CN = 表示的复化
- aliases: complexified
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#first-7d47e5ffbab5d81fba8b-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 反对易子(anticommutator)

- UUID: `5e0b82c6-1611-4f91-b552-11bab1c32916` · okf_url: https://limbo137.top/okf/concepts/5e0b82c6-1611-4f91-b552-11bab1c32916.md
- labels: en = anticommutator；zh-CN = 反对易子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §27.1 · `books/woit-qtgr/2025-10-20/en/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator#first-f5ecaff93e818e885d29-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-f5ecaff93e818e885d29-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 列秩(column rank)

- UUID: `5ec7eeb5-1b45-4838-b12f-549245d4018b` · okf_url: https://limbo137.top/okf/concepts/5ec7eeb5-1b45-4838-b12f-549245d4018b.md
- labels: en = column rank；zh-CN = 列秩
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-column-rank` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md

## 屈内特公式(Künneth formula)

- UUID: `5ed77be4-d851-51e9-b457-f87d7d1b9deb` · okf_url: https://limbo137.top/okf/concepts/5ed77be4-d851-51e9-b457-f87d7d1b9deb.md
- labels: en = Künneth formula；zh-CN = 屈内特公式
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §6.4.3 (pdf_page=258, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §6.4.3给出实de Rham上同调的乘积公式；保留系数及空间假设。 · 原文: `This is rewritten in terms of the Betti numbers as`
  - nakahara-geometry-topology-physics/second-edition-local §6.4.3 (pdf_page=258, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §6.4.3, named Kunneth formula (6.44) for real de Rham cohomology of the product M=M1×M2. This is the book statement of a named formula, not a new definition of cohomology or an assertion of first occurrence. · 原文: `$$ H ^ {r} (M) = \bigoplus_ {p + q = r} [ H ^ {p} (M _ {1}) \otimes H ^ {q} (M _ {2}) ]. \tag {6.44} $$`
  - nakahara-geometry-topology-physics/second-edition-local §6.4.3 (pdf_page=258, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §6.4.3, named Kunneth formula (6.44) for real de Rham cohomology of the product M=M1×M2. This is the book statement of a named formula, not a new definition of cohomology or an assertion of first occurrence. · 原文: `By comparing the LHS with the RHS, we find $\alpha^{p - 1} = \delta^{r - p - 1} = 0$ , hence $\omega_{i}^{p}\wedge \eta_{j}^{r - p} = 0$ in contradiction to our assumption. Thus, $\omega_{i}^{p}\wedge \eta_{j}^{r - p}$ is a nontrivial element of $H^{r}(M)$ . Conversely, any element of $H^{r}(M)$ can be decomposed into a sum of a product of the elements of...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 态空间(state space)

- UUID: `5f2054c6-8388-4d5b-b983-986606916b37` · okf_url: https://limbo137.top/okf/concepts/5f2054c6-8388-4d5b-b983-986606916b37.md
- labels: en = state space；zh-CN = 态空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-e057133f07fc1a01a76d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 群同态的核(kernel of a group homomorphism)

- UUID: `5f7e4cb8-69a7-4933-854b-6e7bbc93a0f1` · okf_url: https://limbo137.top/okf/concepts/5f7e4cb8-69a7-4933-854b-6e7bbc93a0f1.md
- labels: en = kernel of a group homomorphism；zh-CN = 群同态的核
- aliases: kernel；核
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-kernel` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 玻恩定则(Born rule)

- UUID: `5f84a75b-3492-448f-a42d-c54cf4c32da0` · okf_url: https://limbo137.top/okf/concepts/5f84a75b-3492-448f-a42d-c54cf4c32da0.md
- labels: en = Born rule；zh-CN = 玻恩定则
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#concept-1-born-rule` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 平方可积(square integrable)

- UUID: `5fa3bea4-1680-4d31-b2ad-fdff06b948d1` · okf_url: https://limbo137.top/okf/concepts/5fa3bea4-1680-4d31-b2ad-fdff06b948d1.md
- labels: en = square integrable；zh-CN = 平方可积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-square-integrable` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 群范畴(category of groups)

- UUID: `5fca732b-96e6-4605-8410-8351c5d1f96a` · okf_url: https://limbo137.top/okf/concepts/5fca732b-96e6-4605-8410-8351c5d1f96a.md
- labels: en = category of groups；zh-CN = 群范畴
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-category-of-groups` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 量子比特(qubit)

- UUID: `60143061-9c32-4555-8fdc-847e68ce8f16` · okf_url: https://limbo137.top/okf/concepts/60143061-9c32-4555-8fdc-847e68ce8f16.md
- labels: en = qubit；zh-CN = 量子比特
- aliases: qubits
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-87c83517ffc7a6e93810-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.4 · `books/woit-qtgr/2025-10-20/en/chapter-03/04-for-further-reading#first-87c83517ffc7a6e93810-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/04-for-further-reading.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 实数乘法群(multiplicative group of reals)

- UUID: `605a5909-d1a3-4540-a329-5ee89aa7b684` · okf_url: https://limbo137.top/okf/concepts/605a5909-d1a3-4540-a329-5ee89aa7b684.md
- labels: en = multiplicative group of reals；zh-CN = 实数乘法群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-multiplicative-group-of-reals` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 正定(positive definite)

- UUID: `60808942-4f7a-44c9-85de-4b3cf47ec7dd` · okf_url: https://limbo137.top/okf/concepts/60808942-4f7a-44c9-85de-4b3cf47ec7dd.md
- labels: en = positive definite；zh-CN = 正定
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-positive-definite` · translation)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-128346b487443b107031-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-128346b487443b107031-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## Dolbeault上同调群(Dolbeault cohomology group)

- UUID: `6081f08a-0e2e-5a6f-b003-cce54792dd79` · okf_url: https://limbo137.top/okf/concepts/6081f08a-0e2e-5a6f-b003-cce54792dd79.md
- labels: en = Dolbeault cohomology group；zh-CN = Dolbeault上同调群
- aliases: Dolbeault cohomology groups；∂̄-cohomology group；∂̄-上同调群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §8.6 (pdf_page=356, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §8.6 式(8.102)–(8.103)：复流形上双次数(r,s)的∂̄-闭形式模∂̄-恰当形式之商H^{r,s}_{∂̄}(M)。原文明确称∂̄-cohomology group；不是泛指上同调群，也不并入de Rham上同调群。原书无Definition编号；此处为管理员逐字核过的连续正文/公式单元。 证据保留OCR原样：8.102的H/Z/B下标丢失bar，而引句、等价类说明及8.103明确使用∂̄；不重写原文，不据本段声称全书首次出现。 · 原文: `$$ [ \omega ] = \left\{\eta \in \Omega^ {r, s} (M) | \bar {\partial} \eta = 0, \omega - \eta = \bar {\partial} \psi , \psi \in \Omega^ {r, s - 1} (M) \right\}. \tag {8.103} $$`
  - nakahara-geometry-topology-physics/second-edition-local §8.6 (pdf_page=355, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §8.6 式(8.102)–(8.103)：复流形上双次数(r,s)的∂̄-闭形式模∂̄-恰当形式之商H^{r,s}_{∂̄}(M)。原文明确称∂̄-cohomology group；不是泛指上同调群，也不并入de Rham上同调群。原书无Definition编号；此处为管理员逐字核过的连续正文/公式单元。 证据保留OCR原样：8.102的H/Z/B下标丢失bar，而引句、等价类说明及8.103明确使用∂̄；不重写原文，不据本段声称全书首次出现。 · 原文: `$$ H _ {\partial} ^ {r, s} (M) \equiv Z _ {\partial} ^ {r, s} (M) / B _ {\partial} ^ {r, s} (M). \tag {8.102} $$`
  - nakahara-geometry-topology-physics/second-edition-local §8.6 (pdf_page=355, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §8.6 式(8.102)–(8.103)：复流形上双次数(r,s)的∂̄-闭形式模∂̄-恰当形式之商H^{r,s}_{∂̄}(M)。原文明确称∂̄-cohomology group；不是泛指上同调群，也不并入de Rham上同调群。原书无Definition编号；此处为管理员逐字核过的连续正文/公式单元。 证据保留OCR原样：8.102的H/Z/B下标丢失bar，而引句、等价类说明及8.103明确使用∂̄；不重写原文，不据本段声称全书首次出现。 · 原文: `The $(r,s)$ th $\overline{\partial}$ -cohomology group is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §8.6 (pdf_page=356, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §8.6 式(8.102)–(8.103)：复流形上双次数(r,s)的∂̄-闭形式模∂̄-恰当形式之商H^{r,s}_{∂̄}(M)。原文明确称∂̄-cohomology group；不是泛指上同调群，也不并入de Rham上同调群。原书无Definition编号；此处为管理员逐字核过的连续正文/公式单元。 证据保留OCR原样：8.102的H/Z/B下标丢失bar，而引句、等价类说明及8.103明确使用∂̄；不重写原文，不据本段声称全书首次出现。 · 原文: `An element $[\omega] \in H_{\overline{\partial}}^{r,s}(M)$ is an equivalence class of $\overline{\partial}$ -closed forms of bidegree $(r,s)$ which differ from $\omega$ by a $\overline{\partial}$ -exact form,`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 收缩映射(retraction)

- UUID: `60b34861-2f0d-55d9-870b-0057366b7436` · okf_url: https://limbo137.top/okf/concepts/60b34861-2f0d-55d9-870b-0057366b7436.md
- labels: en = retraction；zh-CN = 收缩映射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=148, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.7：收缩映射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Note that the whole of $X$ is mapped onto $R$ keeping points in $R$ fixed. Figure 4.7 is an example of a retract and retraction.`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=148, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.7：收缩映射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.7. Let $R$ ( $\neq \emptyset$ ) be a subspace of $X$ . If there exists a continuous map $f: X \to R$ such that $f|_{R} = \mathrm{id}_{R}$ , $R$ is called a retract of $X$ and $f$ a retraction.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 零模向量(null vector)

- UUID: `60d65a35-60d7-43a2-8535-99cfced50031` · okf_url: https://limbo137.top/okf/concepts/60d65a35-60d7-43a2-8535-99cfced50031.md
- labels: en = null vector；zh-CN = 零模向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-null-vector` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 角动量算符(angular momentum operators)

- UUID: `60d67777-0124-483d-b32a-0fed2ee2d46b` · okf_url: https://limbo137.top/okf/concepts/60d67777-0124-483d-b32a-0fed2ee2d46b.md
- labels: en = angular momentum operators；zh-CN = 角动量算符
- aliases: 角动量算符（一般）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-08dfb4422b908977d4cd-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md

## 分离公理(separation axiom)

- UUID: `60db439c-bb38-47a0-bfe5-c7459f8e6090` · okf_url: https://limbo137.top/okf/concepts/60db439c-bb38-47a0-bfe5-c7459f8e6090.md
- labels: en = separation axiom；zh-CN = 分离公理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.5 · `zh/chapter-10/05-hausdorff-spaces#first-7c8924d891a64b8be7b0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/05-hausdorff-spaces.md

## 德拉姆定理(de Rham’s theorem)

- UUID: `60fba077-e5c2-4328-a0d1-5f983d5cd780` · okf_url: https://limbo137.top/okf/concepts/60fba077-e5c2-4328-a0d1-5f983d5cd780.md
- labels: en = de Rham’s theorem；zh-CN = 德拉姆定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-923e873f2eef96f84eb1-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-923e873f2eef96f84eb1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 升算符与降算符(Raising and lowering operators)

- UUID: `6187bb71-d6f3-4a7c-8765-9320aafd36ad` · okf_url: https://limbo137.top/okf/concepts/6187bb71-d6f3-4a7c-8765-9320aafd36ad.md
- labels: en = Raising and lowering operators；zh-CN = 升算符与降算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §8.1 · `books/woit-qtgr/2025-10-20/zh/chapter-08/01-representations-of-su2-classification#first-3cf89f35c39ca7106813-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §8.1 · `books/woit-qtgr/2025-10-20/en/chapter-08/01-representations-of-su2-classification#first-f0689ce6b239c70be69e-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `Let $$ S _ {+} = S _ {1} + i S _ {2} = \left( \begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array} \right), S _ {-} = S _ {1} - i S _ {2} = \left( \begin{array}{c c} 0 & 0 \\ 1 & 0 \end{array} \right) $$`
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `设 $$ S _ {+} = S _ {1} + i S _ {2} = \left( \begin{array}{c c} 0 & 1 \\ 0 & 0 \end{array} \right), S _ {-} = S _ {1} - i S _ {2} = \left( \begin{array}{c c} 0 & 0 \\ 1 & 0 \end{array} \right) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/01-representations-of-su2-classification.md

## 链(chain)

- UUID: `61fd4411-3d4a-4e1d-8127-2b7adfd4ce07` · okf_url: https://limbo137.top/okf/concepts/61fd4411-3d4a-4e1d-8127-2b7adfd4ce07.md
- labels: en = chain；zh-CN = 链
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-f400ddf3c821a7c31818-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-f400ddf3c821a7c31818-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.2. The $r$ -chain group $C_r(K)$ of a simplicial complex $K$ is a free Abelian group generated by the oriented $r$ -simplexes of $K$ . If $r > \dim K$ , $C_r(K)$ is defined to be 0. An element of $C_r(K)$ is called an $r$ -chain.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let $K$ be an $n$ -dimensional simplicial complex. There exists a sequence of free Abelian groups and homomorphisms,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) = p _ {1} - p _ {0}. \tag {3.19} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c = \sum_ {i = 1} ^ {I _ {r}} c _ {i} \sigma_ {r, i} \quad c _ {i} \in \mathbb {Z}. \tag {3.15} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$\partial_r$ is called the boundary operator. It is easy to see that the boundary operator is a homomorphism.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c + c ^ {\prime} = \sum_ {i} \left(c _ {i} + c _ {i} ^ {\prime}\right) \sigma_ {r, i}. \tag {3.16} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$(p_0p_1) + (p_1p_2) + (p_2p_0)$ . We agree that it has no boundary. If we insisted on the rule $\partial_{1}(p_{0}p_{1}) = p_{0} + p_{1}$ , we would have`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `which contradicts our intuition. If, on the other hand, we take $\partial_1(p_0p_1) = p_1 - p_0$ , we have`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} \sigma_ {r} \equiv \sum_ {i = 0} ^ {r} (- 1) ^ {i} \left(p _ {0} p _ {1} \dots \hat {p} _ {i} \dots p _ {r}\right) \tag {3.20} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r}: C _ {r} (K) \rightarrow C _ {r - 1} (K). \tag {3.22} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For a 1-simplex $(p_0p_1)$ , we define`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {0} + p _ {1} + p _ {1} + p _ {2} + p _ {2} + p _ {0} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $i: 0 \hookrightarrow C_n(K)$ is an inclusion map (0 is regarded as the unit element of $C_n(K)$ ). This sequence is called the chain complex associated with $K$ and is denoted by $C(K)$ . It is interesting to study the image and kernel of the homomorphisms $\partial_r$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `as expected. Hence, we put a plus sign if the first vertex is omitted and a minus sign if the second is omitted. We employ this fact to define the boundary of a general $r$ -simplex.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The reader might wonder about the appearance of a minus sign in front of $p_0$ . This is again related to the orientation. The following examples will clarify this point. In figure 3.7(a), an oriented 1-simplex $(p_0p_2)$ is divided into two, $(p_0p_1)$ and $(p_1p_2)$ . We agree that the boundary of $(p_0p_2)$ is $\{p_0\} \cup \{p_2\}$ and so should be th...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The operator $\partial_r$ acts linearly on an element $c = \sum_{i} c_i \sigma_{r,i}$ of $C_r(K)$ ,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ 0 \xrightarrow {i} C _ {n} (K) \xrightarrow {\partial_ {n}} C _ {n - 1} (K) \xrightarrow {\partial_ {n - 1}} \dots \xrightarrow {\partial_ {2}} C _ {1} (K) \xrightarrow {\partial_ {1}} C _ {0} (K) \xrightarrow {\partial_ {0}} 0 \tag {3.23} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let there be $I_r$ $r$ -simplexes in $K$ . We denote each of them by $\sigma_{r,i}$ ( $1 \leq i \leq I_r$ ). Then $c \in C_r(K)$ is expressed as`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The integers $c_{i}$ are called the coefficients of $c$ . The group structure is given as follows. The addition of two $r$ -chains, $c = \sum_{i} c_{i} \sigma_{r,i}$ and $c' = \sum_{i} c_{i}' \sigma_{r,i}$ , is`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \begin{array}{l} \partial_ {2} \left(p _ {0} p _ {1} p _ {2}\right) = \left(p _ {1} p _ {2}\right) - \left(p _ {0} p _ {2}\right) + \left(p _ {0} p _ {1}\right) \\ \partial_ {3} \left(p _ {0} p _ {1} p _ {2} p _ {3}\right) = \left(p _ {1} p _ {2} p _ {3}\right) - \left(p _ {0} p _ {2} p _ {3}\right) + \left(p _ {0} p _ {1} p _ {3}\right) - \left(p _ {0...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} c = \sum_ {i} c _ {i} \partial_ {r} \sigma_ {r, i}. \tag {3.21} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ C _ {r} (K) \cong \underbrace {\mathbb {Z} \oplus \mathbb {Z} \oplus \cdots \oplus \mathbb {Z}} _ {I _ {r}}. \tag {3.17} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `We formally define $\partial_0\sigma_0 = 0$ for $r = 0$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let $\sigma_r(p_0\ldots p_r)$ ( $r > 0$ ) be an oriented $r$ -simplex. The boundary $\partial_r\sigma_r$ of $\sigma_r$ is an $(r - 1)$ -chain defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {0} p _ {0} = 0. \tag {3.18} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Before we define the cycle group and the boundary group, we need to introduce the boundary operator. Let us denote the boundary of an $r$ -simplex $\sigma_r$ by $\partial_r\sigma_r$ . $\partial_r$ should be understood as an operator acting on $\sigma_r$ to produce its boundary. This point of view will be elaborated later. Let us look at the boundaries of ...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where the point $p_i$ under $\hat{\mathbf{\mu}}$ is omitted. For example,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=122, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The unit element is $0 = \sum_{i} 0 \cdot \sigma_{r,i}$ , while the inverse element of $c$ is $-c = \sum_{i} (-c_i) \sigma_{r,i}$ . [Remark: An oppositely oriented $r$ -simplex $-\sigma_r$ is identified with $(-1) \sigma_r \in C_r(K)$ .] Thus, $C_r(K)$ is a free Abelian group of rank $I_r$ ,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {1} \left(p _ {0} p _ {1}\right) + \partial_ {1} \left(p _ {1} p _ {2}\right) + \partial_ {1} \left(p _ {2} p _ {0}\right) = p _ {1} - p _ {0} + p _ {2} - p _ {1} + p _ {0} - p _ {2} = 0 $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.2：链。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\partial_r$ defines a map`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 括号积(bracket product)

- UUID: `620a0ad6-31da-41aa-a1e4-2973bb1be058` · okf_url: https://limbo137.top/okf/concepts/620a0ad6-31da-41aa-a1e4-2973bb1be058.md
- labels: en = bracket product；zh-CN = 括号积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-eef3afada3a218462345-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-bracket-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 欧拉-庞加莱定理(Euler-Poincaré theorem)

- UUID: `62492e55-1d73-4a9a-8881-81356737d47b` · okf_url: https://limbo137.top/okf/concepts/62492e55-1d73-4a9a-8881-81356737d47b.md
- labels: en = Euler-Poincaré theorem；zh-CN = 欧拉-庞加莱定理
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.7及式3.57，有限单纯复形的交错单纯形数等于交错Betti数；定理与欧拉示性数不合并。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `[Remark: The first equality defines the Euler characteristic of a general polyhedron $|K|$ . Note that this is the generalization of the Euler characteristic defined for surfaces in section 2.4.]`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.7及式3.57，有限单纯复形的交错单纯形数等于交错Betti数；定理与欧拉示性数不合并。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \chi (K) \equiv \sum_ {r = 0} ^ {n} (- 1) ^ {r} I _ {r} = \sum_ {r = 0} ^ {n} (- 1) ^ {r} b _ {r} (K). \tag {3.57} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.7及式3.57，有限单纯复形的交错单纯形数等于交错Betti数；定理与欧拉示性数不合并。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Theorem 3.7. (The Euler-Poincaré theorem) Let $K$ be an $n$ -dimensional simplicial complex and let $I_r$ be the number of $r$ -simplexes in $K$ . Then`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 有界线性泛函(bounded linear functional)

- UUID: `6250cac8-1454-4728-8e48-1890f9a62edf` · okf_url: https://limbo137.top/okf/concepts/6250cac8-1454-4728-8e48-1890f9a62edf.md
- labels: en = bounded linear functional；zh-CN = 有界线性泛函
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.9 · `zh/chapter-10/09-topological-vector-spaces#first-94c71fd50780345cf2d1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md

## 最小零化多项式(minimal annihilating polynomial)

- UUID: `6265e6d8-3613-43e1-bb52-65b36182c218` · okf_url: https://limbo137.top/okf/concepts/6265e6d8-3613-43e1-bb52-65b36182c218.md
- labels: en = minimal annihilating polynomial；zh-CN = 最小零化多项式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-minimal-annihilating-polynomial` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 余切丛(cotangent bundle)

- UUID: `6316e4d5-6175-4c0a-bee4-f7474c095f53` · okf_url: https://limbo137.top/okf/concepts/6316e4d5-6175-4c0a-bee4-f7474c095f53.md
- labels: en = cotangent bundle；zh-CN = 余切丛
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-bf3eb4001356e9567b86-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-cotangent-bundle` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 莫比乌斯变换(Möbius transformation)

- UUID: `6328536c-e4a4-4277-b275-0a8a7146d152` · okf_url: https://limbo137.top/okf/concepts/6328536c-e4a4-4277-b275-0a8a7146d152.md
- labels: en = Möbius transformation；zh-CN = 莫比乌斯变换
- aliases: Möbius transformations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-mobius-transformation` · translation)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-e2a6486a1efc2e914542-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-5ac38295dc438d0557a8-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 平行四边形恒等式(parallelogram law)

- UUID: `635dc93f-838e-447c-aa5a-7f86b575919f` · okf_url: https://limbo137.top/okf/concepts/635dc93f-838e-447c-aa5a-7f86b575919f.md
- labels: en = parallelogram law；zh-CN = 平行四边形恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.1 · `en/chapter-13/01-definitions-and-examples#first-52a19f8f83d310ed13be-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.1 · `zh/chapter-13/01-definitions-and-examples#concept-13-parallelogram-law` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/01-definitions-and-examples.md

## 卡当公式(Cartan’s formulas)

- UUID: `63dd968e-0dae-45a0-98ba-06b83032ed48` · okf_url: https://limbo137.top/okf/concepts/63dd968e-0dae-45a0-98ba-06b83032ed48.md
- labels: en = Cartan’s formulas；zh-CN = 卡当公式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.2 · `en/chapter-16/02-properties-of-exterior-derivative#first-01140c9e890f6fe291ec-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.2 · `zh/chapter-16/02-properties-of-exterior-derivative#first-01140c9e890f6fe291ec-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/02-properties-of-exterior-derivative.md

## 边缘算子(boundary operator)

- UUID: `63e6354d-b1e8-4f50-835c-9b4368891ad8` · okf_url: https://limbo137.top/okf/concepts/63e6354d-b1e8-4f50-835c-9b4368891ad8.md
- labels: en = boundary operator；zh-CN = 边缘算子
- aliases: $\partial_r$
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Let $\sigma_r(p_0\ldots p_r)$ ( $r > 0$ ) be an oriented $r$ -simplex. The boundary $\partial_r\sigma_r$ of $\sigma_r$ is an $(r - 1)$ -chain defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `where the point $p_i$ under $\hat{\mathbf{\mu}}$ is omitted. For example,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `The RHS of (3.21) is an element of $C_{r-1}(K)$ . Accordingly, $\partial_r$ defines a map`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \partial_ {r} \sigma_ {r} \equiv \sum_ {i = 0} ^ {r} (- 1) ^ {i} \left(p _ {0} p _ {1} \dots \hat {p} _ {i} \dots p _ {r}\right) \tag {3.20} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \partial_ {r}: C _ {r} (K) \rightarrow C _ {r - 1} (K). \tag {3.22} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \partial_ {r} c = \sum_ {i} c _ {i} \partial_ {r} \sigma_ {r, i}. \tag {3.21} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `We formally define $\partial_0\sigma_0 = 0$ for $r = 0$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \begin{array}{l} \partial_ {2} \left(p _ {0} p _ {1} p _ {2}\right) = \left(p _ {1} p _ {2}\right) - \left(p _ {0} p _ {2}\right) + \left(p _ {0} p _ {1}\right) \\ \partial_ {3} \left(p _ {0} p _ {1} p _ {2} p _ {3}\right) = \left(p _ {1} p _ {2} p _ {3}\right) - \left(p _ {0} p _ {2} p _ {3}\right) + \left(p _ {0} p _ {1} p _ {3}\right) - \left(p _ {0...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$\partial_r$ is called the boundary operator. It is easy to see that the boundary operator is a homomorphism.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 单纯链复形边缘算子∂r，保留交错面公式、r=0约定及线性延拓；与边界集合、边缘群均不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `The operator $\partial_r$ acts linearly on an element $c = \sum_{i} c_i \sigma_{r,i}$ of $C_r(K)$ ,`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 更粗的拓扑(coarser topology)

- UUID: `640c9ebf-7397-4394-831d-69f5853ac5c8` · okf_url: https://limbo137.top/okf/concepts/640c9ebf-7397-4394-831d-69f5853ac5c8.md
- labels: en = coarser topology；zh-CN = 更粗的拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-5ae3298caec95b3768f4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 功形式(work form)

- UUID: `640cbf2a-bf3f-465e-a7f9-0d97c48f0e1f` · okf_url: https://limbo137.top/okf/concepts/640cbf2a-bf3f-465e-a7f9-0d97c48f0e1f.md
- labels: en = work form；zh-CN = 功形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-bf7505bc87a493212805-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 混合态(mixed state)

- UUID: `64aee7f1-75c2-47b6-90c4-0117e5accf89` · okf_url: https://limbo137.top/okf/concepts/64aee7f1-75c2-47b6-90c4-0117e5accf89.md
- labels: en = mixed state；zh-CN = 混合态
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.4 · `en/chapter-14/04-quantum-statistical-mechanics#first-63cdd481672b0a80aa85-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.4 · `zh/chapter-14/04-quantum-statistical-mechanics#first-63cdd481672b0a80aa85-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/04-quantum-statistical-mechanics.md

## 复标量量子场(Complex scalar quantum field)

- UUID: `64e2d5a7-ae76-40d9-855f-323d58e1ee16` · okf_url: https://limbo137.top/okf/concepts/64e2d5a7-ae76-40d9-855f-323d58e1ee16.md
- labels: en = Complex scalar quantum field；zh-CN = 复标量量子场
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §44.1 · `books/woit-qtgr/2025-10-20/en/chapter-44/01-internal-symmetries#first-32e6e596e353b353501d-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §44.1 · `books/woit-qtgr/2025-10-20/zh/chapter-44/01-internal-symmetries#first-e286bd500e789d9fcb2d-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §44.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `复标量量子场算符是由下式定义的算符值分布 $$ \widehat {\Phi} (\mathbf {x}) = \frac {1}{(2 \pi) ^ {3 / 2}} \int_ {\mathbf {R} ^ {3}} (a (\mathbf {p}) e ^ {i \mathbf {p} \cdot \mathbf {x}} + b ^ {\dagger} (\mathbf {p}) e ^ {- i \mathbf {p} \cdot \mathbf {x}}) \frac {d ^ {3} \mathbf {p}}{\sqrt {2 \omega_ {\mathbf {p}}}} $$ $$ \widehat {\Phi} ^ {\dagger} (\mathbf {x}) = \frac {1}...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-44/01-internal-symmetries.md

## 标准曲线系统(canonical system of curves)

- UUID: `64f25931-1152-4952-86a6-8f098dfc780b` · okf_url: https://limbo137.top/okf/concepts/64f25931-1152-4952-86a6-8f098dfc780b.md
- labels: en = canonical system of curves；zh-CN = 标准曲线系统
- aliases: $\{a_i, b_i\}$
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.5 (pdf_page=134, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara3.3.5中可定向闭曲面Σg上{ai,bi}的2g条标准曲线，语境和原图2.12保留。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Since $\Sigma_g$ is connected, $H_0(\Sigma_g) \cong \mathbb{Z}$ . Observe that $a_i(b_i)$ is homologous to the edge $a_i(b_i)$ of figure 2.12. The $2g$ curves $\{a_i, b_i\}$ are called the canonical system of curves on $\Sigma_g$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 范畴(category)

- UUID: `64ff3c19-0344-41c8-936c-4336e0d0b760` · okf_url: https://limbo137.top/okf/concepts/64ff3c19-0344-41c8-936c-4336e0d0b760.md
- labels: en = category；zh-CN = 范畴
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-33ad6735f7957abbf31f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 传播子(propagator)

- UUID: `65334560-ae6d-466e-803c-78244b705c5b` · okf_url: https://limbo137.top/okf/concepts/65334560-ae6d-466e-803c-78244b705c5b.md
- labels: en = propagator；zh-CN = 传播子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §37.3 · `books/woit-qtgr/2025-10-20/en/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-ad613dd3edd620bb3d47-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §37.3 · `books/woit-qtgr/2025-10-20/zh/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-ad613dd3edd620bb3d47-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory.md

## 简单二向量(simple 2-vector)

- UUID: `6562edc7-ace1-408a-a8bf-2bffcb6572b8` · okf_url: https://limbo137.top/okf/concepts/6562edc7-ace1-408a-a8bf-2bffcb6572b8.md
- labels: en = simple 2-vector；zh-CN = 简单二向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-simple-2-vector` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-3ed15f36991aae8241b7-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md

## 平行的(parallel)

- UUID: `6574dc8c-d1b3-4dcc-96ce-8aa80fbe82b5` · okf_url: https://limbo137.top/okf/concepts/6574dc8c-d1b3-4dcc-96ce-8aa80fbe82b5.md
- labels: en = parallel；zh-CN = 平行的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-322509e7e9906d69d243-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-322509e7e9906d69d243-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 勒贝格积分(Lebesgue integral)

- UUID: `65be7ead-b922-4a43-96c6-d0ecf43ce2b2` · okf_url: https://limbo137.top/okf/concepts/65be7ead-b922-4a43-96c6-d0ecf43ce2b2.md
- labels: en = Lebesgue integral；zh-CN = 勒贝格积分
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-5bd7f95594b12d8e58cf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 哈密顿–雅可比方程(Hamilton–Jacobi equation)

- UUID: `65eda8f9-c136-46f2-bd8c-8b38b2568a34` · okf_url: https://limbo137.top/okf/concepts/65eda8f9-c136-46f2-bd8c-8b38b2568a34.md
- labels: en = Hamilton–Jacobi equation；zh-CN = 哈密顿–雅可比方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-ec6bd6fe60c1f86e447f-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-ec6bd6fe60c1f86e447f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 半群同构(semigroup isomorphism)

- UUID: `65fcf7f9-2b97-4221-82f1-04d6d6e4fcfa` · okf_url: https://limbo137.top/okf/concepts/65fcf7f9-2b97-4221-82f1-04d6d6e4fcfa.md
- labels: en = semigroup isomorphism；zh-CN = 半群同构
- aliases: isomorphism；同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-9be49a86d7ea6d10b7fc-semigroup-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示(Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation)

- UUID: `676173e7-6104-41f9-b2f2-95d235a3a95c` · okf_url: https://limbo137.top/okf/concepts/676173e7-6104-41f9-b2f2-95d235a3a95c.md
- labels: en = Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation；zh-CN = Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §22.3 · `books/woit-qtgr/2025-10-20/zh/chapter-22/03-the-bargmann-fock-representation#first-bb4b3a2dd83c090bc814-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §22.3 · `books/woit-qtgr/2025-10-20/en/chapter-22/03-the-bargmann-fock-representation#first-bfc0d9585e87a8beeba5-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §22.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Bargmann–Fock representation takes the state space to be $\mathcal{H}=\mathcal{F}$, where $\mathcal{F}$ is the space of holomorphic functions on $\mathbb{C}$ (satisfying $\frac{d}{dz}\psi=0$) with finite norm for the inner product $$ \langle\psi_1|\psi_2\rangle=\frac{1}{\pi}\int_{\mathbb{C}}\overline{\psi_1(z)}\psi_2(z)e^{-|z|^2}\,d^2z\tag{22.4} $$ wh...`
  - woit-qtgr/2025-10-20 §22.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `Bargmann–Fock 表示取态空间为 $\mathcal{H}=\mathcal{F}$，其中 $\mathcal{F}$ 是定义在 $\mathbb{C}$ 上、范数有限的全纯函数空间（原书此处印作 $\frac{d}{dz}\psi=0$），其内积为 $$ \langle\psi_1|\psi_2\rangle=\frac{1}{\pi}\int_{\mathbb{C}}\overline{\psi_1(z)}\psi_2(z)e^{-|z|^2}\,d^2z\tag{22.4} $$ 其中 $d^2z=d\operatorname{Re}(z)\,d\operatorname{Im}(z)$。空间 $\mathcal{F}$ 有时称为“福克空间”。定义作用在该空间上的两个算符： $$ a=\f...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-22/03-the-bargmann-fock-representation.md

## 线性映射的核(kernel of a linear map)

- UUID: `6779b152-105a-4fbc-b6e6-aa3246282ea0` · okf_url: https://limbo137.top/okf/concepts/6779b152-105a-4fbc-b6e6-aa3246282ea0.md
- labels: en = kernel of a linear map；zh-CN = 线性映射的核
- aliases: kernel；kernel of a linear mapping；kernel of a linear transformation；核
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-kernel` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 施密特正交归一化(Schmidt orthonormalization)

- UUID: `678d826e-ad2c-493f-8187-25c5734f0232` · okf_url: https://limbo137.top/okf/concepts/678d826e-ad2c-493f-8187-25c5734f0232.md
- labels: en = Schmidt orthonormalization；zh-CN = 施密特正交归一化
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-schmidt-orthonormalization` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 广义克罗内克符号(generalized delta symbol)

- UUID: `67922473-e00b-4db9-ae57-12f963c33087` · okf_url: https://limbo137.top/okf/concepts/67922473-e00b-4db9-ae57-12f963c33087.md
- labels: en = generalized delta symbol；zh-CN = 广义克罗内克符号
- aliases: generalized
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.5 · `en/chapter-08/05-oriented-vector-spaces#first-f5a598a4f196ba658b6b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.5 · `zh/chapter-08/05-oriented-vector-spaces#first-2a4509112d7d6ee05f85-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/05-oriented-vector-spaces.md

## 陪域(codomain)

- UUID: `67953d1f-38a5-442e-8fff-e9ceae6b7a78` · okf_url: https://limbo137.top/okf/concepts/67953d1f-38a5-442e-8fff-e9ceae6b7a78.md
- labels: en = codomain；zh-CN = 陪域
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-53d4a9490538ebd9313d-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=86, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定义条件与符号保留在引文中；不推断跨书等价。 · 原文: `There may be more than two elements in $X$ that correspond to the same $y \in Y$ . A subset of $X$ whose elements are mapped to $y \in Y$ under $f$ is called the inverse image of $y$ , denoted by $f^{-1}(y) = \{x \in X | f(x) = y\}$ . The set $X$ is called the domain of the map while $Y$ is called the range of the map. The image of the map is $f(X) = \{y ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 交换(transposition)

- UUID: `67f5150b-6d3e-43d0-969c-f7387ec86069` · okf_url: https://limbo137.top/okf/concepts/67f5150b-6d3e-43d0-969c-f7387ec86069.md
- labels: en = transposition；zh-CN = 交换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-transposition` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 定义表示(defining representation)

- UUID: `6800cca5-05b3-4571-9b98-979ca6c8a5c1` · okf_url: https://limbo137.top/okf/concepts/6800cca5-05b3-4571-9b98-979ca6c8a5c1.md
- labels: en = defining representation；zh-CN = 定义表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.3 · `books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary#concept-6-3-defining-representation` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.3 · `books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary#concept-6-3-defining-representation` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 稠密的(dense)

- UUID: `68277a25-205a-4619-9a87-590ce1565818` · okf_url: https://limbo137.top/okf/concepts/68277a25-205a-4619-9a87-590ce1565818.md
- labels: en = dense；zh-CN = 稠密的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-7ef8cab0fc33491912d4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-dense` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 非退化厄米形式(nondegenerate Hermitian form)

- UUID: `68b6758f-5f71-444b-8fa4-ff25e8ccdd6b` · okf_url: https://limbo137.top/okf/concepts/68b6758f-5f71-444b-8fa4-ff25e8ccdd6b.md
- labels: en = nondegenerate Hermitian form；zh-CN = 非退化厄米形式
- aliases: Inner product, complex case；complex inner product；complex inner product in Woit；复内积；复内积（Woit约定）
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-326fbf962bb3f72f6083-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-8c9bd1ba0928e81d7e3d-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `$$ \langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle , \quad \langle \alpha u, v \rangle = \overline {{\alpha}} \langle u, v \rangle $$`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（复内积(complex inner product)）。复向量空间 $V$ 上的厄米内积是一个映射 $$ \langle \cdot , \cdot \rangle : V \times V \to \mathbf {C} $$ 即共轭对称 $$ \langle v, w \rangle = \overline {{\langle w , v \rangle}} $$ 在两个变量上均非退化，在第二个变量上线性，在第一个变量上反线性：对于 $\alpha \in \mathbf { C }$ 和 $u , v , w \in V$。 $$ \langle u + v, w \rangle = \langle u, w \rangle + \langle v, w \rangle , \quad \la...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `$$ | | v | | ^ {2} = \langle v, v \rangle $$`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `An inner product gives a notion of length-squared $| | \cdot | | ^ { 2 }$ for vectors, with`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `Note that whether to specify antilinearity in the first or second variable is a matter of convention. The choice we are making is universal among physicists, with the opposite choice common among mathematicians. Our Hermitian inner products will be positive definite ($\|v\|^2>0$ for $v\neq0$) unless specifically noted otherwise (i.e., characterized explic...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `that is conjugate symmetric`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `Definition (Inner product, complex case). A Hermitian inner product on a complex vector space $V$ is a map`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Inner product, complex case). A Hermitian inner product on a complex vector space $V$ is a map $$ \langle \cdot , \cdot \rangle : V \times V \to \mathbf {C} $$ that is conjugate symmetric $$ \langle v, w \rangle = \overline {{\langle w , v \rangle}} $$ non-degenerate in both variables, linear in the second variable, and antilinear in the first...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `non-degenerate in both variables, linear in the second variable, and antilinear in the first variable: for $\alpha \in \mathbf { C }$ and $u , v , w \in V$.`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `$$ \langle v, w \rangle = \overline {{\langle w , v \rangle}} $$`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §4.4 定义中的共轭对称、双变量非退化、第二变量线性第一变量反线性的复形式；定义条件不要求正定。后文说通常默认正定并允许不定例外；与Szekeres §5.2显式要求正定的复内积分开，不声称首次出现。 · 原文: `$$ \langle \cdot , \cdot \rangle : V \times V \to \mathbf {C} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 循环坐标(cyclic coordinate)

- UUID: `68cbad9d-9f9f-49ee-93c4-039489918e96` · okf_url: https://limbo137.top/okf/concepts/68cbad9d-9f9f-49ee-93c4-039489918e96.md
- labels: en = cyclic coordinate；zh-CN = 循环坐标
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-86fee36aadf77e1c7329-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-86fee36aadf77e1c7329-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 转置(transpose)

- UUID: `68e0c96d-dfe2-418d-b27a-df73bf8da23b` · okf_url: https://limbo137.top/okf/concepts/68e0c96d-dfe2-418d-b27a-df73bf8da23b.md
- labels: en = transpose；zh-CN = 转置
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-transpose` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 稳相近似(stationary phase approximation)

- UUID: `68eedf23-0c59-42f4-84e4-dee129e2d813` · okf_url: https://limbo137.top/okf/concepts/68eedf23-0c59-42f4-84e4-dee129e2d813.md
- labels: en = stationary phase approximation；zh-CN = 稳相近似
- 分书定义状态: schwartz-qft/local-experiment = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.3 · `books/woit-qtgr/2025-10-20/zh/chapter-35/03-quantization-and-path-integrals#first-e912908913a6f415c943-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.3 · `books/woit-qtgr/2025-10-20/en/chapter-35/03-quantization-and-path-integrals#first-e912908913a6f415c943-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/03-quantization-and-path-integrals.md

## 生成的西格玛代数(generated sigma-algebra)

- UUID: `6923d159-9120-4de8-aad5-aaf9baa64b37` · okf_url: https://limbo137.top/okf/concepts/6923d159-9120-4de8-aad5-aaf9baa64b37.md
- labels: en = generated sigma-algebra；zh-CN = 生成的西格玛代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-5139e4371bda3b6512b3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 被动(passive transformation)

- UUID: `695999e5-a66d-4ad1-8aab-178887ad28d1` · okf_url: https://limbo137.top/okf/concepts/695999e5-a66d-4ad1-8aab-178887ad28d1.md
- labels: en = passive transformation；zh-CN = 被动
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-passive-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 富比尼定理(Fubini’s theorem)

- UUID: `699abcf0-77c9-4e82-a864-7d6745c8b88d` · okf_url: https://limbo137.top/okf/concepts/699abcf0-77c9-4e82-a864-7d6745c8b88d.md
- labels: en = Fubini’s theorem；zh-CN = 富比尼定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-d30daa45c714c0eae014-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §11.3 · `en/chapter-11/03-lebesgue-integration#first-d30daa45c714c0eae014-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 特征标(character)

- UUID: `69b28db8-59b4-4cec-a6a3-1770de5f7384` · okf_url: https://limbo137.top/okf/concepts/69b28db8-59b4-4cec-a6a3-1770de5f7384.md
- labels: en = character；zh-CN = 特征标
- aliases: Character
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#concept-5-character` · translation)；woit-qtgr/2025-10-20 (zh-CN · §9.4 · `books/woit-qtgr/2025-10-20/zh/chapter-09/04-tensor-products-of-representations#first-b708ccd0bb113bbab077-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §9.4 · `books/woit-qtgr/2025-10-20/en/chapter-09/04-tensor-products-of-representations#first-e9acf34c29cde3db34f8-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §9.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `群$G$的表示$(\pi,V)$的特征标是定义在$G$上的函数： $$ \chi_V(g)=\operatorname{Tr}(\pi(g)) $$`
  - woit-qtgr/2025-10-20 §9.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The character of a representation $(\pi,V)$ of a group $G$ is the function on $G$ defined by $$ \chi_V(g)=\operatorname{Tr}(\pi(g)). $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-09/04-tensor-products-of-representations.md

## 算符范数(operator norm)

- UUID: `69ef56dc-9747-4a55-a4d1-f144c0f887da` · okf_url: https://limbo137.top/okf/concepts/69ef56dc-9747-4a55-a4d1-f144c0f887da.md
- labels: en = operator norm；zh-CN = 算符范数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-operator-norm` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-8e3bd0e538ed12a24621-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 弱收敛(weak convergence)

- UUID: `6a25e9a4-78c0-4b7a-bfe0-2b96d45c1ef2` · okf_url: https://limbo137.top/okf/concepts/6a25e9a4-78c0-4b7a-bfe0-2b96d45c1ef2.md
- labels: en = weak convergence；zh-CN = 弱收敛
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-11b996d2e4613b9a177a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 仿射无关点组(affinely independent points)

- UUID: `6a3fc859-7a0d-492e-ba44-50b4ddc0f20f` · okf_url: https://limbo137.top/okf/concepts/6a3fc859-7a0d-492e-ba44-50b4ddc0f20f.md
- labels: en = affinely independent points；zh-CN = 仿射无关点组
- aliases: independent；independent points；独立点组
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-9665e14337809b625e75-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-7e0cc41005137aad4cb0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 占据数基(occupation number basis)

- UUID: `6a544ce7-2fa1-4b9f-b603-5cfd92463bf6` · okf_url: https://limbo137.top/okf/concepts/6a544ce7-2fa1-4b9f-b603-5cfd92463bf6.md
- labels: en = occupation number basis；zh-CN = 占据数基
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `is finite. This is called the “occupation number” basis of $\mathcal { F } _ { d }$ . In this basis, the annihilation and creation operators are`
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `$$ n = \sum_ {j = 1} ^ {d} n _ {j} $$`
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `$$ \begin{aligned} a _ {j} | n _ {1}, n _ {2}, \dots , n _ {d} \rangle &= \sqrt {n _ {j}} | n _ {1}, n _ {2}, \dots , n _ {j} - 1, \dots , n _ {d} \rangle \\[0.45em] a _ {j} ^ {\dagger} | n _ {1}, n _ {2}, \dots , n _ {d} \rangle &= \sqrt {n _ {j} + 1} | n _ {1}, n _ {2}, \dots , n _ {j} + 1, \dots , n _ {d} \rangle \end{aligned} $$`
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `$\mathcal { F } _ { d }$ has an orthonormal basis`
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `$$ \left| n _ {1}, n _ {2}, \dots , n _ {d} \right> $$`
  - woit-qtgr/2025-10-20 §36.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §36.1.1，玻色Fock空间的正交归一基|n1,...,nd>，ni为非负整数、总占据数有限；定义含升降算符作用。概念是基，不是占据数或数算符，不声称首次出现。 · 原文: `labeled by the eigenvalues $n _ { j }$ of the number operators $N _ { j }$ for $j = 1 , \cdots , d$ Here $n _ { j } \in \{ 0 , 1 , 2 , \cdot \cdot \cdot \}$ and`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 欧拉–拉格朗日场方程(Euler–Lagrange field equations)

- UUID: `6a6a990b-27e1-42a4-8591-59e8db4625bd` · okf_url: https://limbo137.top/okf/concepts/6a6a990b-27e1-42a4-8591-59e8db4625bd.md
- labels: en = Euler–Lagrange field equations；zh-CN = 欧拉–拉格朗日场方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.10 · `en/chapter-18/10-variation-principles-in-space-time#first-36d920412fa399f99205-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.10 · `zh/chapter-18/10-variation-principles-in-space-time#first-36d920412fa399f99205-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/10-variation-principles-in-space-time.md

## 拉格朗日量(Lagrangian)

- UUID: `6af418ab-a35e-47b5-8754-b4fbdaf3ae8e` · okf_url: https://limbo137.top/okf/concepts/6af418ab-a35e-47b5-8754-b4fbdaf3ae8e.md
- labels: en = Lagrangian；zh-CN = 拉格朗日量
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-cfd2d2741d204b299a6b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-70e7db28dd52224668ed-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `位形空间为 $\mathbb R^d$ 的经典系统的[拉格朗日量(Lagrangian)](/concepts/mechanics/lagrangian)是函数 $$ L:(\mathbf q,\mathbf v)\in\mathbb R^d\times\mathbb R^d\longmapsto L(\mathbf q,\mathbf v)\in\mathbb R. $$`
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [Lagrangian](/concepts/mechanics/lagrangian) for a classical system with configuration space $\mathbb R^d$ is a function $$ L:(\mathbf q,\mathbf v)\in\mathbb R^d\times\mathbb R^d\longmapsto L(\mathbf q,\mathbf v)\in\mathbb R. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 轨道角动量算符(orbital angular momentum operators)

- UUID: `6b2e6c15-f414-4480-828a-cc19da05f9fb` · okf_url: https://limbo137.top/okf/concepts/6b2e6c15-f414-4480-828a-cc19da05f9fb.md
- labels: en = orbital angular momentum operators；zh-CN = 轨道角动量算符
- aliases: angular momentum operators
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §38.3 · `books/woit-qtgr/2025-10-20/zh/chapter-38/05-spatial-symmetries#first-b37c22fc0731f70b0874-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §38.3 · `books/woit-qtgr/2025-10-20/en/chapter-38/05-spatial-symmetries#first-1066db9262d0482dcdcb-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/05-spatial-symmetries.md

## 球极投影(stereographic projection)

- UUID: `6b78393e-3876-49d3-b82b-3dd0d4ff3b21` · okf_url: https://limbo137.top/okf/concepts/6b78393e-3876-49d3-b82b-3dd0d4ff3b21.md
- labels: en = stereographic projection；zh-CN = 球极投影
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-d59757d7da819e92dd76-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-d59757d7da819e92dd76-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 自旋二能级系统的态(state of a spin-half two-state system)

- UUID: `6bb96444-d087-4f5c-8398-4f3dc18cac96` · okf_url: https://limbo137.top/okf/concepts/6bb96444-d087-4f5c-8398-4f3dc18cac96.md
- labels: en = state of a spin-half two-state system；zh-CN = 自旋二能级系统的态
- aliases: States；态（自旋二能级系统）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-3608a5d0031de8861919-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 类时向量(timelike vector)

- UUID: `6bcac000-f712-409e-b09d-925136bf96c2` · okf_url: https://limbo137.top/okf/concepts/6bcac000-f712-409e-b09d-925136bf96c2.md
- labels: en = timelike vector；zh-CN = 类时向量
- aliases: timelike；类时
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §40.1 · `books/woit-qtgr/2025-10-20/en/chapter-40/01-minkowski-space#first-adc5c790641d95ac2f09-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 固有时(proper time)

- UUID: `6c0efb7b-2257-4b78-a4a1-65e135a01339` · okf_url: https://limbo137.top/okf/concepts/6c0efb7b-2257-4b78-a4a1-65e135a01339.md
- labels: en = proper time；zh-CN = 固有时
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-b718df73494de0c34bbe-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-b718df73494de0c34bbe-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 位置空间传播子(Position space propagator)

- UUID: `6c13c2cf-973e-4abf-8173-dd90eaf57938` · okf_url: https://limbo137.top/okf/concepts/6c13c2cf-973e-4abf-8173-dd90eaf57938.md
- labels: en = Position space propagator；zh-CN = 位置空间传播子
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §12.5 · `books/woit-qtgr/2025-10-20/zh/chapter-12/05-the-propagator-in-position-space#first-71c09402330fe80076f6-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §12.5 · `books/woit-qtgr/2025-10-20/en/chapter-12/05-the-propagator-in-position-space#first-324a6e15ed01e58de671-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §12.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `位置空间传播子是作用于位置空间波函数的时间演化算符的核 ${ \cal U } ( t , q _ { t } , q _ { 0 } )$。它给出波函数在任意时刻 $t$ 的演化： $$ \psi (q _ {t}, t) = \int_ {- \infty} ^ {+ \infty} U (t, q _ {t}, q _ {0}) \psi (q _ {0}, 0) d q _ {0}\tag{12.4} $$ 其中 $\psi ( q _ { 0 } , 0 )$ 是波函数在时刻 $0$ 的初始值。`
  - woit-qtgr/2025-10-20 §12.5 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The position space propagator is the kernel ${ \cal U } ( t , q _ { t } , q _ { 0 } )$ of the time evolution operator acting on position space wavefunctions. It determines the time evolution of the wavefunction at every time $t$: $$ \psi (q _ {t}, t) = \int_ {- \infty} ^ {+ \infty} U (t, q _ {t}, q _ {0}) \psi (q _ {0}, 0) d q _ {0}\tag{12.4} $$ where $\p...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-12/05-the-propagator-in-position-space.md

## 辛约化(Symplectic reduction)

- UUID: `6c1711a5-ca46-4052-aaed-0e4e27d60636` · okf_url: https://limbo137.top/okf/concepts/6c1711a5-ca46-4052-aaed-0e4e27d60636.md
- labels: en = Symplectic reduction；zh-CN = 辛约化
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §46.3 · `books/woit-qtgr/2025-10-20/en/chapter-46/03-gauss-s-law-and-time-independent-gauge-transformations#first-02b3669bb92f3d81c155-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §46.3 · `books/woit-qtgr/2025-10-20/zh/chapter-46/03-gauss-s-law-and-time-independent-gauge-transformations#first-a12260473d4db81c6692-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §46.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `设群 $G$ 作用在相空间 $M$ 上并保持泊松括号，且该作用具有矩映射 $\mu:M\to\mathfrak g^*$。辛约化是商空间 $M//G=\mu^{-1}(0)/G$。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-46/03-gauss-s-law-and-time-independent-gauge-transformations.md

## 霍奇星算子(Hodge star operator)

- UUID: `6d1cf432-5559-4e89-afa8-dad94bcb4ee2` · okf_url: https://limbo137.top/okf/concepts/6d1cf432-5559-4e89-afa8-dad94bcb4ee2.md
- labels: en = Hodge star operator；zh-CN = 霍奇星算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.6 · `zh/chapter-08/06-the-hodge-dual#first-21844d5d5be719368756-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.6 · `en/chapter-08/06-the-hodge-dual#first-21844d5d5be719368756-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/06-the-hodge-dual.md

## 半直积李代数(Semi-direct product Lie algebra)

- UUID: `6da2baae-8135-47ae-aa6b-d70da67ce130` · okf_url: https://limbo137.top/okf/concepts/6da2baae-8135-47ae-aa6b-d70da67ce130.md
- labels: en = Semi-direct product Lie algebra；zh-CN = 半直积李代数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §18.3 · `books/woit-qtgr/2025-10-20/zh/chapter-18/03-semi-direct-product-lie-algebras#woit-semi-direct-product-lie-algebra` · machine_draft)；woit-qtgr/2025-10-20 (en · §18.3 · `books/woit-qtgr/2025-10-20/en/chapter-18/03-semi-direct-product-lie-algebras#woit-semi-direct-product-lie-algebra` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-18/03-semi-direct-product-lie-algebras.md

## 拓扑子群(topological subgroup)

- UUID: `6dbf77e2-01b5-48e1-8fc5-8ff5b9354b60` · okf_url: https://limbo137.top/okf/concepts/6dbf77e2-01b5-48e1-8fc5-8ff5b9354b60.md
- labels: en = topological subgroup；zh-CN = 拓扑子群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-282ff3f2873eae096c4c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 定向单纯形(oriented simplex)

- UUID: `6de689f3-54b6-42e9-80a2-d2500ccf8629` · okf_url: https://limbo137.top/okf/concepts/6de689f3-54b6-42e9-80a2-d2500ccf8629.md
- labels: en = oriented simplex；zh-CN = 定向单纯形
- aliases: 定向单形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §3.3.1: ordered geometrically independent vertices modulo even permutations define an oriented r-simplex. Preserve both prose halves and the permutation display; no first occurrence is asserted. · 原文: `$$ P = \left( \begin{array}{l l l l} 0 & 1 & \ldots & r \\ i _ {0} & i _ {1} & \ldots & i _ {r} \end{array} \right) $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §3.3.1: ordered geometrically independent vertices modulo even permutations define an oriented r-simplex. Preserve both prose halves and the permutation display; no first occurrence is asserted. · 原文: `It is now easy to construct an oriented $r$ -simplex for any $r \geq 1$ . The formal definition goes as follows. Take $r + 1$ geometrically independent points $p_0, p_1, \ldots, p_r$ in $\mathbb{R}^m$ . Let $\{p_{i_0}, p_{i_1}, \ldots, p_{i_r}\}$ be a sequence of points obtained by a permutation of the points $p_0, \ldots, p_r$ . We define $\{p_0, \ldots,...`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §3.3.1: ordered geometrically independent vertices modulo even permutations define an oriented r-simplex. Preserve both prose halves and the permutation display; no first occurrence is asserted. · 原文: `is an even permutation. Clearly this is an equivalence relation, the equivalence class of which is called an oriented $r$ -simplex. There are two equivalence classes, one consists of even permutations of $p_0, \ldots, p_r$ , the other of odd permutations. The equivalence class (oriented $r$ -simplex) which contains $\{p_0, \ldots, p_r\}$ is denoted by $\s...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 绝热过程(adiabatic process)

- UUID: `6de72ff5-d330-402f-bb23-d6debf12589d` · okf_url: https://limbo137.top/okf/concepts/6de72ff5-d330-402f-bb23-d6debf12589d.md
- labels: en = adiabatic process；zh-CN = 绝热过程
- aliases: adiabatic processes
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-1ae5a26d6c7478e25774-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 反对称部分(antisymmetric part)

- UUID: `6df918ba-11eb-471e-b457-1dbd53fe5c8b` · okf_url: https://limbo137.top/okf/concepts/6df918ba-11eb-471e-b457-1dbd53fe5c8b.md
- labels: en = antisymmetric part；zh-CN = 反对称部分
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.1 · `zh/chapter-08/01-r-vectors-and-r-forms#first-9c254421283ed4a52ea1-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.1 · `en/chapter-08/01-r-vectors-and-r-forms#first-9c254421283ed4a52ea1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/01-r-vectors-and-r-forms.md

## 谱(spectrum)

- UUID: `6e3eb528-17bf-469f-92c2-4bd54698e28d` · okf_url: https://limbo137.top/okf/concepts/6e3eb528-17bf-469f-92c2-4bd54698e28d.md
- labels: en = spectrum；zh-CN = 谱
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-270cf47adc2edc100559-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#concept-13-spectrum` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 测度(measure)

- UUID: `6ea19025-a4ce-4bee-961e-befcd91fd68a` · okf_url: https://limbo137.top/okf/concepts/6ea19025-a4ce-4bee-961e-befcd91fd68a.md
- labels: en = measure；zh-CN = 测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-efbb2ffcf5a8e414faea-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 可微映射(differentiable map)

- UUID: `6ed47c88-27f1-4afd-88e9-df76c8500ef4` · okf_url: https://limbo137.top/okf/concepts/6ed47c88-27f1-4afd-88e9-df76c8500ef4.md
- labels: en = differentiable map；zh-CN = 可微映射
- aliases: smooth map；光滑映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §2.1 · `books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory#first-1298f053f1001f2db6be-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §15.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `If $M$ and $N$ are differentiable manifolds, dimensions $m$ and $n$ respectively, then a map $\alpha : M \to N$ is differentiable at $p \in M$ if for any pair of coordinate charts $( U , \phi ; x ^ { i } )$ ) and $( V , \psi ; y ^ { a } )$ covering $p$ and $\alpha ( p )$ respectively, its coordinate representation $$ \hat {\alpha} = \psi \circ \alpha \cir...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md

## 双边理想(two-sided ideal)

- UUID: `6ee05280-22fc-4e93-a374-8fa890e7552e` · okf_url: https://limbo137.top/okf/concepts/6ee05280-22fc-4e93-a374-8fa890e7552e.md
- labels: en = two-sided ideal；zh-CN = 双边理想
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-two-sided-ideal` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-93ca38fb6cf93275c1ad-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 生成元(generator)

- UUID: `6eff4cba-09bf-4c84-8892-38a06c0c6495` · okf_url: https://limbo137.top/okf/concepts/6eff4cba-09bf-4c84-8892-38a06c0c6495.md
- labels: en = generator；zh-CN = 生成元
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-generator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 映射(mapping)

- UUID: `6f40b78d-cde2-42bd-a896-bdc671395837` · okf_url: https://limbo137.top/okf/concepts/6f40b78d-cde2-42bd-a896-bdc671395837.md
- labels: en = mapping；zh-CN = 映射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-12a65fd3a828e3c6545f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 算符多项式(operator polynomial)

- UUID: `6f5daf0f-3239-4be3-915a-185f62d60224` · okf_url: https://limbo137.top/okf/concepts/6f5daf0f-3239-4be3-915a-185f62d60224.md
- labels: en = operator polynomial；zh-CN = 算符多项式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-operator-polynomial` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 形变收缩核(deformation retract)

- UUID: `6fab2664-bace-5a85-bec4-233f4d75302e` · okf_url: https://limbo137.top/okf/concepts/6fab2664-bace-5a85-bec4-233f4d75302e.md
- labels: en = deformation retract；zh-CN = 形变收缩核
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.8. Let $R$ be a subspace of $X$ . If there exists a continuous map $H: X \times I \to X$ such that`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H (x, t) = x \quad \text {f o r a n y} x \in R \text {a n d a n y} t \in I. \tag {4.6} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `A retract is not necessarily a deformation retract. In figure 4.8, the circle $R$ is a retract of $X$ but not a deformation retract, since the hole in $X$ is an obstruction to continuous deformation of $\mathrm{id}_X$ to the retraction. Since $X$ and $R$ are of the same homotopy type, we have`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \pi_ {1} (X, a) \cong \pi_ {1} (R, a) \quad a \in R. \tag {4.7} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H (x, 0) = x \quad H (x, 1) \in R \quad \text {f o r a n y} x \in X \tag {4.5} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=149, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.8：形变收缩核。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The space $R$ is said to be a deformation retract of $X$ . Note that $H$ is a homotopy between $\mathrm{id}_X$ and a retraction $f: X \to R$ , which leaves all the points in $R$ fixed during deformation.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 可积分的李代数表示(integrable Lie algebra representation)

- UUID: `6fade234-ce94-46a5-9add-2908ca7b39e8` · okf_url: https://limbo137.top/okf/concepts/6fade234-ce94-46a5-9add-2908ca7b39e8.md
- labels: en = integrable Lie algebra representation；zh-CN = 可积分的李代数表示
- aliases: integrable
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-c01f981644cfc30081ca-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 双曲流形(hyperbolic manifold)

- UUID: `6fd2c3b2-de81-4166-9379-10cec9081f7d` · okf_url: https://limbo137.top/okf/concepts/6fd2c3b2-de81-4166-9379-10cec9081f7d.md
- labels: en = hyperbolic manifold；zh-CN = 双曲流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-35631418c6c79a278bc2-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-35631418c6c79a278bc2-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 格拉斯曼代数(Grassmann algebra)

- UUID: `702031d8-e347-4935-9f9b-7a2ddeabb765` · okf_url: https://limbo137.top/okf/concepts/702031d8-e347-4935-9f9b-7a2ddeabb765.md
- labels: en = Grassmann algebra；zh-CN = 格拉斯曼代数
- aliases: Grassmann代数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；schwartz-qft/local-experiment = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-grassmann-algebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-38aedf0733665fced58a-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §30.1 · `books/woit-qtgr/2025-10-20/en/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-a8e32a589399c4dc16a8-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §30.1 · `books/woit-qtgr/2025-10-20/zh/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators#first-a8e32a589399c4dc16a8-zh` · machine_draft)
- 已审定义(definitions):
  - schwartz-qft/local-experiment §— (pdf_page=163, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Schwartz 此处采用复系数；Woit 30.1 的定义采用实数域；不消去基域差别。 · 原文: `A Grassmann algebra is a set of objects $\mathcal{G}$ that are generated by a basis $\{\theta_i\}$ . These $\theta_i$ are Grassmann numbers, which anticommute with each other, $\theta_i\theta_j = -\theta_j\theta_i$ , add commutatively, $\theta_i + \theta_j = \theta_j + \theta_i$ , and can be multiplied by complex numbers, $a\theta \in \mathcal{G}$ for $\t...`
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-30/01-the-grassmann-algebra-of-polynomials-on-anticommuting-generators.md

## 瞬子数(instanton number)

- UUID: `70415427-4cfe-5638-9aa3-16dde8b6ef27` · okf_url: https://limbo137.top/okf/concepts/70415427-4cfe-5638-9aa3-16dde8b6ef27.md
- labels: en = instanton number；zh-CN = 瞬子数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §12.6.2 (pdf_page=491, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.6.2例12.6式12.90及后段：右端表示k∈π3(SU(2))=Z；起句指定丛仅为条件，不把整丛对象当作整数。保留同书第二陈述，不替换11.55。 · 原文: `The RHS represents the instanton number $k \in \pi_3(\mathrm{SU}(2)) = \mathbb{Z}$ . Note that $k > 0$ if $\mathcal{F} = *\mathcal{F}$ while $k 0$ ( $k < 0$ ), see Jackiw and Rebbi (1977). For example, let $\mathcal{F}$ be self-dual. Suppose $\psi^{-} \in \ker D^{\dagger} = \ker DD^{\dagger}$ . From (12.80), we find that`
  - nakahara-geometry-topology-physics/second-edition-local §11.3.1 (pdf_page=452, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara11.3.1式11.55：按该段SU(2)与S4条件的瞬子数积分表达。无作者Definition框；管理员核对的连续正文/公式单元。 · 原文: `$$ \frac {1}{2} \int_ {S ^ {4}} \operatorname {t r} \left(\frac {\mathrm {i} \mathcal {F}}{2 \pi}\right) ^ {2} = \int_ {S ^ {4}} \operatorname {c h} _ {2} (\mathcal {F}). \tag {11.55} $$`
  - nakahara-geometry-topology-physics/second-edition-local §12.6.2 (pdf_page=491, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.6.2例12.6式12.90及后段：右端表示k∈π3(SU(2))=Z；起句指定丛仅为条件，不把整丛对象当作整数。保留同书第二陈述，不替换11.55。 · 原文: `$$ \nu_ {+} - \nu_ {-} = \int_ {S ^ {4}} \operatorname {c h} _ {2} (\mathcal {F}) = \frac {- 1}{8 \pi^ {2}} \int_ {S ^ {4}} \operatorname {t r} \mathcal {F} ^ {2}. \tag {12.90} $$`
  - nakahara-geometry-topology-physics/second-edition-local §11.3.1 (pdf_page=452, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara11.3.1式11.55：按该段SU(2)与S4条件的瞬子数积分表达。无作者Definition框；管理员核对的连续正文/公式单元。 · 原文: `$\mathrm{Ch}(\mathcal{F})$ terminates at $\mathrm{ch}_2(\mathcal{F})$ since $\mathcal{F}^n = 0$ for $n \geq 3$ . Moreover, $\operatorname{tr}\mathcal{F} = 0$ for $G = \mathrm{SU}(2)$ , $n \geq 2$ . As we found in section 10.5, the instanton number is given by`
  - nakahara-geometry-topology-physics/second-edition-local §12.6.2 (pdf_page=491, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara12.6.2例12.6式12.90及后段：右端表示k∈π3(SU(2))=Z；起句指定丛仅为条件，不把整丛对象当作整数。保留同书第二陈述，不替换11.55。 · 原文: `Let $P(S^4, \mathrm{SU}(2))$ be the instanton bundle. Expression (12.88) reads as`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 奇异分布(singular distribution)

- UUID: `704f26e9-640a-4a8e-bdd2-2b6d97745986` · okf_url: https://limbo137.top/okf/concepts/704f26e9-640a-4a8e-bdd2-2b6d97745986.md
- labels: en = singular distribution；zh-CN = 奇异分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-dad6ecca9ad2131a8c8e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 费米子振子(fermionic oscillator)

- UUID: `705ab9c3-10fa-403c-8827-2b2f9368f751` · okf_url: https://limbo137.top/okf/concepts/705ab9c3-10fa-403c-8827-2b2f9368f751.md
- labels: en = fermionic oscillator；zh-CN = 费米子振子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §27 · `books/woit-qtgr/2025-10-20/zh/chapter-27/index#first-732d4f173599fc50341e-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §27 · `books/woit-qtgr/2025-10-20/en/chapter-27/index#first-732d4f173599fc50341e-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/index.md

## 有限集(finite set)

- UUID: `70a8b5c8-179c-4828-a2b2-c579b0ae5301` · okf_url: https://limbo137.top/okf/concepts/70a8b5c8-179c-4828-a2b2-c579b0ae5301.md
- labels: en = finite set；zh-CN = 有限集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-92b93ff7172cec8cebc5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 绝对连续的(absolutely continuous)

- UUID: `70bc7e53-c23f-4dca-8a30-8d362bc9f662` · okf_url: https://limbo137.top/okf/concepts/70bc7e53-c23f-4dca-8a30-8d362bc9f662.md
- labels: en = absolutely continuous；zh-CN = 绝对连续的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#concept-13-absolutely-continuous` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-3b76bb082e1f7ce209aa-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 延拓(extension)

- UUID: `710d8b62-86d3-409e-854a-dd35ef06584b` · okf_url: https://limbo137.top/okf/concepts/710d8b62-86d3-409e-854a-dd35ef06584b.md
- labels: en = extension；zh-CN = 延拓
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-36b2f33676913a26c582-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-extension` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 奇偶性(parity)

- UUID: `71a38b05-9dff-4ccf-a356-9a80cff9b371` · okf_url: https://limbo137.top/okf/concepts/71a38b05-9dff-4ccf-a356-9a80cff9b371.md
- labels: en = parity；zh-CN = 奇偶性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-parity` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-9e3d14610821c354a6a6-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 内部对称性(internal symmetry)

- UUID: `71daf83d-6115-4bde-94bf-237daf170c67` · okf_url: https://limbo137.top/okf/concepts/71daf83d-6115-4bde-94bf-237daf170c67.md
- labels: en = internal symmetry；zh-CN = 内部对称性
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §38.2 · `books/woit-qtgr/2025-10-20/zh/chapter-38/02-internal-symmetries#first-4451a09d81003ddd9c03-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §38.2 · `books/woit-qtgr/2025-10-20/en/chapter-38/02-internal-symmetries#first-4451a09d81003ddd9c03-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/02-internal-symmetries.md

## 切向量(tangent vector)

- UUID: `7206b9e2-70c8-49da-ac09-4d862fa7a1c8` · okf_url: https://limbo137.top/okf/concepts/7206b9e2-70c8-49da-ac09-4d862fa7a1c8.md
- labels: en = tangent vector；zh-CN = 切向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-2c7cfc2a31e723f93edb-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-tangent-vector` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 右理想(right ideal)

- UUID: `72d78a90-b76f-446c-b447-6704c9f5842c` · okf_url: https://limbo137.top/okf/concepts/72d78a90-b76f-446c-b447-6704c9f5842c.md
- labels: en = right ideal；zh-CN = 右理想
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-right-ideal` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-7bbb0ab0da383aa5d473-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 乘积(product)

- UUID: `72d9a1a7-0fe9-42c3-b382-6505d27a9f1b` · okf_url: https://limbo137.top/okf/concepts/72d9a1a7-0fe9-42c3-b382-6505d27a9f1b.md
- labels: en = product；zh-CN = 乘积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 算符的逆(inverse operator)

- UUID: `736457e1-a3e7-4632-a8ee-b461ed484442` · okf_url: https://limbo137.top/okf/concepts/736457e1-a3e7-4632-a8ee-b461ed484442.md
- labels: en = inverse operator；zh-CN = 算符的逆
- aliases: inverse；逆
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-inverse` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-3facc1bc9a0d0163970d-operator-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 三角不等式(Triangle inequality)

- UUID: `737defc6-6789-4cdd-87f7-63952cdef118` · okf_url: https://limbo137.top/okf/concepts/737defc6-6789-4cdd-87f7-63952cdef118.md
- labels: en = Triangle inequality；zh-CN = 三角不等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#first-1cc72a03538ddc6bda1d-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §5.2 · `en/chapter-05/02-complex-inner-product-spaces#first-1cc72a03538ddc6bda1d-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 无限集(infinite set)

- UUID: `739df8ae-bbd0-4c6d-9e4c-4211b47052b4` · okf_url: https://limbo137.top/okf/concepts/739df8ae-bbd0-4c6d-9e4c-4211b47052b4.md
- labels: en = infinite set；zh-CN = 无限集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-10f9d11ac2331c8b32b0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 拓扑积(topological product)

- UUID: `73c98f07-9d4e-4cc9-bae7-adbe213355fe` · okf_url: https://limbo137.top/okf/concepts/73c98f07-9d4e-4cc9-bae7-adbe213355fe.md
- labels: en = topological product；zh-CN = 拓扑积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-f447e5ade7d3cab2b68d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 外尔旋量(Weyl spinor)

- UUID: `73db7415-b73a-58f5-b334-c965edebfb5b` · okf_url: https://limbo137.top/okf/concepts/73db7415-b73a-58f5-b334-c965edebfb5b.md
- labels: en = Weyl spinor；zh-CN = 外尔旋量
- aliases: Weyl spinors；Weyl旋量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §9.4.2 (pdf_page=390, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara9.4.2例9.12：同段明确Weyl旋量是所述SL(2,C)、C²纤维丛的截面。此段尾句转入Dirac旋量；只以其中完整Weyl句作为本身份定义，不把后续Dirac等式归给Weyl。无独立作者Definition编号，不声明首次定义。 · 原文: `The universal covering group of $\mathrm{O}_{\uparrow}^{+}(3,1)$ is $\operatorname {SL}(2,\mathbb{C})$ , see example $5.16(c)$ . The homomorphism $\varphi :\operatorname {SL}(2,\mathbb{C})\to \operatorname{O}_{\uparrow}^{+}(3,1)$ is a $2:1$ map with $\ker \varphi = \{I_2, - I_2\}$ The Weyl spinor is a section of the fibre bundle $(W,\pi ,M,\mathbb{C}^2,\o...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 海森堡李代数，$d=1$(Heisenberg Lie algebra, $d=1$)

- UUID: `7403436a-1d4b-471f-8d7c-af1c5f055118` · okf_url: https://limbo137.top/okf/concepts/7403436a-1d4b-471f-8d7c-af1c5f055118.md
- labels: en = Heisenberg Lie algebra, $d=1$；zh-CN = 海森堡李代数，$d=1$
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.1 · `books/woit-qtgr/2025-10-20/zh/chapter-13/01-the-heisenberg-lie-algebra#first-9645a443334467e77afe-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.1 · `books/woit-qtgr/2025-10-20/en/chapter-13/01-the-heisenberg-lie-algebra#first-ea0bde96372f0ffe655b-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡李代数 $\mathfrak h_3$ 是向量空间 $\mathbf R^3$，其李括号在基 $(X,Y,Z)$ 上的取值规定为 $$ [X,Y]=Z,\qquad [X,Z]=[Y,Z]=0 $$`
  - woit-qtgr/2025-10-20 §13.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Heisenberg Lie algebra $\mathfrak h_3$ is the vector space $\mathbf R^3$ whose Lie bracket is specified on the basis $(X,Y,Z)$ by $$ [X,Y]=Z,\qquad [X,Z]=[Y,Z]=0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/01-the-heisenberg-lie-algebra.md

## 算符值分布(operator-valued distributions)

- UUID: `7466210b-1894-4b14-aa37-e5f27f8f3350` · okf_url: https://limbo137.top/okf/concepts/7466210b-1894-4b14-aa37-e5f27f8f3350.md
- labels: en = operator-valued distributions；zh-CN = 算符值分布
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.3 · `books/woit-qtgr/2025-10-20/en/chapter-36/05-continuum-formalism#first-904896b3439cef9aacbd-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §36.3 · `books/woit-qtgr/2025-10-20/zh/chapter-36/05-continuum-formalism#first-904896b3439cef9aacbd-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/05-continuum-formalism.md

## Groenewold–van Hove不可量子化定理(Groenewold–van Hove no-go theorem)

- UUID: `74ae58d7-65e0-4725-947a-20d3cf498508` · okf_url: https://limbo137.top/okf/concepts/74ae58d7-65e0-4725-947a-20d3cf498508.md
- labels: en = Groenewold–van Hove no-go theorem；zh-CN = Groenewold–van Hove不可量子化定理
- aliases: Groenewold-van Hove；Groenewold–van Hove
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §17.2 · `books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem#first-23a9feb733de92986fe8-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §17.2 · `books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem#first-ca01412bfa07aab6251e-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-17/02-the-groenewold-van-hove-no-go-theorem.md

## 施瓦西半径(Schwarzschild radius)

- UUID: `74e9903d-8775-4f72-a82f-ae88bbbb06ed` · okf_url: https://limbo137.top/okf/concepts/74e9903d-8775-4f72-a82f-ae88bbbb06ed.md
- labels: en = Schwarzschild radius；zh-CN = 施瓦西半径
- aliases: 史瓦西半径；施瓦氏半径
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-d6e56feba7698161e74b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-d6e56feba7698161e74b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 理想流体(perfect fluid)

- UUID: `758889b2-aea5-470b-8b16-c5d25f2b6ba3` · okf_url: https://limbo137.top/okf/concepts/758889b2-aea5-470b-8b16-c5d25f2b6ba3.md
- labels: en = perfect fluid；zh-CN = 理想流体
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.5 · `en/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-88931996366bc6c27969-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.5 · `zh/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-88931996366bc6c27969-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/05-conservation-laws-and-energy-stress-tensors.md

## 局部连通(locally connected)

- UUID: `75b2c385-ada6-413b-bae4-727c2bb7d894` · okf_url: https://limbo137.top/okf/concepts/75b2c385-ada6-413b-bae4-727c2bb7d894.md
- labels: en = locally connected；zh-CN = 局部连通
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-68c618e1b9a820c1ceee-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 非正常正交矩阵(improper orthogonal matrix)

- UUID: `75bc1017-57ee-403b-b328-c878447ef56e` · okf_url: https://limbo137.top/okf/concepts/75bc1017-57ee-403b-b328-c878447ef56e.md
- labels: en = improper orthogonal matrix；zh-CN = 非正常正交矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-improper-orthogonal-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 实射影空间(real projective space)

- UUID: `75e9bd12-9b58-4080-8f6c-d42cdf594c77` · okf_url: https://limbo137.top/okf/concepts/75e9bd12-9b58-4080-8f6c-d42cdf594c77.md
- labels: en = real projective space；zh-CN = 实射影空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-8126cdda28079489caa9-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-6cbaf7163cfb07bdaad4-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-6cbaf7163cfb07bdaad4-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 时间规范(Temporal gauge)

- UUID: `7603b0fb-4649-4039-b83c-aeb2c4c52af9` · okf_url: https://limbo137.top/okf/concepts/7603b0fb-4649-4039-b83c-aeb2c4c52af9.md
- labels: en = Temporal gauge；zh-CN = 时间规范
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §46.2 · `books/woit-qtgr/2025-10-20/en/chapter-46/02-the-hamiltonian-formalism-for-electromagnetic-fields#first-69afab55e802c9f6f688-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §46.2 · `books/woit-qtgr/2025-10-20/zh/chapter-46/02-the-hamiltonian-formalism-for-electromagnetic-fields#first-30df64ea597e8b97f6bb-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §46.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若 $A _ { 0 } = 0$，则称矢势 $A _ { \mu }$ 处于时间规范。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-46/02-the-hamiltonian-formalism-for-electromagnetic-fields.md

## 迹(trace)

- UUID: `761461ce-048a-4c1f-8316-6b32c6043b81` · okf_url: https://limbo137.top/okf/concepts/761461ce-048a-4c1f-8316-6b32c6043b81.md
- labels: en = trace；zh-CN = 迹
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-trace` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 整数加法群(additive group of integers)

- UUID: `764ff34c-80fe-4a25-87b4-e031da4b19a7` · okf_url: https://limbo137.top/okf/concepts/764ff34c-80fe-4a25-87b4-e031da4b19a7.md
- labels: en = additive group of integers；zh-CN = 整数加法群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-additive-group-of-integers` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 降指标(lowering the index)

- UUID: `76bebde3-401f-40cd-bc29-599e3ca0eee0` · okf_url: https://limbo137.top/okf/concepts/76bebde3-401f-40cd-bc29-599e3ca0eee0.md
- labels: en = lowering the index；zh-CN = 降指标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-55f3c6c21080548a7c5b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-55f3c6c21080548a7c5b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 极大元(maximal element)

- UUID: `76dbfbbc-79a6-4b5a-88d3-c31378d27611` · okf_url: https://limbo137.top/okf/concepts/76dbfbbc-79a6-4b5a-88d3-c31378d27611.md
- labels: en = maximal element；zh-CN = 极大元
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-67f091608ab977a826ea-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 特征值(eigenvalue)

- UUID: `76dd2837-afb9-4545-a9b3-37d92e61b023` · okf_url: https://limbo137.top/okf/concepts/76dd2837-afb9-4545-a9b3-37d92e61b023.md
- labels: en = eigenvalue；zh-CN = 特征值
- aliases: eigenvalues；本征值
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-eigenvalue` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-c81046531e55630b809e-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-db508acb9e0ea7c4aab8-zh` · translation)；woit-qtgr/2025-10-20 (en · §7.1 · `books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation#concept-7-1-eigenvalue` · machine_draft)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §13.5 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 非零向量u满足Au=αu；保留算符条件。 · 原文: `As in Chapter 4 a complex number $\alpha$ is an eigenvalue of a bounded linear operator $A$ : $\mathcal { H } \to \mathcal { H }$ if there exists a non-zero vector $u \in \mathcal { H }$ such that $$ A u = \alpha u. $$ $u$ is called the eigenvector of $A$ corresponding to the eigenvalue $\alpha$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 单位元(identity element)

- UUID: `770e9a05-d59d-49d2-9fae-012d9ba3edec` · okf_url: https://limbo137.top/okf/concepts/770e9a05-d59d-49d2-9fae-012d9ba3edec.md
- labels: en = identity element；zh-CN = 单位元
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-fd5d3b063fbc95041194-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 分布(distribution)

- UUID: `7731c967-eca6-43a2-a75d-2070d953614c` · okf_url: https://limbo137.top/okf/concepts/7731c967-eca6-43a2-a75d-2070d953614c.md
- labels: en = distribution；zh-CN = 分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-89a4a784510541d6735a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.6 · `en/chapter-15/06-distributions-and-frobenius-theorem#first-89a4a784510541d6735a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/06-distributions-and-frobenius-theorem.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 占据数(occupation number)

- UUID: `77a3a645-7c60-4692-921e-fa07243c8e43` · okf_url: https://limbo137.top/okf/concepts/77a3a645-7c60-4692-921e-fa07243c8e43.md
- labels: en = occupation number；zh-CN = 占据数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §36.1 · `books/woit-qtgr/2025-10-20/zh/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-6593d51008ae846285f9-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §36.1 · `books/woit-qtgr/2025-10-20/en/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-6593d51008ae846285f9-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 诱导拓扑(induced topology)

- UUID: `77a824d6-26bb-46ec-826c-b7c60478bfea` · okf_url: https://limbo137.top/okf/concepts/77a824d6-26bb-46ec-826c-b7c60478bfea.md
- labels: en = induced topology；zh-CN = 诱导拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-2497c3390c3ef08317a7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 覆盖(covering)

- UUID: `77d78323-eba1-47e1-ada3-16aa434adb16` · okf_url: https://limbo137.top/okf/concepts/77d78323-eba1-47e1-ada3-16aa434adb16.md
- labels: en = covering；zh-CN = 覆盖
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-84ecf433fadcf1b3b273-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 可逆过程(reversible process)

- UUID: `77ded77c-43ae-4c1e-a919-438378bd2042` · okf_url: https://limbo137.top/okf/concepts/77ded77c-43ae-4c1e-a919-438378bd2042.md
- labels: en = reversible process；zh-CN = 可逆过程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-7aea49360df729a2f58c-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-7aea49360df729a2f58c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 闵可夫斯基空间(Minkowski space)

- UUID: `782e15ba-ddb5-4ce8-b352-fc6e31428606` · okf_url: https://limbo137.top/okf/concepts/782e15ba-ddb5-4ce8-b352-fc6e31428606.md
- labels: en = Minkowski space；zh-CN = 闵可夫斯基空间
- aliases: affine Minkowski spacetime；闵可夫斯基仿射时空；闵可夫斯基时空
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-9d46750427b2ec5edde3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-minkowski-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## Bogoliubov变换(Bogoliubov transformations)

- UUID: `78e418b2-6084-4135-979b-115fe4364239` · okf_url: https://limbo137.top/okf/concepts/78e418b2-6084-4135-979b-115fe4364239.md
- labels: en = Bogoliubov transformations；zh-CN = Bogoliubov变换
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §39.1 · `books/woit-qtgr/2025-10-20/en/chapter-39/01-inequivalent-irreducible-representations#first-69f33431470f559279d3-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §39.1 · `books/woit-qtgr/2025-10-20/zh/chapter-39/01-inequivalent-irreducible-representations#first-69f33431470f559279d3-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/01-inequivalent-irreducible-representations.md

## 量子信息论(quantum information theory)

- UUID: `794a1dd4-a2a7-4f6d-9697-5fa32e00a8b4` · okf_url: https://limbo137.top/okf/concepts/794a1dd4-a2a7-4f6d-9697-5fa32e00a8b4.md
- labels: en = quantum information theory；zh-CN = 量子信息论
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-560e7d52f60ee5689df7-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.1 · `books/woit-qtgr/2025-10-20/en/chapter-03/01-the-two-state-quantum-system#first-560e7d52f60ee5689df7-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/04-for-further-reading.md

## 坐标立方区域(coordinate cubical region)

- UUID: `79a53fad-e8c8-467b-a88d-cf89e34cad2c` · okf_url: https://limbo137.top/okf/concepts/79a53fad-e8c8-467b-a88d-cf89e34cad2c.md
- labels: en = coordinate cubical region；zh-CN = 坐标立方区域
- aliases: cubical
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-dd07603497d0997588fd-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 克罗内克符号(Kronecker delta)

- UUID: `79adeeff-76cb-4c6b-a103-039043f484ac` · okf_url: https://limbo137.top/okf/concepts/79adeeff-76cb-4c6b-a103-039043f484ac.md
- labels: en = Kronecker delta；zh-CN = 克罗内克符号
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-kronecker-delta` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 逆变变换律(contravariant)

- UUID: `79d0e08a-17dd-4d77-a721-5b936ec5fc47` · okf_url: https://limbo137.top/okf/concepts/79d0e08a-17dd-4d77-a721-5b936ec5fc47.md
- labels: en = contravariant；zh-CN = 逆变变换律
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-contravariant` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 外测度(outer measure)

- UUID: `79d21424-1731-4347-b63d-0f59a181f111` · okf_url: https://limbo137.top/okf/concepts/79d21424-1731-4347-b63d-0f59a181f111.md
- labels: en = outer measure；zh-CN = 外测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-ce2a8d64eb71f55e39b7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 空集(empty set)

- UUID: `7a17e151-db88-404d-90a2-bc81151a1ad4` · okf_url: https://limbo137.top/okf/concepts/7a17e151-db88-404d-90a2-bc81151a1ad4.md
- labels: en = empty set；zh-CN = 空集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-bd987a52b36d75f7d2ff-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 辛流形(symplectic manifold)

- UUID: `7a317edb-3bbe-49f4-b908-1db5e9edf4bc` · okf_url: https://limbo137.top/okf/concepts/7a317edb-3bbe-49f4-b908-1db5e9edf4bc.md
- labels: en = symplectic manifold；zh-CN = 辛流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-658e3cca18d5c3d420ca-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-658e3cca18d5c3d420ca-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 特殊正交群(special orthogonal group)

- UUID: `7a686007-0c2d-47df-a759-19577bea6f55` · okf_url: https://limbo137.top/okf/concepts/7a686007-0c2d-47df-a759-19577bea6f55.md
- labels: en = special orthogonal group；zh-CN = 特殊正交群
- aliases: proper orthogonal group；rotation group；旋转群；正常正交群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-proper-orthogonal-group` · translation)；woit-qtgr/2025-10-20 (en · §2.2 · `books/woit-qtgr/2025-10-20/en/chapter-02/02-the-group-u-1-and-its-representations#first-387cdcc61313734cef62-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §2.2 · `books/woit-qtgr/2025-10-20/zh/chapter-02/02-the-group-u-1-and-its-representations#first-387cdcc61313734cef62-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §2.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Hence $(\det \mathsf{A})^2=1$ by Eq. (2.7) and the result $\det \mathsf{A}=\pm1$ follows at once. The orthogonal matrices with determinant 1 are called proper orthogonal matrices, while those with determinant −1 are called improper. The proper orthogonal matrices, denoted $S O ( n )$ form a group themselves called the proper orthogonal group of order $n$....`
  - woit-qtgr/2025-10-20 §5.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `- The special orthogonal group $S O ( n ) \subset S L ( n , \mathbf { R } )$ is the subgroup of $O ( n )$ with determinant one. It has the same Lie algebra as $O ( n ) \colon { \mathfrak { s o } } ( n ) = { \mathfrak { o } } ( n )$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 流形边界(boundary of a manifold)

- UUID: `7a875e1f-1834-4097-a32d-6f6e901c812e` · okf_url: https://limbo137.top/okf/concepts/7a875e1f-1834-4097-a32d-6f6e901c812e.md
- labels: en = boundary of a manifold；zh-CN = 流形边界
- aliases: boundary；manifold boundary；边界
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=192, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 带边流形的边界：半空间坐标下最后一坐标为0的点；不同于任意子集的闭包减内部，也不同于链群中∂的像。 · 原文: `Before we give examples, we briefly comment on manifolds with boundaries. So far, we have assumed that the coordinate neighbourhood $U_{i}$ is homeomorphic to an open set of $\mathbb{R}^m$ . In some applications, however, this turns out to be too restrictive and we need to relax this condition. If a topological space $M$ is covered by a family of open set...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 密度(density)

- UUID: `7ac7a4ad-6823-4306-900a-903e3b8e9a27` · okf_url: https://limbo137.top/okf/concepts/7ac7a4ad-6823-4306-900a-903e3b8e9a27.md
- labels: en = density；zh-CN = 密度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-c7cbef42a43b3c941907-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 单元素集(singleton)

- UUID: `7b1b9076-23b8-4ff7-9a71-51931143d125` · okf_url: https://limbo137.top/okf/concepts/7b1b9076-23b8-4ff7-9a71-51931143d125.md
- labels: en = singleton；zh-CN = 单元素集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-b9155981741ffa4049cf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 欧拉–拉格朗日方程(Euler–Lagrange equations)

- UUID: `7b1bcc34-9228-486b-b886-65ee155c86a8` · okf_url: https://limbo137.top/okf/concepts/7b1bcc34-9228-486b-b886-65ee155c86a8.md
- labels: en = Euler–Lagrange equations；zh-CN = 欧拉–拉格朗日方程
- aliases: Euler-Lagrange equation；Euler-Lagrange equations；Euler–Lagrange equation；Euler–Lagrange方程；欧拉-拉格朗日方程
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-e89c7b36b3166d5cd065-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-e89c7b36b3166d5cd065-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-29b663e8c58bcdfd7f29-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-30f11892c0aff6fa7c15-zh` · machine_draft)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §16.5 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 经典力学轨迹q(t)的变分方程（原书式16.25）；不同于场论场方程。 · 原文: `$$ \frac {\partial L}{\partial q ^ {i}} - \frac {\mathrm{d}}{\mathrm{d} t} \left(\frac {\partial L}{\partial \dot {q} ^ {i}}\right) = 0.\tag{16.25} $$ These are known as the Euler–Lagrange equations.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 特征标群(Character group)

- UUID: `7b6409db-7dda-4ee3-8d42-80d4ba31d7de` · okf_url: https://limbo137.top/okf/concepts/7b6409db-7dda-4ee3-8d42-80d4ba31d7de.md
- labels: en = Character group；zh-CN = 特征标群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §20.4 · `books/woit-qtgr/2025-10-20/zh/chapter-20/08-representations-of-nk-n-commutative#first-a644c34da1f1beef7d9e-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §20.4 · `books/woit-qtgr/2025-10-20/en/chapter-20/08-representations-of-nk-n-commutative#first-2cc3bac037e616c2cd40-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §20.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `For a commutative group $N$, let $\widehat{N}$ be its set of characters, that is, functions $\alpha:N\to\mathbf{C}$ satisfying the homomorphism property $$ \alpha(n_1n_2)=\alpha(n_1)\alpha(n_2). $$`
  - woit-qtgr/2025-10-20 §20.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `对于交换群 $N$，令 $\widehat { N }$ 为 $N$ 的特征标集合，即函数 $$ \alpha : N \to \mathbf {C} $$ 满足同态性质 $$ \alpha (n _ {1} n _ {2}) = \alpha (n _ {1}) \alpha (n _ {2}) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-20/08-representations-of-nk-n-commutative.md

## 预解算符(resolvent operator)

- UUID: `7b79ad30-be49-4698-9e54-3d2c3d6fc9c2` · okf_url: https://limbo137.top/okf/concepts/7b79ad30-be49-4698-9e54-3d2c3d6fc9c2.md
- labels: en = resolvent operator；zh-CN = 预解算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-resolvent-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-1464118d832a8f510746-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 里奇标量(Ricci scalar)

- UUID: `7b805cd9-a0c6-4280-9837-760dfece5b6a` · okf_url: https://limbo137.top/okf/concepts/7b805cd9-a0c6-4280-9837-760dfece5b6a.md
- labels: en = Ricci scalar；zh-CN = 里奇标量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.6 · `en/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-c4cd326b58cf9395e8b7-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.6 · `zh/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-c4cd326b58cf9395e8b7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/06-the-riemann-tensor-and-its-symmetries.md

## 第一可数(first countable)

- UUID: `7c674ad2-04ad-4bb5-8434-a4c7b804b02c` · okf_url: https://limbo137.top/okf/concepts/7c674ad2-04ad-4bb5-8434-a4c7b804b02c.md
- labels: en = first countable；zh-CN = 第一可数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-b03c683ecdf83435dd9b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 贝蒂数(Betti number)

- UUID: `7c7c4b86-63b1-451e-b0be-15f02ebca981` · okf_url: https://limbo137.top/okf/concepts/7c7c4b86-63b1-451e-b0be-15f02ebca981.md
- labels: en = Betti number；zh-CN = 贝蒂数
- aliases: Betti数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-fc63cc448792993af149-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-fc63cc448792993af149-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §33.3 · `books/woit-qtgr/2025-10-20/zh/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-9c69bd98843f133f636e-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §33.3 · `books/woit-qtgr/2025-10-20/en/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-9c69bd98843f133f636e-en` · machine_draft)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For example, the Betti numbers of the torus $T^2$ are (see example 3.12)`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ b _ {0} (K) = 1, \qquad b _ {1} (K) = 0, \qquad b _ {2} (K) = 1. $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The following theorem relates the Euler characteristic to the Betti numbers.`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `and those of the sphere $S^2$ are (exercise 3.3)`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.6. Let $K$ be a simplicial complex. The $r$ th Betti number $b_{r}(K)$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `In other words, $b_{r}(K)$ is the rank of the free Abelian part of $H_{r}(K; \mathbb{Z})$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ b _ {0} (K) = 1, \qquad b _ {1} (K) = 2, \qquad b _ {2} (K) = 1 $$`
  - nakahara-geometry-topology-physics/second-edition-local §— (pdf_page=137, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 不得把整数同调的挠部分计入维数。 · 原文: `Definition 3.6. Let $K$ be a simplicial complex. The $r$ th Betti number $b_{r}(K)$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §3.4.3 (pdf_page=137, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.6：贝蒂数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ b _ {r} (K) \equiv \dim H _ {r} (K; \mathbb {R}). \tag {3.56} $$`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 实系数同调空间。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms.md

## 单参数子群(one-parameter subgroup)

- UUID: `7c7d436a-933b-40b7-9fbb-45f0ff035eec` · okf_url: https://limbo137.top/okf/concepts/7c7d436a-933b-40b7-9fbb-45f0ff035eec.md
- labels: en = one-parameter subgroup；zh-CN = 单参数子群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-one-parameter-subgroup` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-b678679c2e7f94246618-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.3 (pdf_page=231, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.12：单参数子群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \phi (t) \phi (s) = \phi (t + s). \tag {5.117} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.3 (pdf_page=231, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.12：单参数子群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.12. A curve $\phi : \mathbb{R} \to G$ is called a one-parameter subgroup of $G$ if it satisfies the condition`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/02-the-exponential-map.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 有限生成阿贝尔群基本定理(fundamental theorem of finitely generated Abelian groups)

- UUID: `7cdfbe87-046b-458c-b6a3-e85e08dab959` · okf_url: https://limbo137.top/okf/concepts/7cdfbe87-046b-458c-b6a3-e85e08dab959.md
- labels: en = fundamental theorem of finitely generated Abelian groups；zh-CN = 有限生成阿贝尔群基本定理
- aliases: Fundamental theorem of finitely generated Abelian groups
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.3 (pdf_page=116, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.2及式3.10，有限生成阿贝尔群分解为无限及有限循环群直和；完整原文假设照录，不补造额外作者条件。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ G \cong \underbrace {\mathbb {Z} \oplus \cdots \oplus \mathbb {Z}} _ {r} \oplus \mathbb {Z} _ {k _ {1}} \oplus \dots \oplus \mathbb {Z} _ {k _ {p}} \tag {3.10} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.1.3 (pdf_page=116, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.2及式3.10，有限生成阿贝尔群分解为无限及有限循环群直和；完整原文假设照录，不补造额外作者条件。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Theorem 3.2. (Fundamental theorem of finitely generated Abelian groups) Let $G$ be a finitely generated Abelian group (not necessarily free) with $m$ generators. Then $G$ is isomorphic to the direct sum of cyclic groups,`
  - nakahara-geometry-topology-physics/second-edition-local §3.1.3 (pdf_page=116, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.2及式3.10，有限生成阿贝尔群分解为无限及有限循环群直和；完整原文假设照录，不补造额外作者条件。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `where $m = r + p$ . The number $r$ is called the rank of $G$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 析取(disjunction)

- UUID: `7d173ecd-6ccd-485b-bbf8-a07c0afcb158` · okf_url: https://limbo137.top/okf/concepts/7d173ecd-6ccd-485b-bbf8-a07c0afcb158.md
- labels: en = disjunction；zh-CN = 析取
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-f0ecedaaf1eea2677a1e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 向量场的流与指数映射(Flow of a vector field and the exponential map)

- UUID: `7d6e05ef-910c-49a8-aebb-ba22dd99946c` · okf_url: https://limbo137.top/okf/concepts/7d6e05ef-910c-49a8-aebb-ba22dd99946c.md
- labels: en = Flow of a vector field and the exponential map；zh-CN = 向量场的流与指数映射
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §15.1 · `books/woit-qtgr/2025-10-20/en/chapter-15/01-vector-fields-and-the-exponential-map#first-6c37c0ee3dabd3ed9eb6-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §15.1 · `books/woit-qtgr/2025-10-20/zh/chapter-15/01-vector-fields-and-the-exponential-map#first-0fe44bd2f2272996f2fd-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §15.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `域 $M$ 上向量场 $X$ 的流是映射 $$ \Phi_ {X}: (t, m) \in \mathbf {R} \times M \to \Phi_ {X} (t, m) \in M $$ 满足 $$ \frac {d}{d t} \Phi_ {X} (t, m) = X (\Phi_ {X} (t, m)) $$ $$ \Phi_ {X} (0, m) = m $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-15/01-vector-fields-and-the-exponential-map.md

## 量子场算符(quantum field operator)

- UUID: `7d7401c6-00ca-4010-901d-09477338cb34` · okf_url: https://limbo137.top/okf/concepts/7d7401c6-00ca-4010-901d-09477338cb34.md
- labels: en = quantum field operator；zh-CN = 量子场算符
- aliases: Quantum fields
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §37.1 · `books/woit-qtgr/2025-10-20/en/chapter-37/01-quantum-field-operators#first-5a6ffa0a0b73cbe59b48-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/01-quantum-field-operators.md

## 拓扑学(topology)

- UUID: `7d7c7c97-dae3-401e-bdc7-6bafcd148048` · okf_url: https://limbo137.top/okf/concepts/7d7c7c97-dae3-401e-bdc7-6bafcd148048.md
- labels: en = topology；zh-CN = 拓扑学
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 复二阶特殊线性群(SL(2,C))

- UUID: `7dbfe73d-968a-4ac8-a84a-e865eecfd3aa` · okf_url: https://limbo137.top/okf/concepts/7dbfe73d-968a-4ac8-a84a-e865eecfd3aa.md
- labels: en = SL(2,C)；zh-CN = 复二阶特殊线性群
- aliases: $SL(2,\mathbb C)$
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §40.4 · `books/woit-qtgr/2025-10-20/en/chapter-40/04-spin-and-the-lorentz-group#first-e84ed108ff72d19c9bb4-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/04-spin-and-the-lorentz-group.md

## 关于测度可积的函数(integrable function)

- UUID: `7dc71b7b-0612-46b1-88a7-e838f7ee6134` · okf_url: https://limbo137.top/okf/concepts/7dc71b7b-0612-46b1-88a7-e838f7ee6134.md
- labels: en = integrable function；zh-CN = 关于测度可积的函数
- aliases: integrable；可积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-929d60427013a6d10773-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 惯性系(inertial frame)

- UUID: `7df51ab3-8816-42e0-8336-5007ee59ce22` · okf_url: https://limbo137.top/okf/concepts/7df51ab3-8816-42e0-8336-5007ee59ce22.md
- labels: en = inertial frame；zh-CN = 惯性系
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-8ba1fcabeeb90487a014-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-8ba1fcabeeb90487a014-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 结合代数(associative algebra)

- UUID: `7e2e6da6-b92e-45a5-b490-c11370b027b4` · okf_url: https://limbo137.top/okf/concepts/7e2e6da6-b92e-45a5-b490-c11370b027b4.md
- labels: en = associative algebra；zh-CN = 结合代数
- aliases: algebra；associative；代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-9f30c583afba7b647f72-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-associative-algebra` · translation)；woit-qtgr/2025-10-20 (zh-CN · §28.1 · `books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-33ec330b5f5555922192-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §28.1 · `books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-33ec330b5f5555922192-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/01-the-complex-weyl-and-cliford-algebras.md

## 自旋极化向量(spin polarization vectors)

- UUID: `7e32fa09-632f-47ec-a8be-e0c229e727da` · okf_url: https://limbo137.top/okf/concepts/7e32fa09-632f-47ec-a8be-e0c229e727da.md
- labels: en = spin polarization vectors；zh-CN = 自旋极化向量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §34.2 · `books/woit-qtgr/2025-10-20/zh/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-78139beaae7ab48617ac-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §34.2 · `books/woit-qtgr/2025-10-20/en/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-78139beaae7ab48617ac-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3.md

## 基(basis)

- UUID: `7e3b88ac-3d0c-45bd-82d8-090fd0c22b05` · okf_url: https://limbo137.top/okf/concepts/7e3b88ac-3d0c-45bd-82d8-090fd0c22b05.md
- labels: en = basis；zh-CN = 基
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-basis` · translation)；woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-basis` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §4.1 · `books/woit-qtgr/2025-10-20/zh/chapter-04/01-vector-spaces-and-linear-maps#first-3f7bad81430eee3c513f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/01-vector-spaces-and-linear-maps.md

## 半群(semigroup)

- UUID: `7e5d331b-ce3d-481b-9afe-f2111adc6389` · okf_url: https://limbo137.top/okf/concepts/7e5d331b-ce3d-481b-9afe-f2111adc6389.md
- labels: en = semigroup；zh-CN = 半群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-2e018671537daaf1de4d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 权与权空间(Weights and weight spaces)

- UUID: `7e89ae8a-6618-4d20-a433-6d3840e1eda5` · okf_url: https://limbo137.top/okf/concepts/7e89ae8a-6618-4d20-a433-6d3840e1eda5.md
- labels: en = Weights and weight spaces；zh-CN = 权与权空间
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §8.1 · `books/woit-qtgr/2025-10-20/en/chapter-08/01-representations-of-su2-classification#first-cc3d76ebb0d582d3c954-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §8.1 · `books/woit-qtgr/2025-10-20/zh/chapter-08/01-representations-of-su2-classification#first-eaf47a83c22ba3b7e6df-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `If $\pi ^ { \prime } ( S _ { 3 } )$ has eigenvalue $\frac { k } { 2 }$, then $k$ is a weight of the representation $(\pi,V)$.`
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若 $\pi ^ { \prime } ( S _ { 3 } )$ 的一个特征值为 $\frac { k } { 2 }$，则称 $k$ 为表示 $(\pi,V)$ 的一个权。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/01-representations-of-su2-classification.md

## 半旋量表示(half-spinor representation)

- UUID: `7e960f07-0efa-4408-a191-ffe41c9395fe` · okf_url: https://limbo137.top/okf/concepts/7e960f07-0efa-4408-a191-ffe41c9395fe.md
- labels: en = half-spinor representation；zh-CN = 半旋量表示
- aliases: half-spinor；半旋量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §31.3 · `books/woit-qtgr/2025-10-20/en/chapter-31/04-spinors-and-the-bargmann-fock-construction#first-a7597379e4449efd95e7-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §31.3 · `books/woit-qtgr/2025-10-20/zh/chapter-31/04-spinors-and-the-bargmann-fock-construction#first-a7597379e4449efd95e7-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/04-spinors-and-the-bargmann-fock-construction.md

## Hilbert–Schmidt算符(Hilbert–Schmidt operator)

- UUID: `7eb52d21-f6f5-4f7f-ad8d-69396d0a851c` · okf_url: https://limbo137.top/okf/concepts/7eb52d21-f6f5-4f7f-ad8d-69396d0a851c.md
- labels: en = Hilbert–Schmidt operator；zh-CN = Hilbert–Schmidt算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §39.2 · `books/woit-qtgr/2025-10-20/en/chapter-39/02-the-restricted-symplectic-group#first-304d5a2460b866d32213-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §39.2 · `books/woit-qtgr/2025-10-20/zh/chapter-39/02-the-restricted-symplectic-group#first-304d5a2460b866d32213-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/02-the-restricted-symplectic-group.md

## 自旋 1 表示(spin-1 representation)

- UUID: `7fc5999a-3153-495e-a9e4-6e2fa4b35466` · okf_url: https://limbo137.top/okf/concepts/7fc5999a-3153-495e-a9e4-6e2fa4b35466.md
- labels: en = spin-1 representation；zh-CN = 自旋 1 表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.3 · `books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary#first-f00ac3ec78b0a932b0c2-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.3 · `books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary#first-f00ac3ec78b0a932b0c2-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 相容复结构(compatible complex structure)

- UUID: `7fff60d8-e117-481c-81fb-4b5b167c4d31` · okf_url: https://limbo137.top/okf/concepts/7fff60d8-e117-481c-81fb-4b5b167c4d31.md
- labels: en = compatible complex structure；zh-CN = 相容复结构
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §26.2 · `books/woit-qtgr/2025-10-20/en/chapter-26/02-compatible-complex-structures-and-positivity#first-416fd553c53d9861e175-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26.2 · `books/woit-qtgr/2025-10-20/zh/chapter-26/02-compatible-complex-structures-and-positivity#first-416fd553c53d9861e175-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/02-compatible-complex-structures-and-positivity.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation.md

## 交集(intersection)

- UUID: `8055f095-5721-452c-a01e-c66adc0bb879` · okf_url: https://limbo137.top/okf/concepts/8055f095-5721-452c-a01e-c66adc0bb879.md
- labels: en = intersection；zh-CN = 交集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-eb2b6fd75047b2b316a3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 逆路径(inverse path)

- UUID: `80778e21-31dc-5f76-b183-bf17d37e6250` · okf_url: https://limbo137.top/okf/concepts/80778e21-31dc-5f76-b183-bf17d37e6250.md
- labels: en = inverse path；zh-CN = 逆路径
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.3：逆路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.3. Let $\alpha : I \to X$ be a path from $x_0$ to $x_1$ . The inverse path $\alpha^{-1}$ of $\alpha$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.3：逆路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \alpha^ {- 1} (s) \equiv \alpha (1 - s) \quad s \in I. \tag {4.2} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=142, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.3：逆路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `It seems that a constant map $c_{x}$ is the unit element. However, it is not: $\alpha * \alpha^{-1}$ is not equal to $c_{x}!$ . We need a concept of homotopy to define a group operation in the space of loops.`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.3：逆路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Since a loop is a special path for which the initial point and end point agree, the product of loops and the inverse of a loop are defined in exactly the same way.`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.3：逆路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `[The inverse path $\alpha^{-1}$ corresponds to traversing the image of $\alpha$ in the opposite direction from $x_{1}$ to $x_0$ .]`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 1-形式(one-form)

- UUID: `80bbdc4e-ab1e-4ceb-b83e-6bab3749f308` · okf_url: https://limbo137.top/okf/concepts/80bbdc4e-ab1e-4ceb-b83e-6bab3749f308.md
- labels: en = one-form；zh-CN = 1-形式
- aliases: 1-form；covector field；one-forms；1形式；一形式；一次微分形式；余向量场；微分1-形式
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-1addb5a5dab37a33237c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-covector-field` · translation)
- 已审定义(definitions):
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §2.3.1 (pdf_page=21, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 流形上的光滑余向量场，即余切丛的光滑截面/微分1-形式。与单点余向量分开；Tong §2.3.1与Szekeres §15.3明确同义。保留作者逐点配对的原文，不声称全书首次出现。 · 原文: `We can also construct fields of cotangent vectors, by picking a member of $T _ { p } ^ { * } ( M )$ for each point $p$ in a smooth manner. Such a cotangent field is better known as a one-form; they map vector fields to real numbers. The set of all one-forms on M is denoted $\Lambda ^ { 1 } ( M )$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 环路(loop)

- UUID: `8117bbb3-9a0c-4046-b36c-15823bdd5b29` · okf_url: https://limbo137.top/okf/concepts/8117bbb3-9a0c-4046-b36c-15823bdd5b29.md
- labels: en = loop；zh-CN = 环路
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.1：环路。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For $x \in X$ , a constant path $c_{x}: I \to X$ is defined by $c_{x}(s) = x, s \in I$ . A constant path is also a constant loop since $c_{x}(0) = c_{x}(1) = x$ . The set of paths or loops in a topological space $X$ may be endowed with an algebraic structure as follows.`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.1：环路。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.1. Let $X$ be a topological space and let $I = [0,1]$ . A continuous map $\alpha : I \to X$ is called a path with an initial point $x_0$ and an end point $x_1$ if $\alpha(0) = x_0$ and $\alpha(1) = x_1$ . If $\alpha(0) = \alpha(1) = x_0$ , the path is called a loop with base point $x_0$ (or a loop at $x_0$ ).`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 平凡表示(trivial representation)

- UUID: `811a9bef-7a24-4f51-a81e-f2924d9b3085` · okf_url: https://limbo137.top/okf/concepts/811a9bef-7a24-4f51-a81e-f2924d9b3085.md
- labels: en = trivial representation；zh-CN = 平凡表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-8c2a6cd4fa94ec5f9396-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 旋磁比(gyromagnetic ratio)

- UUID: `8138ecc5-6158-4786-afec-925599c67b25` · okf_url: https://limbo137.top/okf/concepts/8138ecc5-6158-4786-afec-925599c67b25.md
- labels: en = gyromagnetic ratio；zh-CN = 旋磁比
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.3 · `books/woit-qtgr/2025-10-20/zh/chapter-03/03-dynamics-of-a-two-state-system#first-e018b22fe6d6d650c1fe-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-e018b22fe6d6d650c1fe-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 坐标函数(coordinate function)

- UUID: `8152e9e4-6be6-4b9f-a913-969c920af277` · okf_url: https://limbo137.top/okf/concepts/8152e9e4-6be6-4b9f-a913-969c920af277.md
- labels: en = coordinate function；zh-CN = 坐标函数
- aliases: coordinate functions
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-coordinate-function` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-792fd75ea90223de3bac-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 并集(union)

- UUID: `815f9a8a-fbeb-4518-ae7a-cde1b6dc1631` · okf_url: https://limbo137.top/okf/concepts/815f9a8a-fbeb-4518-ae7a-cde1b6dc1631.md
- labels: en = union；zh-CN = 并集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-9e1d3fe5b00ed59443e5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 分次积(graded product)

- UUID: `81d20637-0d3a-44fb-baa4-abe77a94ade8` · okf_url: https://limbo137.top/okf/concepts/81d20637-0d3a-44fb-baa4-abe77a94ade8.md
- labels: en = graded product；zh-CN = 分次积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.3 · `en/chapter-08/03-exterior-product#first-61d7540ba9c702d1abd5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.3 · `zh/chapter-08/03-exterior-product#first-61d7540ba9c702d1abd5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md

## 同胚(homeomorphism)

- UUID: `8205a1b2-9f2c-420f-852b-e8664f601938` · okf_url: https://limbo137.top/okf/concepts/8205a1b2-9f2c-420f-852b-e8664f601938.md
- labels: en = homeomorphism；zh-CN = 同胚
- aliases: homeomorphic；homeomorphisms；同胚映射；同胚的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-7452ec2bf1b13a249ea6-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.4.1 (pdf_page=105, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.9：同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `In other words, $X_{1}$ is homeomorphic to $X_{2}$ if there exist maps $f: X_{1} \to X_{2}$ and $g: X_{2} \to X_{1}$ such that $f \circ g = \mathrm{id}_{X_{2}}$ , and $g \circ f = \mathrm{id}_{X_{1}}$ . It is easy to show that a homeomorphism is an equivalence relation. Reflectivity follows from the choice $f = \mathrm{id}_{X}$ , while symmetry follows si...`
  - nakahara-geometry-topology-physics/second-edition-local §2.4.1 (pdf_page=105, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.9：同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Figure 2.10 shows some examples of homeomorphisms. It seems impossible to deform the left figure in figure $2.10(b)$ into the right one by continuous deformation. However, this is an artefact of the embedding of these objects in $\mathbb{R}^3$ . In fact, they are continuously deformable in $\mathbb{R}^4$ , see problem 2.3. To distinguish one from the othe...`
  - nakahara-geometry-topology-physics/second-edition-local §2.4.1 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.9：同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.9. Let $X_{1}$ and $X_{2}$ be topological spaces. A map $f: X_{1} \to X_{2}$ is a homeomorphism if it is continuous and has an inverse $f^{-1}: X_{2} \to X_{1}$ which is`
  - nakahara-geometry-topology-physics/second-edition-local §2.4.1 (pdf_page=105, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.9：同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `also continuous. If there exists a homeomorphism between $X_{1}$ and $X_{2}$ , $X_{1}$ is said to be homeomorphic to $X_{2}$ and vice versa.`
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书逐字条件保留；不同书路径分别记录。 · 原文: `Let $X$ and $Y$ be two topological spaces. Theorem 10.1 motivates the following definition: a function $f : X \to Y$ is said to be continuous if the inverse image $f ^ { - 1 } ( U )$ of every open set $U$ in $Y$ is open in $X$. If $f$ is one-to-one and its inverse $f ^ { - 1 } : Y \to X$ is continuous, the function is called a homeomorphism and the topolo...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 连通空间(connected space)

- UUID: `82292311-3a67-40cb-98ef-cfcad48c9f28` · okf_url: https://limbo137.top/okf/concepts/82292311-3a67-40cb-98ef-cfcad48c9f28.md
- labels: en = connected space；zh-CN = 连通空间
- aliases: connected；connectedness；连通；连通性；连通的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.7 · `zh/chapter-10/07-connected-spaces#first-dea27de8694dddeed0ef-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。原书定义未明确排除空分解，原文缺陷保留；不擅加非空假设。 · 原文: `Definition 2.8. (a) A topological space $X$ is connected if it cannot be written as $X = X_{1} \cup X_{2}$ , where $X_{1}$ and $X_{2}$ are both open and $X_{1} \cap X_{2} = \emptyset$ . Otherwise $X$ is called disconnected.`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。原书定义未明确排除空分解，原文缺陷保留；不擅加非空假设。 · 原文: `(b) A topological space $X$ is called arcwise connected if, for any points $x, y \in X$ , there exists a continuous map $f: [0,1] \to X$ such that $f(0) = x$ and $f(1) = y$ . With a few pathological exceptions, arcwise connectedness is practically equivalent to connectedness. (c) A loop in a topological space $X$ is a continuous map $f: [0,1] \to X$ such ...`
  - szekeres/source-snapshot-bc3404156f6e §10.7 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书逐字条件保留；不同书路径分别记录。 · 原文: `Intuitively, we can think of a topological space $X$ as being ‘disconnected’ if it can be decomposed into two disjoint subsets $X = A \cup B$ without these sets having any boundary points in common. Since the boundary of a set is at the same time the boundary of the complement of that set, the only way such a decomposition can occur is if there exists a s...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/07-connected-spaces.md

## 主压力(principal pressure)

- UUID: `82330dcc-48f7-4c62-89e7-748d6cbf0849` · okf_url: https://limbo137.top/okf/concepts/82330dcc-48f7-4c62-89e7-748d6cbf0849.md
- labels: en = principal pressure；zh-CN = 主压力
- aliases: principal pressures
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.5 · `zh/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-1ec2d01950718b824cc6-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.5 · `en/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-0fd2d5978d50d4f97763-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/05-conservation-laws-and-energy-stress-tensors.md

## 矩映射反常(moment-map anomaly)

- UUID: `82788b92-e7e2-458d-aa00-d14e5448427a` · okf_url: https://limbo137.top/okf/concepts/82788b92-e7e2-458d-aa00-d14e5448427a.md
- labels: en = moment-map anomaly；zh-CN = 矩映射反常
- aliases: 反常
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §15.3 · `books/woit-qtgr/2025-10-20/zh/chapter-15/03-group-actions-on-m-and-the-moment-map#first-47bbbd204887dc38bdee-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-15/03-group-actions-on-m-and-the-moment-map.md

## 上界(upper bound)

- UUID: `831ad068-9b9d-44d5-bf37-2ad01936ba60` · okf_url: https://limbo137.top/okf/concepts/831ad068-9b9d-44d5-bf37-2ad01936ba60.md
- labels: en = upper bound；zh-CN = 上界
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-96bc35f3513f48ffce16-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 挠率张量(torsion tensor)

- UUID: `83b3b28d-e254-4344-acab-c55b1176cdcd` · okf_url: https://limbo137.top/okf/concepts/83b3b28d-e254-4344-acab-c55b1176cdcd.md
- labels: en = torsion tensor；zh-CN = 挠率张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-420b7873852d76b4348b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-420b7873852d76b4348b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 面(face)

- UUID: `83c6cbb2-d720-4dd5-8186-1ceb458dfe53` · okf_url: https://limbo137.top/okf/concepts/83c6cbb2-d720-4dd5-8186-1ceb458dfe53.md
- labels: en = face；zh-CN = 面
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-aaf267790520ced38388-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-aaf267790520ced38388-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 卡西米尔算符(Casimir operator)

- UUID: `83d350f3-74f0-404e-8258-e3c819f0b558` · okf_url: https://limbo137.top/okf/concepts/83d350f3-74f0-404e-8258-e3c819f0b558.md
- labels: en = Casimir operator；zh-CN = 卡西米尔算符
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §42.2 · `books/woit-qtgr/2025-10-20/zh/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-d8fc60a9c20c14b1427c-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §42.2 · `books/woit-qtgr/2025-10-20/en/chapter-42/02-irreducible-representations-of-the-poincar-e-group#first-9bba0179fd53343d3bb8-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §42.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `庞加莱群的卡西米尔（或第一卡西米尔）算符是算符 $$ P ^ {2} = - P _ {0} ^ {2} + P _ {1} ^ {2} + P _ {2} ^ {2} + P _ {3} ^ {2} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-42/02-irreducible-representations-of-the-poincar-e-group.md

## 转置变换(Transpose transformation)

- UUID: `842e3529-eaac-49b5-977c-78c9ea84570c` · okf_url: https://limbo137.top/okf/concepts/842e3529-eaac-49b5-977c-78c9ea84570c.md
- labels: en = Transpose transformation；zh-CN = 转置变换
- aliases: transpose transformation
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.2 · `books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces#first-d494d1429558b73ddb9f-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-d494d1429558b73ddb9f-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Transpose transformation). The transpose of $L$ is the linear transformation $$ L ^ {t}: V ^ {*} \to V ^ {*} $$ given by $$ (L ^ {t} l) (v) = l (L v)\tag{4.1} $$ for $l \in V ^ { * } , v \in V .$`
  - woit-qtgr/2025-10-20 §4.2 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（[转置变换(transpose transformation)](/concepts/linear-algebra/transpose-transformation)）。$L$ 的转置是线性变换 $$ L ^ {t}: V ^ {*} \to V ^ {*} $$ 由下式给出： $$ (L ^ {t} l) (v) = l (L v)\tag{4.1} $$ 对于 $l \in V ^ { * } , v \in V .$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md

## 离散动力系统(discrete dynamical system)

- UUID: `84677aa0-3aca-48a1-8a10-3da6b818b8b9` · okf_url: https://limbo137.top/okf/concepts/84677aa0-3aca-48a1-8a10-3da6b818b8b9.md
- labels: en = discrete dynamical system；zh-CN = 离散动力系统
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-d6863c1b4be5c2fb600e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 拉格朗日密度(Lagrangian density)

- UUID: `846bba7a-6161-4a61-addd-19b60032e968` · okf_url: https://limbo137.top/okf/concepts/846bba7a-6161-4a61-addd-19b60032e968.md
- labels: en = Lagrangian density；zh-CN = 拉格朗日密度
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §37.4 · `books/woit-qtgr/2025-10-20/zh/chapter-37/04-interacting-quantum-fields#first-bde437854c9faeec03e6-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §37.4 · `books/woit-qtgr/2025-10-20/en/chapter-37/04-interacting-quantum-fields#first-bde437854c9faeec03e6-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/04-interacting-quantum-fields.md

## 定向向量空间(oriented vector space)

- UUID: `84fdb7c6-ad08-4e44-851b-c0fbfd67193b` · okf_url: https://limbo137.top/okf/concepts/84fdb7c6-ad08-4e44-851b-c0fbfd67193b.md
- labels: en = oriented vector space；zh-CN = 定向向量空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §8.5 · `en/chapter-08/05-oriented-vector-spaces#first-e02cdd7886d0c6f30a01-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.5 · `zh/chapter-08/05-oriented-vector-spaces#first-e02cdd7886d0c6f30a01-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/05-oriented-vector-spaces.md

## 纤维(fiber)

- UUID: `850fd218-36a7-4024-b096-39443088c1ec` · okf_url: https://limbo137.top/okf/concepts/850fd218-36a7-4024-b096-39443088c1ec.md
- labels: en = fiber；zh-CN = 纤维
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-81061e294ac10574c590-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 相空间(phase space)

- UUID: `85694223-0c9f-4798-a693-c9ae294130f3` · okf_url: https://limbo137.top/okf/concepts/85694223-0c9f-4798-a693-c9ae294130f3.md
- labels: en = phase space；zh-CN = 相空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-8c18c8afa671277219b5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-8c18c8afa671277219b5-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-3ed50b2c89c22e69f977-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 实内积空间(real inner product space)

- UUID: `858a3ad2-68c0-4b1b-936e-0524ef5b00a1` · okf_url: https://limbo137.top/okf/concepts/858a3ad2-68c0-4b1b-936e-0524ef5b00a1.md
- labels: en = real inner product space；zh-CN = 实内积空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-real-inner-product-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 对合分布(involutive distribution)

- UUID: `85919c87-f1a4-437e-a42c-b843dc737232` · okf_url: https://limbo137.top/okf/concepts/85919c87-f1a4-437e-a42c-b843dc737232.md
- labels: en = involutive distribution；zh-CN = 对合分布
- aliases: involutive
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.6 · `en/chapter-15/06-distributions-and-frobenius-theorem#first-8c18d6b571786d75bffe-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/06-distributions-and-frobenius-theorem.md

## 向量场(vector field)

- UUID: `85e79af6-5c6d-4e9b-b6b4-4ed3060bfd23` · okf_url: https://limbo137.top/okf/concepts/85e79af6-5c6d-4e9b-b6b4-4ed3060bfd23.md
- labels: en = vector field；zh-CN = 向量场
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-9c7d421876a7463f4c22-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-vector-field` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 矩阵(matrix)

- UUID: `860ec6cb-b0fa-4738-8ece-bdbebb0bdddf` · okf_url: https://limbo137.top/okf/concepts/860ec6cb-b0fa-4738-8ece-bdbebb0bdddf.md
- labels: en = matrix；zh-CN = 矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 莫比乌斯群(Möbius group)

- UUID: `861e1ce9-8826-4d7f-b756-fead75d051ac` · okf_url: https://limbo137.top/okf/concepts/861e1ce9-8826-4d7f-b756-fead75d051ac.md
- labels: en = Möbius group；zh-CN = 莫比乌斯群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-mobius-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 元辛群(metaplectic group)

- UUID: `862f56d4-0241-4c8e-897a-15830afd1941` · okf_url: https://limbo137.top/okf/concepts/862f56d4-0241-4c8e-897a-15830afd1941.md
- labels: en = metaplectic group；zh-CN = 元辛群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-99b9bbdd43b91e227dcf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 对称关系(symmetric relation)

- UUID: `8634dc6a-08e3-49e9-941d-590b27f88cfe` · okf_url: https://limbo137.top/okf/concepts/8634dc6a-08e3-49e9-941d-590b27f88cfe.md
- labels: en = symmetric relation；zh-CN = 对称关系
- aliases: symmetric；对称
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-760eb7e9150caa9c7ef3-relation-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 薛定谔方程(Schrödinger equation)

- UUID: `868c8408-74e1-42b6-9b32-3267a97bd88b` · okf_url: https://limbo137.top/okf/concepts/868c8408-74e1-42b6-9b32-3267a97bd88b.md
- labels: en = Schrödinger equation；zh-CN = 薛定谔方程
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-7d5ee1d3418e9aa885bf-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-7d5ee1d3418e9aa885bf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 紫外截断(ultraviolet cutoff)

- UUID: `86b49294-3197-40c3-94d8-1af3e1c723bc` · okf_url: https://limbo137.top/okf/concepts/86b49294-3197-40c3-94d8-1af3e1c723bc.md
- labels: en = ultraviolet cutoff；zh-CN = 紫外截断
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §36.2 · `books/woit-qtgr/2025-10-20/zh/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-081dacab86af793b9dd3-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §36.2 · `books/woit-qtgr/2025-10-20/en/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-081dacab86af793b9dd3-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism.md

## 雅可比恒等式(Jacobi identity)

- UUID: `87107086-71e4-4b8b-899e-10c24486072b` · okf_url: https://limbo137.top/okf/concepts/87107086-71e4-4b8b-899e-10c24486072b.md
- labels: en = Jacobi identity；zh-CN = 雅可比恒等式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-5a798c9794bd7dbacc1a-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-5a798c9794bd7dbacc1a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 可观测量算符(observable operator)

- UUID: `87fd3d74-b7dd-4de8-af55-08545b8f224c` · okf_url: https://limbo137.top/okf/concepts/87fd3d74-b7dd-4de8-af55-08545b8f224c.md
- labels: en = observable operator；zh-CN = 可观测量算符
- aliases: observable operators
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.3 · `books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture#concept-7-3-quantum-observable` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 相位(phase)

- UUID: `88352bef-51d3-40b4-83c8-5d8f96aae1dc` · okf_url: https://limbo137.top/okf/concepts/88352bef-51d3-40b4-83c8-5d8f96aae1dc.md
- labels: en = phase；zh-CN = 相位
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-4dfbeb9a4e39699c15de-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-4dfbeb9a4e39699c15de-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 笛卡尔积(Cartesian product)

- UUID: `88b4f4ce-cde9-4985-b6f7-0dd2d2e234a9` · okf_url: https://limbo137.top/okf/concepts/88b4f4ce-cde9-4985-b6f7-0dd2d2e234a9.md
- labels: en = Cartesian product；zh-CN = 笛卡尔积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-5d67eba8206f55f38506-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 合取(conjunction)

- UUID: `88b73e49-36cf-4222-b64a-abe13ee42ee5` · okf_url: https://limbo137.top/okf/concepts/88b73e49-36cf-4222-b64a-abe13ee42ee5.md
- labels: en = conjunction；zh-CN = 合取
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-336c9f02aa41317bf09f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 不变子空间(invariant subspace)

- UUID: `88d6ad6c-87e4-4223-b3e8-c33c364f0070` · okf_url: https://limbo137.top/okf/concepts/88d6ad6c-87e4-4223-b3e8-c33c364f0070.md
- labels: en = invariant subspace；zh-CN = 不变子空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-invariant-subspace` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 实内积(real inner product in Woit)

- UUID: `897523c4-d8fb-48f8-8a4c-e059d6fb978b` · okf_url: https://limbo137.top/okf/concepts/897523c4-d8fb-48f8-8a4c-e059d6fb978b.md
- labels: en = real inner product in Woit；zh-CN = 实内积
- aliases: Inner product, real case；实内积（Woit约定）
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-30023cefe1a0aa6b0bbc-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Inner product, real case). An inner product on a real vector space $V$ is a symmetric $( ( v , w ) = ( w , v ) )$ map $$ (\cdot , \cdot): V \times V \to \mathbf {R} $$ that is non-degenerate and linear in both variables.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 正则反对易关系(canonical anticommutation relations)

- UUID: `89b2c869-2305-4cc2-8424-af0fde56c114` · okf_url: https://limbo137.top/okf/concepts/89b2c869-2305-4cc2-8424-af0fde56c114.md
- labels: en = canonical anticommutation relations；zh-CN = 正则反对易关系
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §27.2 · `books/woit-qtgr/2025-10-20/en/chapter-27/02-multiple-degrees-of-freedom#first-b34a5089bdb75166b16f-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §27.2 · `books/woit-qtgr/2025-10-20/zh/chapter-27/02-multiple-degrees-of-freedom#first-b34a5089bdb75166b16f-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/02-multiple-degrees-of-freedom.md

## 狄拉克记号(Dirac notation)

- UUID: `89b660bc-0d4b-4aad-8a17-de594b19d0dc` · okf_url: https://limbo137.top/okf/concepts/89b660bc-0d4b-4aad-8a17-de594b19d0dc.md
- labels: en = Dirac notation；zh-CN = 狄拉克记号
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-788cbaf33477cfe0ea44-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-788cbaf33477cfe0ea44-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## 达朗贝尔算子(d’Alembertian)

- UUID: `89f68268-4ac6-4fb9-bad7-bc164c2ec3ac` · okf_url: https://limbo137.top/okf/concepts/89f68268-4ac6-4fb9-bad7-bc164c2ec3ac.md
- labels: en = d’Alembertian；zh-CN = 达朗贝尔算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-19a56f0aa7224f5d2547-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-19a56f0aa7224f5d2547-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 黎曼流形(Riemannian manifold)

- UUID: `89f7a365-87fe-475c-a00b-27b4892ce4c3` · okf_url: https://limbo137.top/okf/concepts/89f7a365-87fe-475c-a00b-27b4892ce4c3.md
- labels: en = Riemannian manifold；zh-CN = 黎曼流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-2bbe90746e718753fb9d-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-2bbe90746e718753fb9d-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 微分形式上的拉普拉斯算子(Laplacian operator on differential forms)

- UUID: `8a3364eb-be9a-4280-9185-16e79b63e191` · okf_url: https://limbo137.top/okf/concepts/8a3364eb-be9a-4280-9185-16e79b63e191.md
- labels: en = Laplacian operator on differential forms；zh-CN = 微分形式上的拉普拉斯算子
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33.3 · `books/woit-qtgr/2025-10-20/en/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-07e3492ab6eae1bf3ee5-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms.md

## 量子态(quantum state)

- UUID: `8a86f3a5-8b52-4b89-ab39-b297da0ffe2d` · okf_url: https://limbo137.top/okf/concepts/8a86f3a5-8b52-4b89-ab39-b297da0ffe2d.md
- labels: en = quantum state；zh-CN = 量子态
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#concept-1-quantum-state` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §1.2.2 (pdf_page=29, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定义条件与符号保留在引文中；不推断跨书等价。 · 原文: `A1. There exists a Hilbert space $\mathcal{H}$ for a quantum system and the state of the system is required to be described by a vector $|\psi \rangle \in \mathcal{H}$ . In this sense, $|\psi \rangle$ is also called the state or a state vector. Moreover, two states $|\psi \rangle$ and $c|\psi \rangle$ ( $c \in \mathbb{C}, c \neq 0$ ) describe the same sta...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 自旋群(spin group)

- UUID: `8adadd91-0a76-4463-a472-62ff66c5bfb3` · okf_url: https://limbo137.top/okf/concepts/8adadd91-0a76-4463-a472-62ff66c5bfb3.md
- labels: en = spin group；zh-CN = 自旋群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-spin` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-spin` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 酉表示(unitary representation)

- UUID: `8adb7f10-55df-4db3-bbdc-c44765232a11` · okf_url: https://limbo137.top/okf/concepts/8adb7f10-55df-4db3-bbdc-c44765232a11.md
- labels: en = unitary representation；zh-CN = 酉表示
- aliases: Unitary representation；unitary group representation；unitary group representations；unitary representations；群的酉表示；酉群表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#concept-5-unitary-representation` · translation)；woit-qtgr/2025-10-20 (en · §1.3 · `books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations#first-851e0548d24298e19cdf-group-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-851e0548d24298e19cdf-group-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §1.3 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Unitary representation). A representation $( \pi , V )$ on a complex vector space $V$ with Hermitian inner product $\langle \cdot , \cdot \rangle$ is a unitary representation if it preserves the inner product, i.e. $$ \langle \pi (g) v _ {1}, \pi (g) v _ {2} \rangle = \langle v _ {1}, v _ {2} \rangle $$ for all $g \in G$ and $v _ { 1 } , v _ {...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/01-the-symplectic-group.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 开映射(open map)

- UUID: `8b3f9405-d392-41a2-9fc5-9a4dc5f7b0bc` · okf_url: https://limbo137.top/okf/concepts/8b3f9405-d392-41a2-9fc5-9a4dc5f7b0bc.md
- labels: en = open map；zh-CN = 开映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-3b4442d7900b8959b435-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 瞬子丛(instanton bundle)

- UUID: `8b5a77df-1450-5779-a699-807898c57194` · okf_url: https://limbo137.top/okf/concepts/8b5a77df-1450-5779-a699-807898c57194.md
- labels: en = instanton bundle；zh-CN = 瞬子丛
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 不定厄米形式(indefinite Hermitian form)

- UUID: `8ba0c8f9-deb3-4652-aedf-98f9fda4a22e` · okf_url: https://limbo137.top/okf/concepts/8ba0c8f9-deb3-4652-aedf-98f9fda4a22e.md
- labels: en = indefinite Hermitian form；zh-CN = 不定厄米形式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §26.2 · `books/woit-qtgr/2025-10-20/en/chapter-26/02-compatible-complex-structures-and-positivity#first-b87bf06a1eb2147cfa37-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26.2 · `books/woit-qtgr/2025-10-20/zh/chapter-26/02-compatible-complex-structures-and-positivity#first-b87bf06a1eb2147cfa37-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/02-compatible-complex-structures-and-positivity.md

## 非齐次线性变换(inhomogeneous linear transformation)

- UUID: `8bd3bc1a-11d0-46f7-81ea-1188f7ddb154` · okf_url: https://limbo137.top/okf/concepts/8bd3bc1a-11d0-46f7-81ea-1188f7ddb154.md
- labels: en = inhomogeneous linear transformation；zh-CN = 非齐次线性变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-inhomogeneous-linear-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 积拓扑(product topology)

- UUID: `8bed2d93-396b-4829-9d37-d34ecc7e93b7` · okf_url: https://limbo137.top/okf/concepts/8bed2d93-396b-4829-9d37-d34ecc7e93b7.md
- labels: en = product topology；zh-CN = 积拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-dc3eb7b09be2b856c40b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 李群(Lie group)

- UUID: `8c364498-0e05-4c91-8fd7-1730d6e8656e` · okf_url: https://limbo137.top/okf/concepts/8c364498-0e05-4c91-8fd7-1730d6e8656e.md
- labels: en = Lie group；zh-CN = 李群
- aliases: Lie groups
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-lie-group` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-5b8eee65f4e6910b5a7b-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-e952a0fbafb6add3a4d3-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.6 · `books/woit-qtgr/2025-10-20/en/chapter-05/06-for-further-reading#first-906755697a9d77bc2a2b-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.1 (pdf_page=226, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.8：李群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.8. A Lie group $G$ is a differentiable manifold which is endowed with a group structure such that the group operations`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.1 (pdf_page=226, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.8：李群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The unit element of a Lie group is written as $e$ . The dimension of a Lie group $G$ is defined to be the dimension of $G$ as a manifold. The product symbol may be omitted and $g_{1} \cdot g_{2}$ is usually written as $g_{1}g_{2}$ . For example, let $\mathbb{R}^{*} \equiv \mathbb{R} - \{0\}$ . Take three elements $x, y, z \in \mathbb{R}^{*}$ such that $xy...`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.1 (pdf_page=226, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.8：李群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $\cdot : G \times G \to G$ , $(g_1, g_2) \mapsto g_1 \cdot g_2$ (ii) ${}^{-1}: G \to G$ , $g \mapsto g^{-1}$`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.1 (pdf_page=226, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.8：李群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `are differentiable. [Remark: It can be shown that $G$ has a unique analytic structure with which the product and the inverse operations are written as convergent power series.]`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/06-for-further-reading.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 子流形(submanifold)

- UUID: `8c4df920-879e-5e17-a913-24dd9344969a` · okf_url: https://limbo137.top/okf/concepts/8c4df920-879e-5e17-a913-24dd9344969a.md
- labels: en = submanifold；zh-CN = 子流形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.3. (Immersion, submanifold, embedding) Let $f: M \to N$ be a smooth map and let $\dim M \leq \dim N$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) The map $f$ is called an immersion of $M$ into $N$ if $f_{*}: T_{p}M \to T_{f(p)}N$ is an injection (one to one), that is $\operatorname{rank} f_{*} = \dim M$ . (b) The map $f$ is called an **embedding** if $f$ is an injection and an immersion. The image $f(M)$ is called a **submanifold** of $N$ . [In practice, $f(M)$ thus defined is diffeomorphic to ...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：子流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If $f$ is an immersion, $f^{*}$ maps $T_{p}M$ isomorphically to an $m$ -dimensional vector subspace of $T_{f(p)}N$ since $\mathrm{rank}f_{*} = \dim M$ . From theorem 2.1, we also find $\ker f_{*} = \{0\}$ . If $f$ is an embedding, $M$ is diffeomorphic to $f(M)$ . Examples will clarify these rather technical points. Consider a map $f: S^1 \to \mathbb{R}^2$...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 普法夫方程组(Pfaffian system)

- UUID: `8c5e05bd-6774-4afe-aae0-27aa273bad37` · okf_url: https://limbo137.top/okf/concepts/8c5e05bd-6774-4afe-aae0-27aa273bad37.md
- labels: en = Pfaffian system；zh-CN = 普法夫方程组
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.3 · `en/chapter-16/03-frobenius-theorem-dual-form#first-8d5abeaa322326ce650d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.3 · `zh/chapter-16/03-frobenius-theorem-dual-form#first-8d5abeaa322326ce650d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/03-frobenius-theorem-dual-form.md

## 相空间路径积分(phase space path integral)

- UUID: `8c7b5380-5128-4333-9ea2-4fde37887085` · okf_url: https://limbo137.top/okf/concepts/8c7b5380-5128-4333-9ea2-4fde37887085.md
- labels: en = phase space path integral；zh-CN = 相空间路径积分
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.3 · `books/woit-qtgr/2025-10-20/en/chapter-35/03-quantization-and-path-integrals#first-efe83d38d4761e050e7d-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.3 · `books/woit-qtgr/2025-10-20/zh/chapter-35/03-quantization-and-path-integrals#first-efe83d38d4761e050e7d-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/03-quantization-and-path-integrals.md

## 里斯表示定理(Riesz representation theorem)

- UUID: `8cc47160-e281-4f43-93d4-177f7b999134` · okf_url: https://limbo137.top/okf/concepts/8cc47160-e281-4f43-93d4-177f7b999134.md
- labels: en = Riesz representation theorem；zh-CN = 里斯表示定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.3 · `en/chapter-13/03-linear-functionals#first-226eaa7f139e78020d8c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.3 · `zh/chapter-13/03-linear-functionals#first-226eaa7f139e78020d8c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/03-linear-functionals.md

## 自旋 1/2 表示(spin-1/2 representation)

- UUID: `8d16bffc-ce2e-49f4-9683-6eabeb80e54b` · okf_url: https://limbo137.top/okf/concepts/8d16bffc-ce2e-49f4-9683-6eabeb80e54b.md
- labels: en = spin-1/2 representation；zh-CN = 自旋 1/2 表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.3 · `books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary#concept-6-3-spinor-representation` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.3 · `books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary#concept-6-3-spinor-representation` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 传递群作用(transitive group action)

- UUID: `8d238fdd-b982-43b3-9c41-7de28e84a222` · okf_url: https://limbo137.top/okf/concepts/8d238fdd-b982-43b3-9c41-7de28e84a222.md
- labels: en = transitive group action；zh-CN = 传递群作用
- aliases: transitive；transitive (group action)；transitive action；transitive group actions；传递；传递作用；传递的；传递的（群作用）；遍历的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-transitive` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-16cddd3179afedd4e486-action-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 群作用只有一个轨道；不表示二元关系的传递律 · 原文: `(a) transitive if, for any $p_1, p_2 \in M$ , there exists an element $g \in G$ such that $\sigma(g, p_1) = p_2$ ; (b) free if every non-trivial element $g \neq e$ of $G$ has no fixed points in $M$ , that is, if there exists an element $p \in M$ such that $\sigma(g, p) = p$ , then $g$ must be the unit element $e$ ; and (c) effective if the unit element $e...`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：传递的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.15. Let $G$ be a Lie group that acts on a manifold $M$ by $\sigma : G \times M \to M$ . The action $\sigma$ is said to be`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：传递的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) transitive if, for any $p_1, p_2 \in M$ , there exists an element $g \in G$ such that $\sigma(g, p_1) = p_2$ ; (b) free if every non-trivial element $g \neq e$ of $G$ has no fixed points in $M$ , that is, if there exists an element $p \in M$ such that $\sigma(g, p) = p$ , then $g$ must be the unit element $e$ ; and (c) effective if the unit element $e...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md

## p-形式(p-form)

- UUID: `8d36a7f4-ce03-5309-a7c0-73b6f2b87ee0` · okf_url: https://limbo137.top/okf/concepts/8d36a7f4-ce03-5309-a7c0-73b6f2b87ee0.md
- labels: en = p-form；zh-CN = p-形式
- aliases: differential p-form；differential r-form；p-forms；p形式；r-形式；微分p-形式
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §2.4 (pdf_page=28, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 固定阶数p的光滑、完全反对称(0,p)张量场，即流形上的微分p-形式；p/r为同一阶数参数的不同记号。不是单点切空间上的交替张量，也不是各阶混合的外代数元素。单点形式与光滑形式场须区分。 · 原文: `Some tensors are more interesting than others. A particularly interesting class are totally anti-symmetric $( 0 , p )$ tensors fields. These are called p-forms. The set of all p-forms over a manifold M is denoted $\Lambda ^ { p } ( M )$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 压缩态(squeezed state)

- UUID: `8d9cedbe-155f-47e3-b48e-edf05ba7b6ed` · okf_url: https://limbo137.top/okf/concepts/8d9cedbe-155f-47e3-b48e-edf05ba7b6ed.md
- labels: en = squeezed state；zh-CN = 压缩态
- aliases: squeezed states
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §26.5 · `books/woit-qtgr/2025-10-20/zh/chapter-26/05-complex-structures-for-d-1-and-squeezed-states#first-62c5a6c16de6e35407f2-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §26.5 · `books/woit-qtgr/2025-10-20/en/chapter-26/05-complex-structures-for-d-1-and-squeezed-states#first-753bad2e30d0efb8b0f7-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/05-complex-structures-for-d-1-and-squeezed-states.md

## 闭的(closed)

- UUID: `8da45318-5944-4aa7-afd5-245f6894ccb2` · okf_url: https://limbo137.top/okf/concepts/8da45318-5944-4aa7-afd5-245f6894ccb2.md
- labels: en = closed；zh-CN = 闭的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-df2cde65096927db28d1-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-closed` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 标准拓扑(standard topology)

- UUID: `8df99f4d-d7ad-420b-9593-8e737c9fdcd1` · okf_url: https://limbo137.top/okf/concepts/8df99f4d-d7ad-420b-9593-8e737c9fdcd1.md
- labels: en = standard topology；zh-CN = 标准拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-7cbe5935e635816411ab-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 微分同胚(diffeomorphism)

- UUID: `8e3f52f8-c41d-44c8-8648-f6ee019acffe` · okf_url: https://limbo137.top/okf/concepts/8e3f52f8-c41d-44c8-8648-f6ee019acffe.md
- labels: en = diffeomorphism；zh-CN = 微分同胚
- aliases: diffeomorphic；diffeomorphisms；微分同胚映射；微分同胚的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.2 · `zh/chapter-15/02-differentiable-maps-and-curves#concept-15-diffeomorphism` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.2 · `en/chapter-15/02-differentiable-maps-and-curves#first-c90ee60ec510dcc909d0-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=200, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `A function $f$ on $M$ is a smooth map from $M$ to $\mathbb{R}$ , see figure 5.8. On a chart $(U, \varphi)$ , the coordinate presentation of $f$ is given by $f \circ \varphi^{-1} : \mathbb{R}^m \to \mathbb{R}$ which is a real-valued function of $m$ variables. We denote the set of smooth functions on $M$ by $\mathcal{F}(M)$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=199, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.2. Let $f: M \to N$ be a homeomorphism and $\psi$ and $\varphi$ be coordinate functions as previously defined. If $\psi \circ f \circ \varphi^{-1}$ is invertible (that is, there exists a map $\varphi \circ f^{-1} \circ \psi^{-1}$ ) and both $y = \psi \circ f \circ \varphi^{-1}(x)$ and $x = \varphi \circ f^{-1} \circ \psi^{-1}(y)$ are $C^\inft...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=199, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Clearly $\dim M = \dim N$ if $M \equiv N$ . In chapter 2, we noted that homeomorphisms classify spaces according to whether it is possible to deform one space into another continuously. Diffeomorphisms classify spaces into equivalence classes according to whether it is possible to deform one space to another smoothly. Two diffeomorphic spaces are regarded...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=199, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Now we look at special classes of mappings, namely curves and functions. An open curve in an $m$ -dimensional manifold $M$ is a map $c: (a, b) \to M$ where $(a, b)$ is an open interval such that $a < 0 < b$ . We assume that the curve does not intersect with itself (figure 5.7). The number $a(b)$ may be $-\infty (+\infty)$ and we have included 0 in the int...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=199, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The set of diffeomorphisms $f: M \to M$ is a group denoted by $\operatorname{Diff}(M)$ . Take a point $p$ in a chart $(U, \varphi)$ such that $\varphi(p) = x^{\mu}(p)$ . Under $f \in \operatorname{Diff}(M)$ , $p$ is mapped to $f(p)$ whose coordinates are $\varphi(f(p)) = y^{\mu}(f(p))$ (we have assumed $f(p) \in U$ ). Clearly $y$ is a differentiable funct...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.1 (pdf_page=200, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.2：微分同胚。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `regarded as a map $c: S^1 \to M$ . In both cases, $c$ is locally a map from an open interval to $M$ . On a chart $(U, \varphi)$ , a curve $c(t)$ has the coordinate presentation $x = \varphi \circ c: \mathbb{R} \to \mathbb{R}^m$ .`
  - szekeres/source-snapshot-bc3404156f6e §15.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 两个光滑流形之间存在光滑且逆也光滑的双射 · 原文: `A diffeomorphism is a map $\alpha ; M \to N$ that is one-to-one and both $\alpha$ and $\alpha ^ { - 1 } : N \to M$ are differentiable. Two manifolds $M$ and $N$ are said to be diffeomorphic, written $M \cong N _ { \mathrm { : } }$ if there exists a diffeomorphism $\alpha : M \to N ;$ ; the dimensions of the two manifolds must of course be equal, $m = n$ ....`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/02-differentiable-maps-and-curves.md

## 连续映射(continuous map)

- UUID: `8e591054-5de3-450f-95bc-7ec9335b2228` · okf_url: https://limbo137.top/okf/concepts/8e591054-5de3-450f-95bc-7ec9335b2228.md
- labels: en = continuous map；zh-CN = 连续映射
- aliases: continuity；continuous；continuous function；continuous functions；continuous mapping；continuous mappings；continuous maps；连续；连续函数；连续性；连续的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-f07bd807fba8ab155af7-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.2 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.4：连续的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `We take the usual topology in $\mathbb{R}$ , hence any open interval $(a, b)$ is an open set. In the usual calculus, $f$ is said to have a discontinuity at $x = 0$ . For an open set $(3/2, 2) \subset Y$ , we find $f^{-1}((3/2, 2)) = (-1, -1/2)$ which is an open set in $X$ . If we take an open set $(1 - 1/4, 1 + 1/4) \subset Y$ , however, we find $f^{-1}((...`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.2 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 同一连续映射条件的性质名词 · 原文: `Definition 2.4. Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.2 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.4：连续的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ f (x) = \left\{ \begin{array}{l l} - x + 1 & x \leq 0 \\ - x + \frac {1}{2} & x > 0. \end{array} \right. \tag {2.27} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.2 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.4：连续的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.4. Let $X$ and $Y$ be topological spaces. A map $f: X \to Y$ is continuous if the inverse image of an open set in $Y$ is an open set in $X$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.2 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.4：连续的。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `This definition is in agreement with our intuitive notion of continuity. For instance, let $f: \mathbb{R} \to \mathbb{R}$ be defined by`
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 映射满足开集逆像为开集的连续性条件 · 原文: `Let $X$ and $Y$ be two topological spaces. Theorem 10.1 motivates the following definition: a function $f : X \to Y$ is said to be continuous if the inverse image $f ^ { - 1 } ( U )$ of every open set $U$ in $Y$ is open in $X$. If $f$ is one-to-one and its inverse $f ^ { - 1 } : Y \to X$ is continuous, the function is called a homeomorphism and the topolo...`
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 拓扑空间之间的映射；连续的、连续性、连续映射按同一开集逆像条件查询，词性按标签保留。 · 原文: `Let $X$ and $Y$ be two topological spaces. Theorem 10.1 motivates the following definition: a function $f : X \to Y$ is said to be continuous if the inverse image $f ^ { - 1 } ( U )$ of every open set $U$ in $Y$ is open in $X$. If $f$ is one-to-one and its inverse $f ^ { - 1 } : Y \to X$ is continuous, the function is called a homeomorphism and the topolo...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 积分因子(integrating factor)

- UUID: `8e9b8a96-dfaf-446c-a30f-fa4a86514d82` · okf_url: https://limbo137.top/okf/concepts/8e9b8a96-dfaf-446c-a30f-fa4a86514d82.md
- labels: en = integrating factor；zh-CN = 积分因子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.3 · `en/chapter-16/03-frobenius-theorem-dual-form#first-bc971362295b187c52c0-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.3 · `zh/chapter-16/03-frobenius-theorem-dual-form#first-bc971362295b187c52c0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/03-frobenius-theorem-dual-form.md

## 反对易关系(anticommutation relations)

- UUID: `8ec85552-7145-4f89-8ce0-72304c15edc8` · okf_url: https://limbo137.top/okf/concepts/8ec85552-7145-4f89-8ce0-72304c15edc8.md
- labels: en = anticommutation relations；zh-CN = 反对易关系
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-06ac1ac5c9af06d63d4f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 欧拉示性数(Euler characteristic)

- UUID: `8edfdb84-325b-48e9-8526-006d8053914f` · okf_url: https://limbo137.top/okf/concepts/8edfdb84-325b-48e9-8526-006d8053914f.md
- labels: en = Euler characteristic；zh-CN = 欧拉示性数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-2b00b5e9bcab738f196a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-2b00b5e9bcab738f196a-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.4.4 (pdf_page=107, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.10：欧拉示性数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \begin{array}{l} \chi (X) = (\text {n u m b e r o f v e r t i c i e s i n} K) - (\text {n u m b e r o f e d g e s i n} K) \\ + (\text {n u m b e r o f f a c e s i n} K). \tag {2.31} \\ \end{array} $$`
  - nakahara-geometry-topology-physics/second-edition-local §2.4.4 (pdf_page=108, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.10：欧拉示性数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The reader might wonder if $\chi(X)$ depends on the polyhedron $K$ or not. The following theorem due to Poincaré and Alexander guarantees that it is, in fact, independent of the polyhedron $K$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.4.4 (pdf_page=107, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.10：欧拉示性数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.10. Let $X$ be a subset of $\mathbb{R}^3$ , which is homeomorphic to a polyhedron $K$ . Then the Euler characteristic $\chi(X)$ of $X$ is defined by`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Szekeres 此处未证明拓扑不变性。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 阶梯函数(step function)

- UUID: `8ef2b38f-1f13-42b5-bdfd-e9450c87b6e8` · okf_url: https://limbo137.top/okf/concepts/8ef2b38f-1f13-42b5-bdfd-e9450c87b6e8.md
- labels: en = step function；zh-CN = 阶梯函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-604ae42777cd5b35c692-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 可测函数(measurable function)

- UUID: `8fe45994-d5c0-4ad0-af56-77339775e681` · okf_url: https://limbo137.top/okf/concepts/8fe45994-d5c0-4ad0-af56-77339775e681.md
- labels: en = measurable function；zh-CN = 可测函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-b0b90660ad04da6d8ff2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 可三角剖分(triangulable)

- UUID: `9016d19c-ddbb-49ae-8612-be3a5e5d0e5f` · okf_url: https://limbo137.top/okf/concepts/9016d19c-ddbb-49ae-8612-be3a5e5d0e5f.md
- labels: en = triangulable；zh-CN = 可三角剖分
- aliases: $(K, f)$；triangulation
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.2 (pdf_page=119, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 拓扑空间X存在单纯复形K及同胚f:|K|→X；性质与三角剖分这对数据分别理解。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Let $X$ be a topological space. If there exists a simplicial complex $K$ and a homeomorphism $f: |K| \to X$ , $X$ is said to be triangulable and the pair $(K, f)$ is called a triangulation of $X$ . Given a topological space $X$ , its triangulation is far from unique. We will be concerned with triangulable spaces only.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 一般代数的结构常数(structure constants of an algebra)

- UUID: `903209da-385d-464a-8d57-61b71d54e2e8` · okf_url: https://limbo137.top/okf/concepts/903209da-385d-464a-8d57-61b71d54e2e8.md
- labels: en = structure constants of an algebra；zh-CN = 一般代数的结构常数
- aliases: structure constants；结构常数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-structure-constants` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-f998b514210726440578-algebra-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 类空向量(spacelike vector)

- UUID: `903eac38-d4eb-4e4a-b7d4-a25741bba472` · okf_url: https://limbo137.top/okf/concepts/903eac38-d4eb-4e4a-b7d4-a25741bba472.md
- labels: en = spacelike vector；zh-CN = 类空向量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.1 · `books/woit-qtgr/2025-10-20/zh/chapter-40/01-minkowski-space#first-f5841724552749dfac35-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 闭链群(cycle group)

- UUID: `908f0abd-5194-447e-8401-aca500fcdbf6` · okf_url: https://limbo137.top/okf/concepts/908f0abd-5194-447e-8401-aca500fcdbf6.md
- labels: en = cycle group；zh-CN = 闭链群
- aliases: p-cycle group；r-cycle group；$p$-闭链群；$r$-闭链群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-5452aa02de58e0354cf4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-5452aa02de58e0354cf4-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$c$ is called an $r$ -cycle. The set of $r$ -cycles $Z_{r}(K)$ is a subgroup of $C_{r}(K)$ and is called the $r$ -cycle group. Note that $Z_{r}(K) = \ker \partial_{r}$ . [Remark: If $r = 0$ , $\partial_0 c$ vanishes identically and $Z_{0}(K) = C_{0}(K)$ , see (3.23).]`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.3. If $c \in C_r(K)$ satisfies`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.3：闭链群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ \partial_ {r} c = 0 \tag {3.24} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 准静态过程(quasi-static process)

- UUID: `909aa045-9701-4f8a-a442-95178cb2906d` · okf_url: https://limbo137.top/okf/concepts/909aa045-9701-4f8a-a442-95178cb2906d.md
- labels: en = quasi-static process；zh-CN = 准静态过程
- aliases: quasi-static
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-7523d4e23be471ccee71-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 无界的(unbounded)

- UUID: `909ec15d-c535-4149-b8d5-bf92d3f9da33` · okf_url: https://limbo137.top/okf/concepts/909ec15d-c535-4149-b8d5-bf92d3f9da33.md
- labels: en = unbounded；zh-CN = 无界的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-4842899ecabbb9d75a7d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-unbounded` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 傅里叶变换(Fourier transform)

- UUID: `9108ce8e-1818-467c-a25f-8675415492dd` · okf_url: https://limbo137.top/okf/concepts/9108ce8e-1818-467c-a25f-8675415492dd.md
- labels: en = Fourier transform；zh-CN = 傅里叶变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.3 · `zh/chapter-12/03-fourier-transforms#first-8580ce4b7e4e61afb836-zh` · translation)；woit-qtgr/2025-10-20 (en · §11.2 · `books/woit-qtgr/2025-10-20/en/chapter-11/02-the-group-r-and-the-fourier-transform#first-3c91f74bd22d187234b0-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §11.2 · `books/woit-qtgr/2025-10-20/zh/chapter-11/02-the-group-r-and-the-fourier-transform#first-175c6f3f3d84afe07eca-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §11.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Fourier transform of a function $\psi$ is denoted by $\mathcal F\psi$ or $\widetilde\psi$, where $$ (\mathcal {F} \psi) (k) = \widetilde {\psi} (k) = \frac {1}{\sqrt {2 \pi}} \int_ {- \infty} ^ {\infty} e ^ {- i k q} \psi (q) d q\tag{11.3} $$`
  - woit-qtgr/2025-10-20 §11.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `函数 $\psi$ 的傅里叶变换记作 $\mathcal F\psi$ 或 $\widetilde\psi$，其定义为 $$ (\mathcal {F} \psi) (k) = \widetilde {\psi} (k) = \frac {1}{\sqrt {2 \pi}} \int_ {- \infty} ^ {\infty} e ^ {- i k q} \psi (q) d q\tag{11.3} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/03-fourier-transforms.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/03-the-fourier-transform-in-minkowski-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-11/02-the-group-r-and-the-fourier-transform.md

## 旋量(spinor)

- UUID: `91589de7-bd73-55ae-b166-4cd0949b5b14` · okf_url: https://limbo137.top/okf/concepts/91589de7-bd73-55ae-b166-4cd0949b5b14.md
- labels: en = spinor；zh-CN = 旋量
- aliases: spinors；旋量(spinor)
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 截面(section)

- UUID: `91a7526b-ab87-4e11-b9c7-913c316d0cef` · okf_url: https://limbo137.top/okf/concepts/91a7526b-ab87-4e11-b9c7-913c316d0cef.md
- labels: en = section；zh-CN = 截面
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-01f327f577c7d80318ba-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.5 · `books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere#first-01f327f577c7d80318ba-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 有限群(finite group)

- UUID: `91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e` · okf_url: https://limbo137.top/okf/concepts/91c6b7de-78fd-4b5a-9d1e-2b69ee97ba8e.md
- labels: en = finite group；zh-CN = 有限群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-finite-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-bbdbf409a34a3551c1be-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## Peierls括号(Peierls bracket)

- UUID: `91e3781b-f12b-42af-bf58-371780386623` · okf_url: https://limbo137.top/okf/concepts/91e3781b-f12b-42af-bf58-371780386623.md
- labels: en = Peierls bracket；zh-CN = Peierls括号
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-6c57bd3b8c40600387c5-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-6c57bd3b8c40600387c5-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 上边界链(coboundary)

- UUID: `91f394ee-b446-4502-be7d-9d5169bdd21e` · okf_url: https://limbo137.top/okf/concepts/91f394ee-b446-4502-be7d-9d5169bdd21e.md
- labels: en = coboundary；zh-CN = 上边界链
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-0b1865b872f69033467f-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-0b1865b872f69033467f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## Hirzebruch符号差(Hirzebruch signature)

- UUID: `920c8477-59c0-571c-8a82-6a8150c21f69` · okf_url: https://limbo137.top/okf/concepts/920c8477-59c0-571c-8a82-6a8150c21f69.md
- labels: en = Hirzebruch signature；zh-CN = Hirzebruch符号差
- aliases: Hirzebruch 符号差；希策布鲁赫符号差
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara例11.8完整假设与相交形式式11.97、定义τ(M)=p-q式11.98；本证据为四维情形，不把证明此数等于L类积分的定理当数值本身。 · 原文: `$$ \sigma ([ \alpha ], [ \beta ]) = \int_ {M} \alpha \wedge \beta . \tag {11.97} $$`
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara例11.8完整假设与相交形式式11.97、定义τ(M)=p-q式11.98；本证据为四维情形，不把证明此数等于L类积分的定理当数值本身。 · 原文: `Example 11.8. Let $M$ be a compact connected and orientable four-dimensional manifold. Let us consider the symmetric bilinear form $\sigma : H^2(M; \mathbb{R}) \times H^2(M; \mathbb{R}) \to \mathbb{R}$ defined by`
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara例11.8完整假设与相交形式式11.97、定义τ(M)=p-q式11.98；本证据为四维情形，不把证明此数等于L类积分的定理当数值本身。 · 原文: `$$ \tau (M) \equiv p - q. \tag {11.98} $$`
  - nakahara-geometry-topology-physics/second-edition-local §11.4.3 (pdf_page=462, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara例11.8完整假设与相交形式式11.97、定义τ(M)=p-q式11.98；本证据为四维情形，不把证明此数等于L类积分的定理当数值本身。 · 原文: `$\sigma$ is a $b^{2} \times b^{2}$ symmetric matrix where $b^{2} = \dim H^{2}(M; \mathbb{R})$ is the Betti number. Clearly $\sigma$ is non-degenerate since $\sigma([\alpha], [\beta]) = 0$ for any $[\alpha] \in H^{2}(M; \mathbb{R})$ implies $[\beta] = 0$ . Let $p(q)$ be the number of positive (negative) eigenvalues of $\sigma$ . The Hirzebruch signature of...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 叉积(cross product)

- UUID: `9217da23-ed1d-4016-b7e9-9a1d7c04a546` · okf_url: https://limbo137.top/okf/concepts/9217da23-ed1d-4016-b7e9-9a1d7c04a546.md
- labels: en = cross product；zh-CN = 叉积
- aliases: 向量叉积
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.1 · `books/woit-qtgr/2025-10-20/zh/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-cross-product` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-cross-product` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 波函数(wavefunction)

- UUID: `928ab6a9-7e9a-4beb-adff-65f984589c86` · okf_url: https://limbo137.top/okf/concepts/928ab6a9-7e9a-4beb-adff-65f984589c86.md
- labels: en = wavefunction；zh-CN = 波函数
- aliases: wavefunctions
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §1.4 · `books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics#concept-1-wavefunction` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.4 · `books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics#concept-1-wavefunction` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md

## 里斯–费舍尔定理(Riesz–Fischer theorem)

- UUID: `929c9263-d054-49be-a645-da3f363ec87a` · okf_url: https://limbo137.top/okf/concepts/929c9263-d054-49be-a645-da3f363ec87a.md
- labels: en = Riesz–Fischer theorem；zh-CN = 里斯–费舍尔定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.1 · `en/chapter-13/01-definitions-and-examples#first-f29febbce570ecbf7dac-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.1 · `zh/chapter-13/01-definitions-and-examples#first-f29febbce570ecbf7dac-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/01-definitions-and-examples.md

## 向量部(vector part)

- UUID: `92e8c070-3f75-4df4-86dd-3fdebe66df03` · okf_url: https://limbo137.top/okf/concepts/92e8c070-3f75-4df4-86dd-3fdebe66df03.md
- labels: en = vector part；zh-CN = 向量部
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-vector-part` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-df47b972d3aaaacd4ca7-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md

## 集合(set)

- UUID: `93057b90-f764-44c7-b119-a2c29c5e144e` · okf_url: https://limbo137.top/okf/concepts/93057b90-f764-44c7-b119-a2c29c5e144e.md
- labels: en = set；zh-CN = 集合
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-dec42533b63e3964a8f9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 测地偏离方程(equation of geodesic deviation)

- UUID: `93315aa2-8062-4644-a14e-60ba46bc2a42` · okf_url: https://limbo137.top/okf/concepts/93315aa2-8062-4644-a14e-60ba46bc2a42.md
- labels: en = equation of geodesic deviation；zh-CN = 测地偏离方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.5 · `zh/chapter-18/05-equation-of-geodesic-deviation#first-1cbfb7663b0780f12200-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.5 · `en/chapter-18/05-equation-of-geodesic-deviation#first-1cbfb7663b0780f12200-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/05-equation-of-geodesic-deviation.md

## 行秩(row rank)

- UUID: `933abc9a-7725-4444-8639-3a20824a352e` · okf_url: https://limbo137.top/okf/concepts/933abc9a-7725-4444-8639-3a20824a352e.md
- labels: en = row rank；zh-CN = 行秩
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-row-rank` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md

## 洛伦兹群(Lorentz group)

- UUID: `937e2225-ecf0-4854-ab8b-d17f476f7021` · okf_url: https://limbo137.top/okf/concepts/937e2225-ecf0-4854-ab8b-d17f476f7021.md
- labels: en = Lorentz group；zh-CN = 洛伦兹群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-lorentz-group` · translation)；woit-qtgr/2025-10-20 (en · §29.2 · `books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry#first-762a1e60efdf89af31a4-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §29.2 · `books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry#first-762a1e60efdf89af31a4-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §40.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[洛伦兹群(Lorentz group)](/concepts/lie-theory/lorentz-group) $O(3,1)$ 是保持 $\mathbb R^4$ 上闵可夫斯基内积的线性变换群。`
  - woit-qtgr/2025-10-20 §40.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[Lorentz group](/concepts/lie-theory/lorentz-group) $O(3,1)$ is the group of linear transformations preserving the Minkowski inner product on $\mathbb R^4$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/02-the-lorentz-group-and-its-lie-algebra.md

## 全对称张量(totally symmetric tensor)

- UUID: `9386ac37-5b97-4db1-aabf-051fc76c47d5` · okf_url: https://limbo137.top/okf/concepts/9386ac37-5b97-4db1-aabf-051fc76c47d5.md
- labels: en = totally symmetric tensor；zh-CN = 全对称张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-90934793b24752d4accc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 时间膨胀(time dilatation)

- UUID: `9391a1e9-edb1-4fe3-82c9-3be87013075a` · okf_url: https://limbo137.top/okf/concepts/9391a1e9-edb1-4fe3-82c9-3be87013075a.md
- labels: en = time dilatation；zh-CN = 时间膨胀
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.2 · `zh/chapter-09/02-relativistic-kinematics#first-6ee6a5bdbfa01d4fb7f7-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.2 · `en/chapter-09/02-relativistic-kinematics#first-6ee6a5bdbfa01d4fb7f7-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/02-relativistic-kinematics.md

## 不相交(disjoint)

- UUID: `9393d6e8-2f1f-48db-b75c-81af6880c852` · okf_url: https://limbo137.top/okf/concepts/9393d6e8-2f1f-48db-b75c-81af6880c852.md
- labels: en = disjoint；zh-CN = 不相交
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-d72cebbfdfba94362687-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 初等因子(elementary divisor)

- UUID: `9395d494-7be0-4d0f-8429-30b4f311266c` · okf_url: https://limbo137.top/okf/concepts/9395d494-7be0-4d0f-8429-30b4f311266c.md
- labels: en = elementary divisor；zh-CN = 初等因子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-elementary-divisor` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 楞次向量(Lenz vector)

- UUID: `93db54df-1884-4aa0-bba4-11221ec183a6` · okf_url: https://limbo137.top/okf/concepts/93db54df-1884-4aa0-bba4-11221ec183a6.md
- labels: en = Lenz vector；zh-CN = 楞次向量
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §21.2 · `books/woit-qtgr/2025-10-20/zh/chapter-21/02-so-4-symmetry-and-the-coulomb-potential#first-fa95a4c5bf3e31b1d664-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §21.2 · `books/woit-qtgr/2025-10-20/en/chapter-21/02-so-4-symmetry-and-the-coulomb-potential#first-89aa14c7918bcea2fb01-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §21.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `楞次向量是相空间 $\dot { \mathbf { R } } ^ { 6 }$ 上的向量值函数，由下式给出 $$ \mathbf {w} = \frac {1}{m} (\mathbf {l} \times \mathbf {p}) + e ^ {2} \frac {\mathbf {q}}{| \mathbf {q} |} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-21/02-so-4-symmetry-and-the-coulomb-potential.md

## 狄拉克伽马矩阵(Dirac gamma matrices)

- UUID: `9445e0c4-8ed1-4567-844b-c66f13045539` · okf_url: https://limbo137.top/okf/concepts/9445e0c4-8ed1-4567-844b-c66f13045539.md
- labels: en = Dirac gamma matrices；zh-CN = 狄拉克伽马矩阵
- aliases: Dirac $\gamma$-matrices
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §29.2 · `books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry#first-73ede448c89ff1a9dee8-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md

## 辛矩阵(symplectic matrix)

- UUID: `9452675b-d56a-4a38-90c9-9efa189f3d94` · okf_url: https://limbo137.top/okf/concepts/9452675b-d56a-4a38-90c9-9efa189f3d94.md
- labels: en = symplectic matrix；zh-CN = 辛矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-symplectic-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 拉普拉斯算符(Laplacian operator)

- UUID: `94acb024-2f20-4221-93e3-8560f3208f93` · okf_url: https://limbo137.top/okf/concepts/94acb024-2f20-4221-93e3-8560f3208f93.md
- labels: en = Laplacian operator；zh-CN = 拉普拉斯算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §33.3 · `books/woit-qtgr/2025-10-20/zh/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-957f0615e74d4b6adf75-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms.md

## 半群同态(semigroup homomorphism)

- UUID: `94ed061d-4936-4c6b-81d0-953dbc6bdb3d` · okf_url: https://limbo137.top/okf/concepts/94ed061d-4936-4c6b-81d0-953dbc6bdb3d.md
- labels: en = semigroup homomorphism；zh-CN = 半群同态
- aliases: homomorphism；semigroup homomorphisms；同态
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-65ac5289df72c51c1bee-semigroup-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 向量表示(vector representation)

- UUID: `94ff4c93-90de-45ed-a904-70b63d9c30cf` · okf_url: https://limbo137.top/okf/concepts/94ff4c93-90de-45ed-a904-70b63d9c30cf.md
- labels: en = vector representation；zh-CN = 向量表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-vector-representation` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.1 · `books/woit-qtgr/2025-10-20/zh/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-vector-representation` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 同伦群(homotopy group)

- UUID: `95386eb0-a5f1-5b60-a15a-fa0614cd04bb` · okf_url: https://limbo137.top/okf/concepts/95386eb0-a5f1-5b60-a15a-fa0614cd04bb.md
- labels: en = homotopy group；zh-CN = 同伦群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=166, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.11：同伦群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.11. Let $X$ be a topological space. The set of homotopy classes of $n$ -loops ( $n \geq 1$ ) at $x_0 \in X$ is denoted by $\pi_n(X, x_0)$ and called the $n$ th homotopy group at $x_0$ . $\pi_n(x, x_0)$ is called the higher homotopy group if $n \geq 2$ .`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=166, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.11：同伦群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The product $\alpha * \beta$ just defined naturally induces a product of homotopy classes defined by`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=166, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.11：同伦群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ [ \alpha ] * [ \beta ] \equiv [ \alpha * \beta ] \tag {4.41} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.5.1 (pdf_page=166, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.11：同伦群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `where $\alpha$ and $\beta$ are $n$ -loops at $x_0$ . The following exercises verify that this product is well defined and satisfies the group axioms.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## Hilbert空间的标准正交基(Hilbert-space orthonormal basis)

- UUID: `95c3db47-c6cf-46ca-bf3d-3fd42ed1e6eb` · okf_url: https://limbo137.top/okf/concepts/95c3db47-c6cf-46ca-bf3d-3fd42ed1e6eb.md
- labels: en = Hilbert-space orthonormal basis；zh-CN = Hilbert空间的标准正交基
- aliases: orthonormal bases；orthonormal basis；标准正交基；正交归一基；正交标准基
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-1f1dfb34be33497204d3-hilbert-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-orthonormal-basis` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 麦克斯韦应力张量(Maxwell stress tensor)

- UUID: `96348a4e-25ce-493c-a4bb-dc9b59e18d07` · okf_url: https://limbo137.top/okf/concepts/96348a4e-25ce-493c-a4bb-dc9b59e18d07.md
- labels: en = Maxwell stress tensor；zh-CN = 麦克斯韦应力张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.5 · `zh/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-db3b92c5d3114c7e2bb3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.5 · `en/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-db3b92c5d3114c7e2bb3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/05-conservation-laws-and-energy-stress-tensors.md

## 弗里德曼模型(Friedmann model)

- UUID: `96689ef3-6ecf-45d5-9f2d-db37a9d82122` · okf_url: https://limbo137.top/okf/concepts/96689ef3-6ecf-45d5-9f2d-db37a9d82122.md
- labels: en = Friedmann model；zh-CN = 弗里德曼模型
- aliases: Friedmann models
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-bb2fc2024a29e9cbf7c5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-252c0c423b07a38dc06d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 可测空间(measurable space)

- UUID: `9679e150-e342-4615-98e6-84014c53d162` · okf_url: https://limbo137.top/okf/concepts/9679e150-e342-4615-98e6-84014c53d162.md
- labels: en = measurable space；zh-CN = 可测空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-dad37d95bc8cf0c1d993-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 混合张量(mixed tensor)

- UUID: `96fbecaf-6992-4f99-9242-47463c445948` · okf_url: https://limbo137.top/okf/concepts/96fbecaf-6992-4f99-9242-47463c445948.md
- labels: en = mixed tensor；zh-CN = 混合张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-e5651aa5916f5733639f-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-e5651aa5916f5733639f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 向量场的协变导数(covariant derivative of a vector field)

- UUID: `97514113-7cb5-48b4-9541-733ea93565c0` · okf_url: https://limbo137.top/okf/concepts/97514113-7cb5-48b4-9541-733ea93565c0.md
- labels: en = covariant derivative of a vector field；zh-CN = 向量场的协变导数
- aliases: covariant derivative；协变导数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-34eace036055d44a85b5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-34eace036055d44a85b5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 原像(preimage)

- UUID: `9778c275-047c-4ba5-88d8-10a3f4eb2a5a` · okf_url: https://limbo137.top/okf/concepts/9778c275-047c-4ba5-88d8-10a3f4eb2a5a.md
- labels: en = preimage；zh-CN = 原像
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-cca7ba4bbd70474a1b10-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 旋量表示(spinor representation)

- UUID: `9784dcc7-9e6e-4545-b544-9c8e88f90bf7` · okf_url: https://limbo137.top/okf/concepts/9784dcc7-9e6e-4545-b544-9c8e88f90bf7.md
- labels: en = spinor representation；zh-CN = 旋量表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.3 · `books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary#first-a5b0282092358cde57b2-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.3 · `books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary#first-a5b0282092358cde57b2-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md

## 局部可积(locally integrable)

- UUID: `978636a4-0d2c-40e2-8800-0037500324a9` · okf_url: https://limbo137.top/okf/concepts/978636a4-0d2c-40e2-8800-0037500324a9.md
- labels: en = locally integrable；zh-CN = 局部可积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-1e78b459360a0b74fac7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 李群的李代数(Lie algebra of a Lie group)

- UUID: `978dbe89-0e8f-46c3-b80a-20f64bb067fc` · okf_url: https://limbo137.top/okf/concepts/978dbe89-0e8f-46c3-b80a-20f64bb067fc.md
- labels: en = Lie algebra of a Lie group；zh-CN = 李群的李代数
- aliases: Lie algebra of the Lie group
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-ef34c11150218cf44b2b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 恒等映射(identity mapping)

- UUID: `97b3a40b-9b3f-4b28-b8ce-5cd7d2a62e46` · okf_url: https://limbo137.top/okf/concepts/97b3a40b-9b3f-4b28-b8ce-5cd7d2a62e46.md
- labels: en = identity mapping；zh-CN = 恒等映射
- aliases: identity map
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-f94561d3a8a555355d4f-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-c103b1aa760687fcb609-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §1.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `A bijective map $\varphi : X \to X$ from $X$ onto itself is called a transformation of $X$. The most trivial transformation of all is the identity map $\mathrm{id}$ defined by $$ \mathrm{id} _ {X} (x) = x, \quad \forall x \in X. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 玻色子(boson)

- UUID: `97d3aaf4-eaff-44bf-9f79-29b5b6749422` · okf_url: https://limbo137.top/okf/concepts/97d3aaf4-eaff-44bf-9f79-29b5b6749422.md
- labels: en = boson；zh-CN = 玻色子
- aliases: bosons
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-e6107de2af9b0ac4db43-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-9b1589d89dfd5a52ab6d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 稠密集(dense set)

- UUID: `97f428fa-4790-4aff-9bc7-7928fe489d3c` · okf_url: https://limbo137.top/okf/concepts/97f428fa-4790-4aff-9bc7-7928fe489d3c.md
- labels: en = dense set；zh-CN = 稠密集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-121d77c9c61c7b6844dd-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 退相干(decoherence)

- UUID: `97fb932d-a189-418b-9bc4-efac0ef32ed6` · okf_url: https://limbo137.top/okf/concepts/97fb932d-a189-418b-9bc4-efac0ef32ed6.md
- labels: en = decoherence；zh-CN = 退相干
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-be96bf58a467a989042d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/06-for-further-reading.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 覆叠空间(covering space)

- UUID: `980eae96-7af8-5311-a7e2-f65c0020e8c6` · okf_url: https://limbo137.top/okf/concepts/980eae96-7af8-5311-a7e2-f65c0020e8c6.md
- labels: en = covering space；zh-CN = 覆叠空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ p ^ {- 1} (U) = \bigcup_ {n \in \mathbb {Z}} (x _ {0} - 0. 1 + n, x _ {0} + 0. 1 + n) $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `which is a disjoint union of open sets of $\mathbb{R}$ . It is easy to show that $p$ is also a homomorphism with respect to addition in $\mathbb{R}$ and multiplication in $\mathrm{U}(1)$ . Hence, $(\mathbb{R}, p)$ is the universal covering group of $\mathrm{U}(1) = S^1$ .`
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.12. Let $X$ and $\widetilde{X}$ be connected topological spaces. The pair $(\widetilde{X}, p)$ , or simply $\widetilde{X}$ , is called the covering space of $X$ if there exists a continuous map $p: \widetilde{X} \to X$ such that`
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For example, $\mathbb{R}$ is the universal covering space of $S^1$ , see section 4.3. Since $S^1$ is identified with $\mathrm{U}(1)$ , $\mathbb{R}$ is a universal covering group of $\mathrm{U}(1)$ if $\mathbb{R}$ is regarded as an additive group. The map $p: \mathbb{R} \to \mathrm{U}(1)$ may be $p: x \to \mathrm{e}^{\mathrm{i}2\pi x}$ . Clearly $p$ is sur...`
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `In particular, if $\widetilde{X}$ is simply connected, $(\widetilde{X}, p)$ is called the universal covering space of $X$ . [Remarks: Certain groups are known to be topological spaces. They are called topological groups. For example $\mathrm{SO}(n)$ and $\mathrm{SU}(n)$ are topological groups. If $X$ and $\widetilde{X}$ in definition 4.12 happen to be top...`
  - nakahara-geometry-topology-physics/second-edition-local §4.6.5 (pdf_page=168, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.12：覆叠空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(1) $p$ is surjective (onto) (2) for each $x \in X$ , there exists a connected open set $U \subset X$ containing $x$ , such that $p^{-1}(U)$ is a disjoint union of open sets in $\widetilde{X}$ , each of which is mapped homeomorphically onto $U$ by $p$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 权(weight)

- UUID: `983a40d9-c3f5-4be7-bca7-0561ba9fab62` · okf_url: https://limbo137.top/okf/concepts/983a40d9-c3f5-4be7-bca7-0561ba9fab62.md
- labels: en = weight；zh-CN = 权
- aliases: weights
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §8.1 · `books/woit-qtgr/2025-10-20/zh/chapter-08/01-representations-of-su2-classification#first-a12d394f714291926271-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §8.1 · `books/woit-qtgr/2025-10-20/en/chapter-08/01-representations-of-su2-classification#first-3e137bf5a178a24a2a2b-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/01-representations-of-su2-classification.md

## 代数同态(algebra homomorphism)

- UUID: `984484bb-484a-47b5-89ac-6eb7a35f9f7c` · okf_url: https://limbo137.top/okf/concepts/984484bb-484a-47b5-89ac-6eb7a35f9f7c.md
- labels: en = algebra homomorphism；zh-CN = 代数同态
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-45da2b432d6f6d5a2ab4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-algebra-homomorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 子集(subset)

- UUID: `98a72c65-8dcc-43d1-b199-ee3daa22cb2c` · okf_url: https://limbo137.top/okf/concepts/98a72c65-8dcc-43d1-b199-ee3daa22cb2c.md
- labels: en = subset；zh-CN = 子集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-0866e3b696afe03917dc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 酉群(unitary group)

- UUID: `98d3a096-7153-450f-9725-091ca1eea20c` · okf_url: https://limbo137.top/okf/concepts/98d3a096-7153-450f-9725-091ca1eea20c.md
- labels: en = unitary group；zh-CN = 酉群
- aliases: Unitary group
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-unitary-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-7a6137edbe02f84c9bd4-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.6 · `books/woit-qtgr/2025-10-20/en/chapter-04/06-orthogonal-and-unitary-transformations#first-7a6137edbe02f84c9bd4-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.6 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Unitary group). The unitary group $U ( n )$ in $n$ dimensions is the group of invertible transformations preserving a Hermitian inner product on a complex $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ complex invertible matrices satisfying $$ L ^ {- 1} = \overline {{L}} ^ {T} = L ^ {\dagger} $$`
  - woit-qtgr/2025-10-20 §4.6 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（酉群(unitary group)）。$n$维的酉群$U ( n )$是保持复$n$维向量空间$V$上厄米内积的可逆变换构成的群。它同构于满足以下条件的$n$乘$n$复可逆矩阵构成的群： $$ L ^ {- 1} = \overline {{L}} ^ {T} = L ^ {\dagger} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/06-orthogonal-and-unitary-transformations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/03-a-summary.md

## 线性近似(linearized approximation)

- UUID: `98f26388-ced3-42da-808f-8c4fcb33803d` · okf_url: https://limbo137.top/okf/concepts/98f26388-ced3-42da-808f-8c4fcb33803d.md
- labels: en = linearized approximation；zh-CN = 线性近似
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-5909adf110e9fae4c729-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-5909adf110e9fae4c729-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 费米子(fermion)

- UUID: `98f78d2b-8094-41b8-a6bb-713325f793a0` · okf_url: https://limbo137.top/okf/concepts/98f78d2b-8094-41b8-a6bb-713325f793a0.md
- labels: en = fermion；zh-CN = 费米子
- aliases: fermions
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-e85a10c313130665f222-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-19533ec6c0c0ffb4239c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 曲率张量(curvature tensor)

- UUID: `99710f21-885f-448b-a6e8-4048c38b658c` · okf_url: https://limbo137.top/okf/concepts/99710f21-885f-448b-a6e8-4048c38b658c.md
- labels: en = curvature tensor；zh-CN = 曲率张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-193b746269ba35c30dca-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-193b746269ba35c30dca-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 二元运算(binary operation)

- UUID: `997dc740-d5b3-48e6-ad69-9d4d90ed540f` · okf_url: https://limbo137.top/okf/concepts/997dc740-d5b3-48e6-ad69-9d4d90ed540f.md
- labels: en = binary operation；zh-CN = 二元运算
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-ac4fc9349aaa8a0cc158-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 不可分拓扑(indiscrete topology)

- UUID: `99d3a8fc-4856-47a6-824e-8ee0083dec99` · okf_url: https://limbo137.top/okf/concepts/99d3a8fc-4856-47a6-824e-8ee0083dec99.md
- labels: en = indiscrete topology；zh-CN = 不可分拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-e5c6d3c4a6e37e4a4ed5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 带交叉帽的球面(sphere with crosscaps)

- UUID: `99d3c359-bd83-4b6a-be0d-432df132bc0e` · okf_url: https://limbo137.top/okf/concepts/99d3c359-bd83-4b6a-be0d-432df132bc0e.md
- labels: en = sphere with crosscaps；zh-CN = 带交叉帽的球面
- aliases: crosscap
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 测度空间(measure space)

- UUID: `99e2c957-1871-4274-8461-a308d395c420` · okf_url: https://limbo137.top/okf/concepts/99e2c957-1871-4274-8461-a308d395c420.md
- labels: en = measure space；zh-CN = 测度空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-3e5c2f97b6919a9f7eab-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 仿紧的(paracompact)

- UUID: `9a348175-31ec-4401-afa6-cb89bf54511c` · okf_url: https://limbo137.top/okf/concepts/9a348175-31ec-4401-afa6-cb89bf54511c.md
- labels: en = paracompact；zh-CN = 仿紧的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.1 · `zh/chapter-17/01-partitions-of-unity#first-e0b213c4c786619739d5-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.1 · `en/chapter-17/01-partitions-of-unity#first-e0b213c4c786619739d5-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/01-partitions-of-unity.md

## 一般线性群(general linear group)

- UUID: `9a7432a7-4b27-41b4-83d5-d9b6ff0ce028` · okf_url: https://limbo137.top/okf/concepts/9a7432a7-4b27-41b4-83d5-d9b6ff0ce028.md
- labels: en = general linear group；zh-CN = 一般线性群
- aliases: general linear groups；广义线性群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-general-linear-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-d7a574bbe40adfb64e6d-zh` · translation)；woit-qtgr/2025-10-20 (en · §1.3 · `books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations#first-d7a574bbe40adfb64e6d-en` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §3.3 (pdf_page=not-recorded, status=librarian-source-reviewed; pending-independent-review, dependency_status=not-recorded) · 条件: 线性自同构构造；向量空间 V 上的可逆线性算符，运算为映射复合。 · 原文: `Linear maps $S : V \to V$ are called linear operators on $V$. They form the vector space $L ( V , V )$ . If $S$ is an invertible linear operator on $V$ it is called a linear transformation on $V$. It may be thought of as a vector space isomorphism of $V$ onto itself, or an automorphism of $V$. The linear transformations of $V$ form a group with respect to...`
  - szekeres/source-snapshot-bc3404156f6e §2.3 (pdf_page=not-recorded, status=librarian-source-reviewed; pending-independent-review, dependency_status=not-recorded) · 条件: 矩阵构造；实数或复数域上的有限阶非奇异方阵，运算为矩阵乘法。 · 原文: `The set of all $n \times n$ non-singular real matrices is a group, denoted $G L ( n , \mathbb { R } )$ . The key to this result is the product law of determinants $$ \det (\mathsf {A B}) = \det (\mathsf {A}) \det (\mathsf {B}).\tag{2.5} $$ Closure: this follows from the fact that $\det \mathsf { A } \neq 0$ and $\det \mathtt { B } \neq 0$ implies that $\b...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 纯四元数(pure quaternion)

- UUID: `9a93e3ee-75af-4603-8af3-5487ba52f0ae` · okf_url: https://limbo137.top/okf/concepts/9a93e3ee-75af-4603-8af3-5487ba52f0ae.md
- labels: en = pure quaternion；zh-CN = 纯四元数
- aliases: Pure quaternions
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-pure-quaternion` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-75d39da5969f8f84b4bf-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md

## 可测集(measurable set)

- UUID: `9aae2266-af5b-4543-9ecd-4efcddaf22f8` · okf_url: https://limbo137.top/okf/concepts/9aae2266-af5b-4543-9ecd-4efcddaf22f8.md
- labels: en = measurable set；zh-CN = 可测集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-ae0bf7c54b9f9325e947-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## Legendre变换(Legendre transform)

- UUID: `9ab99bcf-0f4e-4b96-bafa-fc34e2cb40de` · okf_url: https://limbo137.top/okf/concepts/9ab99bcf-0f4e-4b96-bafa-fc34e2cb40de.md
- labels: en = Legendre transform；zh-CN = Legendre变换
- 分书定义状态: schwartz-qft/local-experiment = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-42ecaf6e1d80a94e1594-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-42ecaf6e1d80a94e1594-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
- 显式依赖组(dependency_groups): schwartz-qft/local-experiment (2 条, completeness=recorded-only) · 摘录: `Using Eq. (14.12), each term in Eq. (14.10) becomes just a number and the product reduces to`；schwartz-qft/local-experiment (1 条, completeness=recorded-only) · 摘录: `Now we can use the Gaussian integral in Eq. (14.6), $\int dp\exp \left(-\frac{1}{2} ap^2 +Jp\right) = \sqrt{\frac{2\pi}{a}}\exp \left(\frac{J^2}{2a}\right)$ , w...`
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 泛性质(universal property)

- UUID: `9ac8f727-4eb8-406f-8ac4-006f5b7d7a7f` · okf_url: https://limbo137.top/okf/concepts/9ac8f727-4eb8-406f-8ac4-006f5b7d7a7f.md
- labels: en = universal property；zh-CN = 泛性质
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-315dbfc6265865430b46-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-315dbfc6265865430b46-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 欧氏等距群(full Euclidean group)

- UUID: `9ada65b6-1571-46df-a3d0-bb2763594eff` · okf_url: https://limbo137.top/okf/concepts/9ada65b6-1571-46df-a3d0-bb2763594eff.md
- labels: en = full Euclidean group；zh-CN = 欧氏等距群
- aliases: Euclidean group；欧氏等距群（含反射）；欧氏群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-euclidean-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 经验熵(empirical entropy)

- UUID: `9b0f1e4c-11ed-41bf-bfbb-42b4279c4916` · okf_url: https://limbo137.top/okf/concepts/9b0f1e4c-11ed-41bf-bfbb-42b4279c4916.md
- labels: en = empirical entropy；zh-CN = 经验熵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-da370a7bd4185b5f423f-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-da370a7bd4185b5f423f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 切丛(tangent bundle)

- UUID: `9b37ab33-699e-4407-9ec2-2f983ae360ed` · okf_url: https://limbo137.top/okf/concepts/9b37ab33-699e-4407-9ec2-2f983ae360ed.md
- labels: en = tangent bundle；zh-CN = 切丛
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-tangent-bundle` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-fb22533a97ceec737628-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 交替群(alternating group)

- UUID: `9b61e2c0-8342-477e-aaf1-6d5cb829583d` · okf_url: https://limbo137.top/okf/concepts/9b61e2c0-8342-477e-aaf1-6d5cb829583d.md
- labels: en = alternating group；zh-CN = 交替群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-alternating-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 等势(equinumerous)

- UUID: `9b99a8bf-735d-4de5-a8d9-ce80d7f1104a` · okf_url: https://limbo137.top/okf/concepts/9b99a8bf-735d-4de5-a8d9-ce80d7f1104a.md
- labels: en = equinumerous；zh-CN = 等势
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-419bc247d1bf514dc858-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 对合分布(involutive distribution)

- UUID: `9bcdf532-9fed-4389-8edd-48cb894e1be9` · okf_url: https://limbo137.top/okf/concepts/9bcdf532-9fed-4389-8edd-48cb894e1be9.md
- labels: en = involutive distribution；zh-CN = 对合分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.6 · `zh/chapter-15/06-distributions-and-frobenius-theorem#concept-15-involutive-distribution` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/06-distributions-and-frobenius-theorem.md

## 群作用(group action)

- UUID: `9bd75d12-0eb0-4d97-ab55-23361589bdab` · okf_url: https://limbo137.top/okf/concepts/9bd75d12-0eb0-4d97-ab55-23361589bdab.md
- labels: en = group action；zh-CN = 群作用
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-2114245f91c19109af2f-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-2114245f91c19109af2f-en` · machine_draft)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=236, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.14：作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `[Remark: We often use the notation $gp$ instead of $\sigma(g, p)$ . The second condition in this notation is $g_1(g_2p) = (g_1g_2)p$ .]`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=235, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.14：作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.14. Let $G$ be a Lie group and $M$ be a manifold. The action of $G$ on $M$ is a differentiable map $\sigma : G \times M \to M$ which satisfies the conditions`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=235, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.14：作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $\sigma (e,p) = p$ for any $p\in M$ (5.141a) (ii) $\sigma (g_1,\sigma (g_2,p)) = \sigma (g_1g_2,p).$ (5.141b)`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/01-the-symplectic-group.md

## 正交群(orthogonal group)

- UUID: `9c0cba6a-c2b4-4c44-b15e-b04e84d239af` · okf_url: https://limbo137.top/okf/concepts/9c0cba6a-c2b4-4c44-b15e-b04e84d239af.md
- labels: en = orthogonal group；zh-CN = 正交群
- aliases: Orthogonal group；orthogonal groups
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-orthogonal-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-bbb06fbb334ea918614f-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.6 · `books/woit-qtgr/2025-10-20/en/chapter-04/06-orthogonal-and-unitary-transformations#first-bbb06fbb334ea918614f-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.6 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Orthogonal group). The orthogonal group $O ( n )$ in $n$ dimensions is the group of invertible transformations preserving an inner product on a real $n$ dimensional vector space $V$ . This is isomorphic to the group of $n$ by $n$ real invertible matrices $L$ satisfying $$ L ^ {- 1} = L ^ {T} $$`
  - woit-qtgr/2025-10-20 §4.6 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（正交群(orthogonal group)）。$n$维正交群$O ( n )$是保持实$n$维向量空间$V$上内积的可逆变换构成的群。它同构于满足下式的 $n$ 乘 $n$ 实可逆矩阵 $L$ 所构成的群 $$ L ^ {- 1} = L ^ {T} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/06-orthogonal-and-unitary-transformations.md

## 磁矩算符(magnetic moment operator)

- UUID: `9c252b25-011e-4b3d-8bf1-b54966fe2881` · okf_url: https://limbo137.top/okf/concepts/9c252b25-011e-4b3d-8bf1-b54966fe2881.md
- labels: en = magnetic moment operator；zh-CN = 磁矩算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.2 · `books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#concept-7-magnetic-moment-operator` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-eaf57b445651d135ec4a-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 单粒子算符的局域性(locality of single-particle operators)

- UUID: `9c364158-a7d0-454f-95b3-5175b7142824` · okf_url: https://limbo137.top/okf/concepts/9c364158-a7d0-454f-95b3-5175b7142824.md
- labels: en = locality of single-particle operators；zh-CN = 单粒子算符的局域性
- aliases: local；局域性（单粒子算符）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §38.1 · `books/woit-qtgr/2025-10-20/en/chapter-38/01-unitary-transformations-on-h1#first-4ee982f576db118729e5-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/01-unitary-transformations-on-h1.md

## 总角动量(total angular momentum)

- UUID: `9c61b51f-4d17-4106-b147-7257f22762ec` · okf_url: https://limbo137.top/okf/concepts/9c61b51f-4d17-4106-b147-7257f22762ec.md
- labels: en = total angular momentum；zh-CN = 总角动量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §34.2 · `books/woit-qtgr/2025-10-20/en/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-d833930ed5edd37f90d0-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §34.2 · `books/woit-qtgr/2025-10-20/zh/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-d833930ed5edd37f90d0-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3.md

## 单调收敛定理(monotone convergence theorem)

- UUID: `9c65e0ef-4b4f-4119-8cd8-d4144a929324` · okf_url: https://limbo137.top/okf/concepts/9c65e0ef-4b4f-4119-8cd8-d4144a929324.md
- labels: en = monotone convergence theorem；zh-CN = 单调收敛定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-66225a5512559fb6f8eb-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 事件视界(event horizon)

- UUID: `9c94cffc-a155-4a23-9c2f-298515ef4d50` · okf_url: https://limbo137.top/okf/concepts/9c94cffc-a155-4a23-9c2f-298515ef4d50.md
- labels: en = event horizon；zh-CN = 事件视界
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-aae3a818b4ab5afa13ff-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-aae3a818b4ab5afa13ff-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 定向2-单纯形(oriented 2-simplex)

- UUID: `9ca732a9-0bb8-478c-bb77-52dafdd55473` · okf_url: https://limbo137.top/okf/concepts/9ca732a9-0bb8-478c-bb77-52dafdd55473.md
- labels: en = oriented 2-simplex；zh-CN = 定向2-单纯形
- aliases: $\sigma_2 = (p_0p_1p_2)$；定向 2-单形；定向 2-单纯形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=120, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 三顶点三角区域附边方向，偶排列同向、奇排列反向；保留维数2特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Similarly, an oriented 2-simplex $\sigma_{2} = (p_{0}p_{1}p_{2})$ is a triangular region $p_0p_1p_2$ with a prescribed orientation along the edges (figure 3.6(b)). Observe that the orientation given by $p_0p_1p_2$ is the same as that given by $p_2p_0p_1$ or $p_1p_2p_0$ but opposite to $p_0p_2p_1$ , $p_2p_1p_0$ or $p_1p_0p_2$ . We require that`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=120, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 三顶点三角区域附边方向，偶排列同向、奇排列反向；保留维数2特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \begin{array}{l} (p _ {0} p _ {1} p _ {2}) = (p _ {2} p _ {0} p _ {1}) = (p _ {1} p _ {2} p _ {0}) \\ = - \left(p _ {0} p _ {2} p _ {1}\right) = - \left(p _ {2} p _ {1} p _ {0}\right) = - \left(p _ {1} p _ {0} p _ {2}\right). \\ \end{array} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 有限子覆盖(finite subcovering)

- UUID: `9cd293e5-41f8-41f6-bc80-04d11b5fe31b` · okf_url: https://limbo137.top/okf/concepts/9cd293e5-41f8-41f6-bc80-04d11b5fe31b.md
- labels: en = finite subcovering；zh-CN = 有限子覆盖
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-b1489a82892540b65f5f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 同态基本定理(fundamental theorem of homomorphism)

- UUID: `9cd6de15-4314-4caf-b479-24e7015985ae` · okf_url: https://limbo137.top/okf/concepts/9cd6de15-4314-4caf-b479-24e7015985ae.md
- labels: en = fundamental theorem of homomorphism；zh-CN = 同态基本定理
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.1 (pdf_page=114, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.1及式3.6，群同态f的G1/ker f≅im f；不按泛化词推广到其他范畴。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Theorem 3.1. (Fundamental theorem of homomorphism) Let $f: G_1 \to G_2$ be a homomorphism. Then`
  - nakahara-geometry-topology-physics/second-edition-local §3.1.1 (pdf_page=114, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定理3.1及式3.6，群同态f的G1/ker f≅im f；不按泛化词推广到其他范畴。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ G _ {1} / \ker f \cong \operatorname {i m} f. \tag {3.6} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 二重向量(bivector)

- UUID: `9cf8dd01-d877-4d37-a1ea-6edadf79b415` · okf_url: https://limbo137.top/okf/concepts/9cf8dd01-d877-4d37-a1ea-6edadf79b415.md
- labels: en = bivector；zh-CN = 二重向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-20ec19259fce5297812c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-bivector` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md

## 线性变换(linear transformation)

- UUID: `9d1874b8-6c8c-465e-849a-f2ddc348d3d0` · okf_url: https://limbo137.top/okf/concepts/9d1874b8-6c8c-465e-849a-f2ddc348d3d0.md
- labels: en = linear transformation；zh-CN = 线性变换
- aliases: linear transformations
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §16.1 · `books/woit-qtgr/2025-10-20/en/chapter-16/01-the-symplectic-group#first-8e4012da08917067c3b4-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-5e64326d1ddbf841b8da-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/03-change-of-basis.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/01-the-symplectic-group.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 张量场(tensor field)

- UUID: `9d3097aa-7a26-4f11-a534-22f5389499f0` · okf_url: https://limbo137.top/okf/concepts/9d3097aa-7a26-4f11-a534-22f5389499f0.md
- labels: en = tensor field；zh-CN = 张量场
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-tensor-field` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-e7ca85ef91659b4c5911-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 挠率2-形式(torsion 2-form)

- UUID: `9d30b85b-344c-429f-b827-475935331862` · okf_url: https://limbo137.top/okf/concepts/9d30b85b-344c-429f-b827-475935331862.md
- labels: en = torsion 2-form；zh-CN = 挠率2-形式
- aliases: torsion 2-forms；挠率二形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.7 · `zh/chapter-18/07-cartan-formalism#first-aa6c8a8deea6b25d0dcb-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.7 · `en/chapter-18/07-cartan-formalism#first-cd42f3066d40aa7b729a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/07-cartan-formalism.md

## 费米子Fock空间(fermionic Fock space)

- UUID: `9d30f001-86e0-4e06-9631-c199d9cc5ae4` · okf_url: https://limbo137.top/okf/concepts/9d30f001-86e0-4e06-9631-c199d9cc5ae4.md
- labels: en = fermionic Fock space；zh-CN = 费米子Fock空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §31.3 · `books/woit-qtgr/2025-10-20/zh/chapter-31/04-spinors-and-the-bargmann-fock-construction#first-1c8d82155046b0e8ad06-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §31.3 · `books/woit-qtgr/2025-10-20/en/chapter-31/04-spinors-and-the-bargmann-fock-construction#first-1c8d82155046b0e8ad06-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/04-spinors-and-the-bargmann-fock-construction.md

## 欧几里得空间(Euclidean space)

- UUID: `9d440069-537c-483e-8258-cc51a6c2ac9c` · okf_url: https://limbo137.top/okf/concepts/9d440069-537c-483e-8258-cc51a6c2ac9c.md
- labels: en = Euclidean space；zh-CN = 欧几里得空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-5fff31f866001ff37147-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-5fff31f866001ff37147-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 结合律(associative law)

- UUID: `9da874db-a055-441b-9133-1a2d9c5350ce` · okf_url: https://limbo137.top/okf/concepts/9da874db-a055-441b-9133-1a2d9c5350ce.md
- labels: en = associative law；zh-CN = 结合律
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-associative-law` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 复内积(complex inner product)

- UUID: `9dc809a9-95f3-4c45-bb82-06dd391d1f6b` · okf_url: https://limbo137.top/okf/concepts/9dc809a9-95f3-4c45-bb82-06dd391d1f6b.md
- labels: en = complex inner product；zh-CN = 复内积
- aliases: inner product；内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-inner-product` · translation)；woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-7fbfc15c27cd8776f6cd-complex-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 流形(manifold)

- UUID: `9df678fa-3e31-41da-9c0c-ae284360d723` · okf_url: https://limbo137.top/okf/concepts/9df678fa-3e31-41da-9c0c-ae284360d723.md
- labels: en = manifold；zh-CN = 流形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 选择公理(axiom of choice)

- UUID: `9e1d5449-45f6-40f9-8d47-0cf092efa242` · okf_url: https://limbo137.top/okf/concepts/9e1d5449-45f6-40f9-8d47-0cf092efa242.md
- labels: en = axiom of choice；zh-CN = 选择公理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-dae11b63504294bf66fd-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 子表示(subrepresentation)

- UUID: `9e24a1e3-d6ef-4cdf-8871-7731a246b44d` · okf_url: https://limbo137.top/okf/concepts/9e24a1e3-d6ef-4cdf-8871-7731a246b44d.md
- labels: en = subrepresentation；zh-CN = 子表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-subrepresentation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md

## 复结构(complex structure)

- UUID: `9e862121-659d-4ec4-8d9f-204bdd709d9f` · okf_url: https://limbo137.top/okf/concepts/9e862121-659d-4ec4-8d9f-204bdd709d9f.md
- labels: en = complex structure；zh-CN = 复结构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.2 · `zh/chapter-06/02-complex-numbers-and-complex-structures#concept-6-complex-structure` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.2 · `en/chapter-06/02-complex-numbers-and-complex-structures#first-cc6b390cf641456f2750-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §26 · `books/woit-qtgr/2025-10-20/en/chapter-26/index#first-da487ccbf2475101f64c-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26 · `books/woit-qtgr/2025-10-20/zh/chapter-26/index#first-da487ccbf2475101f64c-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/02-complex-numbers-and-complex-structures.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/01-complex-structures-and-phase-space.md

## 紧支集(compact support)

- UUID: `9ea30020-de0b-4e68-a69f-2702d7b9caa8` · okf_url: https://limbo137.top/okf/concepts/9ea30020-de0b-4e68-a69f-2702d7b9caa8.md
- labels: en = compact support；zh-CN = 紧支集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-8508c05f312930cdb661-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 度量空间(metric space)

- UUID: `9ecea840-2cf6-4b0e-9f6c-8d08670a4b94` · okf_url: https://limbo137.top/okf/concepts/9ecea840-2cf6-4b0e-9f6c-8d08670a4b94.md
- labels: en = metric space；zh-CN = 度量空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-43847cf4b94d42e36279-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 时间平移(time translation)

- UUID: `9ed047f9-1af4-4fff-bb3d-d3ded6b43252` · okf_url: https://limbo137.top/okf/concepts/9ed047f9-1af4-4fff-bb3d-d3ded6b43252.md
- labels: en = time translation；zh-CN = 时间平移
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.4 · `books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics#concept-1-time-translation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md

## 自伴算符(self-adjoint operator in Woit representation theory)

- UUID: `9f2dec8c-99ca-42d9-be8f-4f6c039a058a` · okf_url: https://limbo137.top/okf/concepts/9f2dec8c-99ca-42d9-be8f-4f6c039a058a.md
- labels: en = self-adjoint operator in Woit representation theory；zh-CN = 自伴算符
- aliases: self-adjoint operators；自伴算符（Woit表象论语境）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §1.4 · `books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics#concept-1-self-adjoint-operator` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md

## 拓扑向量空间(topological vector space)

- UUID: `9f888a8d-f2d1-435c-9a64-9e5410e6148c` · okf_url: https://limbo137.top/okf/concepts/9f888a8d-f2d1-435c-9a64-9e5410e6148c.md
- labels: en = topological vector space；zh-CN = 拓扑向量空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.9 · `zh/chapter-10/09-topological-vector-spaces#first-6a55cd98bb64fb1c08ed-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md

## 变换群(transformation group)

- UUID: `a0008efe-3fef-4617-9ff0-159a32b75bde` · okf_url: https://limbo137.top/okf/concepts/a0008efe-3fef-4617-9ff0-159a32b75bde.md
- labels: en = transformation group；zh-CN = 变换群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-transformation-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-134ed74213a223ada75d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 类空向量(spacelike vector)

- UUID: `a013ad57-8c7e-4a67-8001-ba939965b6c8` · okf_url: https://limbo137.top/okf/concepts/a013ad57-8c7e-4a67-8001-ba939965b6c8.md
- labels: en = spacelike vector；zh-CN = 类空向量
- aliases: spacelike
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §40.1 · `books/woit-qtgr/2025-10-20/en/chapter-40/01-minkowski-space#first-afbfa88720c82c722652-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 固有电荷密度(proper charge density)

- UUID: `a02550bd-936b-4152-a46e-65678fcc71d3` · okf_url: https://limbo137.top/okf/concepts/a02550bd-936b-4152-a46e-65678fcc71d3.md
- labels: en = proper charge density；zh-CN = 固有电荷密度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-cc489f6e49ed53e7a5f2-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-cc489f6e49ed53e7a5f2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 可微流形(differentiable manifold)

- UUID: `a0553dab-d811-417e-8a79-67ec620792e7` · okf_url: https://limbo137.top/okf/concepts/a0553dab-d811-417e-8a79-67ec620792e7.md
- labels: en = differentiable manifold；zh-CN = 可微流形
- aliases: smooth manifold；光滑流形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-00aa8b52b61298eda3c9-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-differentiable-manifold` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-668c74fa248813950d61-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=191, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(iv) given $U_i$ and $U_j$ such that $U_i \cap U_j \neq \emptyset$ , the map $\psi_{ij} = \varphi_i \circ \varphi_j^{-1}$ from $\varphi_j(U_i \cap U_j)$ to $\varphi_i(U_i \cap U_j)$ is infinitely differentiable.`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=190, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.1. $M$ is an $m$ -dimensional differentiable manifold if`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=192, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If $U_i$ and $U_j$ overlap, two coordinate systems are assigned to a point in $U_i \cap U_j$ . Axiom (iv) asserts that the transition from one coordinate system to another be smooth ( $C^\infty$ ). The map $\varphi_i$ assigns $m$ coordinate values $x^\mu$ ( $1 \leq \mu \leq m$ ) to a point $p \in U_i \cap U_j$ , while $\varphi_j$ assigns $y^\nu$ ( $1 \leq...`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=192, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `If the union of two atlases $\{(U_i, \varphi_i)\}$ and $\{(V_j, \psi_j)\}$ is again an atlas, these two atlases are said to be compatible. The compatibility is an equivalence relation, the equivalence class of which is called the differentiable structure. It is also said that mutually compatible atlases define the same differentiable structure on $M$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=192, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Before we give examples, we briefly comment on manifolds with boundaries. So far, we have assumed that the coordinate neighbourhood $U_{i}$ is homeomorphic to an open set of $\mathbb{R}^m$ . In some applications, however, this turns out to be too restrictive and we need to relax this condition. If a topological space $M$ is covered by a family of open set...`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=191, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `The pair $(U_i, \varphi_i)$ is called a chart while the whole family $\{(U_i, \varphi_i)\}$ is called, for obvious reasons, an atlas. The subset $U_i$ is called the coordinate neighbourhood while $\varphi_i$ is the coordinate function or, simply, the coordinate. The homeomorphism $\varphi_i$ is represented by $m$ functions $\{x^1(p), \ldots, x^m(p)\}$ . T...`
  - nakahara-geometry-topology-physics/second-edition-local §5.1.2 (pdf_page=190, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.1：可微流形。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $M$ is a topological space; (ii) $M$ is provided with a family of pairs $\{(U_i,\varphi_i)\}$ ; (iii) $\{U_i\}$ is a family of open sets which covers $M$ , that is, $\cup_{i} U_i = M$ . $\varphi_i$ is a homeomorphism from $U_i$ onto an open subset $U_i'$ of $\mathbb{R}^m$ (figure 5.2); and`
  - szekeres/source-snapshot-bc3404156f6e §15.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Any atlas $\boldsymbol { A }$ may thus be extended to a maximal atlas by adding to it all charts that are $C ^ { \infty }$ -compatible with the charts of . This maximal atlas is called a differentiable structure on $M$. A pair $( M , A )$ , where $M$, is an $n$-dimensional topological manifold and is a differentiable structure on $M ,$ , is called a diffe...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md

## 分支群(group of components)

- UUID: `a0657410-f98f-4bcb-9554-b3cc8a38759a` · okf_url: https://limbo137.top/okf/concepts/a0657410-f98f-4bcb-9554-b3cc8a38759a.md
- labels: en = group of components；zh-CN = 分支群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-e01c979bb31a09870c08-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 莫比乌斯带(Möbius band)

- UUID: `a09ffd88-cf37-40f6-a917-bff05e5ec894` · okf_url: https://limbo137.top/okf/concepts/a09ffd88-cf37-40f6-a917-bff05e5ec894.md
- labels: en = Möbius band；zh-CN = 莫比乌斯带
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.1 · `en/chapter-17/01-partitions-of-unity#first-777cca8da3dca9e69fc3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.1 · `zh/chapter-17/01-partitions-of-unity#first-777cca8da3dca9e69fc3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/01-partitions-of-unity.md

## 双射(bijection)

- UUID: `a0c09675-2669-4494-a712-c29017a7b391` · okf_url: https://limbo137.top/okf/concepts/a0c09675-2669-4494-a712-c29017a7b391.md
- labels: en = bijection；zh-CN = 双射
- aliases: bijective；bijective map；bijective mapping；bijective maps；双射映射；双射的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-9fdad7a27f354934cb46-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：双射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) A map $f: X \to Y$ is called injective (or one to one) if $x \neq x'$ implies $f(x) \neq f(x')$ . (b) A map $f: X \to Y$ is called surjective (or onto) if for each $y \in Y$ there exists at least one element $x \in X$ such that $f(x) = y$ . (c) A map $f: X \to Y$ is called bijective if it is both injective and surjective.`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.1 (pdf_page=87, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.1：双射。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.1. If a map satisfies a certain condition it bears a special name.`
  - szekeres/source-snapshot-bc3404156f6e §1.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 映射同时单射且满射 · 原文: `A map $\varphi$ that is injective and surjective, or equivalently one-to-one and onto, is called bijective or a bijection. In this and only this case can one define the inverse map $\varphi ^ { - 1 }$ $Y\to X$ having the property`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 入射波条件(ingoing wave condition)

- UUID: `a0f33d5b-4bdb-46e8-ae40-393c840c6f62` · okf_url: https://limbo137.top/okf/concepts/a0f33d5b-4bdb-46e8-ae40-393c840c6f62.md
- labels: en = ingoing wave condition；zh-CN = 入射波条件
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.4 · `zh/chapter-12/04-green-s-functions#first-27bc4e8abae23305209d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/04-green-s-functions.md

## 向量叉积(vector cross product)

- UUID: `a10b148f-7a19-4720-b9bc-78e287270624` · okf_url: https://limbo137.top/okf/concepts/a10b148f-7a19-4720-b9bc-78e287270624.md
- labels: en = vector cross product；zh-CN = 向量叉积
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#first-520fa7e8a1d606a122a8-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md

## 表示论(representation theory)

- UUID: `a16683df-6af0-4a23-b60d-31ff355a5ae6` · okf_url: https://limbo137.top/okf/concepts/a16683df-6af0-4a23-b60d-31ff355a5ae6.md
- labels: en = representation theory；zh-CN = 表示论
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-0d1f73d33008027b167a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/01-introduction.md

## 群同构(group isomorphism)

- UUID: `a21851fd-d421-4e32-9a3b-46fc03a5df9a` · okf_url: https://limbo137.top/okf/concepts/a21851fd-d421-4e32-9a3b-46fc03a5df9a.md
- labels: en = group isomorphism；zh-CN = 群同构
- aliases: isomorphism；同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-isomorphism` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-8ef151dcbf5aa11fbb49-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 第一积分(first integral)

- UUID: `a22bc13f-5935-4db9-875e-3485a213df7d` · okf_url: https://limbo137.top/okf/concepts/a22bc13f-5935-4db9-875e-3485a213df7d.md
- labels: en = first integral；zh-CN = 第一积分
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.3 · `en/chapter-16/03-frobenius-theorem-dual-form#first-60872f1afed48622a255-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.3 · `zh/chapter-16/03-frobenius-theorem-dual-form#first-60872f1afed48622a255-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/03-frobenius-theorem-dual-form.md

## 费米子场(fermionic fields)

- UUID: `a23b30c2-bd9f-4a80-b0da-1266559ffb7a` · okf_url: https://limbo137.top/okf/concepts/a23b30c2-bd9f-4a80-b0da-1266559ffb7a.md
- labels: en = fermionic fields；zh-CN = 费米子场
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §37.5 · `books/woit-qtgr/2025-10-20/en/chapter-37/05-fermion-fields#first-7dd2d4650bcb3e07da41-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §37.5 · `books/woit-qtgr/2025-10-20/zh/chapter-37/05-fermion-fields#first-7dd2d4650bcb3e07da41-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/05-fermion-fields.md

## 里奇恒等式(Ricci identities)

- UUID: `a2bccb82-6af3-444e-8e0b-264f69e220a2` · okf_url: https://limbo137.top/okf/concepts/a2bccb82-6af3-444e-8e0b-264f69e220a2.md
- labels: en = Ricci identities；zh-CN = 里奇恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-f78a31fb69e60acc7c50-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-f78a31fb69e60acc7c50-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 完备性关系(completeness relation)

- UUID: `a3fbacd4-6311-47d0-ba10-1410d55788ce` · okf_url: https://limbo137.top/okf/concepts/a3fbacd4-6311-47d0-ba10-1410d55788ce.md
- labels: en = completeness relation；zh-CN = 完备性关系
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-abe99102c93a3d612e1a-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-abe99102c93a3d612e1a-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 涡点(vortex point)

- UUID: `a4292ed1-76f5-4176-966f-5c9b3c78a5be` · okf_url: https://limbo137.top/okf/concepts/a4292ed1-76f5-4176-966f-5c9b3c78a5be.md
- labels: en = vortex point；zh-CN = 涡点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-vortex-point` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 标量部(scalar part)

- UUID: `a43e572c-cd06-4bc8-a90f-c177c9017645` · okf_url: https://limbo137.top/okf/concepts/a43e572c-cd06-4bc8-a90f-c177c9017645.md
- labels: en = scalar part；zh-CN = 标量部
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-812e6b9bc765a8ce699b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-scalar-part` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md

## 切映射(tangent map)

- UUID: `a44b2072-d453-40a7-a2d1-62be9964e197` · okf_url: https://limbo137.top/okf/concepts/a44b2072-d453-40a7-a2d1-62be9964e197.md
- labels: en = tangent map；zh-CN = 切映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.4 · `en/chapter-15/04-tangent-map-and-submanifolds#first-fd2eb6a1579ff4675386-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.4 · `zh/chapter-15/04-tangent-map-and-submanifolds#concept-15-tangent-map` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/04-tangent-map-and-submanifolds.md

## 链复形(chain complex)

- UUID: `a498f7ed-f758-4e27-8853-2e5a2c36d203` · okf_url: https://limbo137.top/okf/concepts/a498f7ed-f758-4e27-8853-2e5a2c36d203.md
- labels: en = chain complex；zh-CN = 链复形
- aliases: $C(K)$
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara3.3.2的单纯链群及边缘映射序列C(K)，完整式3.23及命名句；不是任意对象的链。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `where $i: 0 \hookrightarrow C_n(K)$ is an inclusion map (0 is regarded as the unit element of $C_n(K)$ ). This sequence is called the chain complex associated with $K$ and is denoted by $C(K)$ . It is interesting to study the image and kernel of the homomorphisms $\partial_r$ .`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara3.3.2的单纯链群及边缘映射序列C(K)，完整式3.23及命名句；不是任意对象的链。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Let $K$ be an $n$ -dimensional simplicial complex. There exists a sequence of free Abelian groups and homomorphisms,`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=123, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara3.3.2的单纯链群及边缘映射序列C(K)，完整式3.23及命名句；不是任意对象的链。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ 0 \xrightarrow {i} C _ {n} (K) \xrightarrow {\partial_ {n}} C _ {n - 1} (K) \xrightarrow {\partial_ {n - 1}} \dots \xrightarrow {\partial_ {2}} C _ {1} (K) \xrightarrow {\partial_ {1}} C _ {0} (K) \xrightarrow {\partial_ {0}} 0 \tag {3.23} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 路径的连接(path concatenation)

- UUID: `a51762fe-b5fc-5c7f-8802-7a6c7fec4299` · okf_url: https://limbo137.top/okf/concepts/a51762fe-b5fc-5c7f-8802-7a6c7fec4299.md
- labels: en = path concatenation；zh-CN = 路径的连接
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.2：路径的连接。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。这是端点相容路径的连接，不是一般代数乘积。 · 原文: `$$ \alpha * \beta (s) = \left\{ \begin{array}{l l} \alpha (2 s) & 0 \leq s \leq \frac {1}{2} \\ \beta (2 s - 1) & \frac {1}{2} \leq s \leq 1 \end{array} \right. \tag {4.1} $$`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.2：路径的连接。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。这是端点相容路径的连接，不是一般代数乘积。 · 原文: `see figure 4.2. Since $\alpha(1) = \beta(0)$ , $\alpha * \beta$ is a continuous map from $I$ to $X$ . [Geometrically, $\alpha * \beta$ corresponds to traversing the image $\alpha(I)$ , in the first half, then followed by $\beta(I)$ in the remaining half. Note that the velocity is doubled.]`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.2：路径的连接。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。这是端点相容路径的连接，不是一般代数乘积。 · 原文: `Definition 4.2. Let $\alpha, \beta: I \to X$ be paths such that $\alpha(1) = \beta(0)$ . The product of $\alpha$ and $\beta$ , denoted by $\alpha * \beta$ , is a path in $X$ defined by`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 四维加速度(4-acceleration)

- UUID: `a538059a-fc53-4e69-9b8b-e636980189c1` · okf_url: https://limbo137.top/okf/concepts/a538059a-fc53-4e69-9b8b-e636980189c1.md
- labels: en = 4-acceleration；zh-CN = 四维加速度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-5e3d5f6c601bc465725e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-5e3d5f6c601bc465725e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 循环(cycle)

- UUID: `a548ce48-3340-4d0d-a18e-ca75f604678e` · okf_url: https://limbo137.top/okf/concepts/a548ce48-3340-4d0d-a18e-ca75f604678e.md
- labels: en = cycle；zh-CN = 循环
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-cycle` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 闭集(closed set)

- UUID: `a54a08d4-ec2f-4235-9680-8694b869f6ef` · okf_url: https://limbo137.top/okf/concepts/a54a08d4-ec2f-4235-9680-8694b869f6ef.md
- labels: en = closed set；zh-CN = 闭集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-da67b99a25415507f253-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 伽利略群(Galilean group)

- UUID: `a56ff642-210c-4826-b97b-1990fb6e62a3` · okf_url: https://limbo137.top/okf/concepts/a56ff642-210c-4826-b97b-1990fb6e62a3.md
- labels: en = Galilean group；zh-CN = 伽利略群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-galilean-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 正相容复结构(positive compatible complex structure)

- UUID: `a584147f-8858-4fc2-a721-a9db0b16bf71` · okf_url: https://limbo137.top/okf/concepts/a584147f-8858-4fc2-a721-a9db0b16bf71.md
- labels: en = positive compatible complex structure；zh-CN = 正相容复结构
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §26.2 · `books/woit-qtgr/2025-10-20/en/chapter-26/02-compatible-complex-structures-and-positivity#first-b01b61886d3111abc485-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26.2 · `books/woit-qtgr/2025-10-20/zh/chapter-26/02-compatible-complex-structures-and-positivity#first-b01b61886d3111abc485-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/02-compatible-complex-structures-and-positivity.md

## 陪集(coset)

- UUID: `a62c9f7e-718e-4a9b-8f98-17ec505fd6e9` · okf_url: https://limbo137.top/okf/concepts/a62c9f7e-718e-4a9b-8f98-17ec505fd6e9.md
- labels: en = coset；zh-CN = 陪集
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-coset` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 独立事件(independent events)

- UUID: `a687cd39-e408-4054-9a16-38a0e5adc054` · okf_url: https://limbo137.top/okf/concepts/a687cd39-e408-4054-9a16-38a0e5adc054.md
- labels: en = independent events；zh-CN = 独立事件
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-f6b25c3582e0421bfcd9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 单纯形(simplex)

- UUID: `a69f6d9c-3138-4b13-9c2f-bacc1e99d3aa` · okf_url: https://limbo137.top/okf/concepts/a69f6d9c-3138-4b13-9c2f-bacc1e99d3aa.md
- labels: en = simplex；zh-CN = 单纯形
- aliases: q-simplex；r-simplex；simplexes；simplices；单形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=117, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `$(c_0, \ldots, c_r)$ is called the barycentric coordinate of $x$ . Since $\sigma_r$ is a bounded and closed subset of $\mathbb{R}^m$ , it is compact.`
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=117, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §3.2.1: geometrically independent vertices in R^m, m>=r, and their convex combinations (3.11). Original archived node order; the live display interposes a figure and caption, which are not definition evidence. No author-numbered Definition environment or first occurrence is asserted. · 原文: `Simplexes are building blocks of a polyhedron. A 0-simplex $\langle p_0\rangle$ is a point, or a vertex, and a 1-simplex $\langle p_0p_1\rangle$ is a line, or an edge. A 2-simplex $\langle p_0p_1p_2\rangle$ is defined to be a triangle with its interior included and a 3-simplex $\langle p_0p_1p_2p_3\rangle$ is a solid tetrahedron (figure 3.2). It is common...`
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=117, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §3.2.1: geometrically independent vertices in R^m, m>=r, and their convex combinations (3.11). Original archived node order; the live display interposes a figure and caption, which are not definition evidence. No author-numbered Definition environment or first occurrence is asserted. · 原文: `$$ \sigma^ {r} = \left\{x \in \mathbb {R} ^ {m} \mid x = \sum_ {i = 0} ^ {r} c _ {i} p _ {i}, c _ {i} \geq 0, \sum_ {i = 0} ^ {r} c _ {i} = 1 \right\}. \tag {3.11} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 拓扑流形(topological manifold)

- UUID: `a6a7c409-75c4-4cde-8470-bb4153ad9c9e` · okf_url: https://limbo137.top/okf/concepts/a6a7c409-75c4-4cde-8470-bb4153ad9c9e.md
- labels: en = topological manifold；zh-CN = 拓扑流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-d646764cbb8c4587d792-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-topological-manifold` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-d3bdf382f0f62d23edc2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 交织算符(intertwining operator)

- UUID: `a6bac404-ae9c-41fe-b818-5b13db52a897` · okf_url: https://limbo137.top/okf/concepts/a6bac404-ae9c-41fe-b818-5b13db52a897.md
- labels: en = intertwining operator；zh-CN = 交织算符
- aliases: Intertwining operator
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-intertwining-operator` · translation)；woit-qtgr/2025-10-20 (zh-CN · §20.1 · `books/woit-qtgr/2025-10-20/zh/chapter-20/01-intertwining-operators-and-the-metaplectic-representation#first-a4245e9319e1af4c2b65-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §20.1 · `books/woit-qtgr/2025-10-20/en/chapter-20/01-intertwining-operators-and-the-metaplectic-representation#first-24946370770da8070b96-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §20.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `If $(\pi_1,V_1)$ and $(\pi_2,V_2)$ are two representations of a group $G$, an intertwining operator between them is an operator $U$ such that $$ \pi_2(g)U = U\pi_1(g), \qquad \forall g\in G. $$`
  - woit-qtgr/2025-10-20 §20.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `设 $(\pi_1,V_1)$ 和 $(\pi_2,V_2)$ 是群 $G$ 的两个表示。若算符 $U$ 满足 $$ \pi_2(g)U = U\pi_1(g), \qquad \forall g\in G, $$ 则称 $U$ 为这两个表示之间的交织算符。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-20/01-intertwining-operators-and-the-metaplectic-representation.md

## 连通分支(connected component)

- UUID: `a6c5c20e-3e73-4361-880e-3e496f4f25b2` · okf_url: https://limbo137.top/okf/concepts/a6c5c20e-3e73-4361-880e-3e496f4f25b2.md
- labels: en = connected component；zh-CN = 连通分支
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.7 · `zh/chapter-10/07-connected-spaces#first-eb366184c6f1502e7f0d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/07-connected-spaces.md

## 正常正交矩阵(proper orthogonal matrix)

- UUID: `a723cc18-0199-4733-acfe-3fc04d0940d3` · okf_url: https://limbo137.top/okf/concepts/a723cc18-0199-4733-acfe-3fc04d0940d3.md
- labels: en = proper orthogonal matrix；zh-CN = 正常正交矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-proper-orthogonal-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 张量的反对称性(antisymmetry of a bilinear tensor)

- UUID: `a7683258-d9d8-48c2-a1c7-5a4a489c7aea` · okf_url: https://limbo137.top/okf/concepts/a7683258-d9d8-48c2-a1c7-5a4a489c7aea.md
- labels: en = antisymmetry of a bilinear tensor；zh-CN = 张量的反对称性
- aliases: antisymmetric
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-aeb2d833b7edb5357ab1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/01-r-vectors-and-r-forms.md

## 自伴线性算符(self-adjoint linear operator)

- UUID: `a7971b9e-8f7b-49a0-a412-0ae28a65c35b` · okf_url: https://limbo137.top/okf/concepts/a7971b9e-8f7b-49a0-a412-0ae28a65c35b.md
- labels: en = self-adjoint linear operator；zh-CN = 自伴线性算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-cea3b2fa67351cb42cfb-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-cea3b2fa67351cb42cfb-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/01-introduction.md

## 反对称性(antisymmetry)

- UUID: `a7988952-2738-40a2-aedf-6fb49ac99800` · okf_url: https://limbo137.top/okf/concepts/a7988952-2738-40a2-aedf-6fb49ac99800.md
- labels: en = antisymmetry；zh-CN = 反对称性
- aliases: skew-symmetric；skew-symmetry；关系的反对称性；斜对称性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-173d266f4e5fec67345f-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.2 · `books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-8f10d4109c454c2ba592-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.2 · `books/woit-qtgr/2025-10-20/en/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-8f10d4109c454c2ba592-en` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §1.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 二元关系：aRb且bRa蕴含a=b；不是交换后变号的条件。 · 原文: `The characteristic features of an ‘order relation’ have been discussed in Example 1.2, specifically for the case of the real numbers. More generally, a relation $R$ on a set $S$ is said to be a partial order on $S$ if it is reflexive and transitive, and in place of the symmetric property it satisfies the ‘antisymmetric’ property $$ a R b \text { and } b R...`
  - woit-qtgr/2025-10-20 §5.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 交换自变量后变号；双线性形式与矩阵的表达为X^T=-X；不指二元关系的反对称律。 · 原文: `These give a path connecting $\Omega$ to the identity (taking $e^{sX},s\in[0,t]$). We saw above that the condition $\Omega^T=\Omega^{-1}$ corresponds to skew-symmetry of the matrix $X$ $$ X ^ {T} = - X $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md

## 向量的分量(vector component)

- UUID: `a7f1665c-c377-4cb5-8778-59e8e0abffc1` · okf_url: https://limbo137.top/okf/concepts/a7f1665c-c377-4cb5-8778-59e8e0abffc1.md
- labels: en = vector component；zh-CN = 向量的分量
- aliases: component；分量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-component` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 超前格林函数(advanced Green’s function)

- UUID: `a7f6804f-360a-47ea-85b6-df05f924dc3c` · okf_url: https://limbo137.top/okf/concepts/a7f6804f-360a-47ea-85b6-df05f924dc3c.md
- labels: en = advanced Green’s function；zh-CN = 超前格林函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.4 · `zh/chapter-12/04-green-s-functions#first-9683ea08d13476321ae6-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/04-green-s-functions.md

## 不可数集(uncountable set)

- UUID: `a846c946-b159-4fed-8f94-14622a9bc324` · okf_url: https://limbo137.top/okf/concepts/a846c946-b159-4fed-8f94-14622a9bc324.md
- labels: en = uncountable set；zh-CN = 不可数集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-aeaefe1490ef17d14690-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 曲率2-形式(curvature 2-form)

- UUID: `a873bf82-2f8f-4df9-bdfb-2637dc96b182` · okf_url: https://limbo137.top/okf/concepts/a873bf82-2f8f-4df9-bdfb-2637dc96b182.md
- labels: en = curvature 2-form；zh-CN = 曲率2-形式
- aliases: curvature 2-forms；曲率二形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.7 · `en/chapter-18/07-cartan-formalism#first-55299f8835e79db857ed-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.7 · `zh/chapter-18/07-cartan-formalism#first-e570ad5911d4812b405d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/07-cartan-formalism.md

## 粒子数密度算符(number density operator)

- UUID: `a8db106c-e9e6-4eef-8c0c-f49fb18fb037` · okf_url: https://limbo137.top/okf/concepts/a8db106c-e9e6-4eef-8c0c-f49fb18fb037.md
- labels: en = number density operator；zh-CN = 粒子数密度算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §37.2 · `books/woit-qtgr/2025-10-20/en/chapter-37/02-quadratic-operators-and-dynamics#first-cac812e54e407d642011-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §37.2 · `books/woit-qtgr/2025-10-20/zh/chapter-37/02-quadratic-operators-and-dynamics#first-cac812e54e407d642011-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/02-quadratic-operators-and-dynamics.md

## 最大元(greatest element)

- UUID: `a9c977eb-c3c5-4c7a-9c4f-0eb222bec3c6` · okf_url: https://limbo137.top/okf/concepts/a9c977eb-c3c5-4c7a-9c4f-0eb222bec3c6.md
- labels: en = greatest element；zh-CN = 最大元
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-3702a271341dea7573ff-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 典范映射(canonical mapping)

- UUID: `aa2120c3-751f-4f90-944f-26ec8d4c046f` · okf_url: https://limbo137.top/okf/concepts/aa2120c3-751f-4f90-944f-26ec8d4c046f.md
- labels: en = canonical mapping；zh-CN = 典范映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-86a675776f33545ca770-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 交换图(commutative diagram)

- UUID: `aa2761c6-6e45-430f-9851-bb20306ec735` · okf_url: https://limbo137.top/okf/concepts/aa2761c6-6e45-430f-9851-bb20306ec735.md
- labels: en = commutative diagram；zh-CN = 交换图
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-c5795682d2843eca3a4b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 泡利原理(Pauli principle)

- UUID: `aaa0a3c5-6818-4769-b378-186b91122f2b` · okf_url: https://limbo137.top/okf/concepts/aaa0a3c5-6818-4769-b378-186b91122f2b.md
- labels: en = Pauli principle；zh-CN = 泡利原理
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.1 · `books/woit-qtgr/2025-10-20/en/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-790b798a85aeca00946f-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 能量–应力张量(energy–stress tensor)

- UUID: `aaa78f2a-eb86-4143-83c9-e88e2bbab2bb` · okf_url: https://limbo137.top/okf/concepts/aaa78f2a-eb86-4143-83c9-e88e2bbab2bb.md
- labels: en = energy–stress tensor；zh-CN = 能量–应力张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.5 · `zh/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-5e9d23eb84f621a2cc04-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.5 · `en/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-5e9d23eb84f621a2cc04-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/05-conservation-laws-and-energy-stress-tensors.md

## 向量加法(vector addition)

- UUID: `aaec29ac-0cc2-4aa6-aa59-fe4a1cfbe6c5` · okf_url: https://limbo137.top/okf/concepts/aaec29ac-0cc2-4aa6-aa59-fe4a1cfbe6c5.md
- labels: en = vector addition；zh-CN = 向量加法
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-vector-addition` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 斜伴随算符(skew-adjoint operator)

- UUID: `aaee3ca4-0875-466a-ba37-18936cefa633` · okf_url: https://limbo137.top/okf/concepts/aaee3ca4-0875-466a-ba37-18936cefa633.md
- labels: en = skew-adjoint operator；zh-CN = 斜伴随算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.5 · `books/woit-qtgr/2025-10-20/zh/chapter-04/05-adjoint-operators#first-724d4a2187c8da1944d4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/05-adjoint-operators.md

## 内积运算(interior product)

- UUID: `ab0e2528-c5ad-4abe-bc3a-3011f27b7d65` · okf_url: https://limbo137.top/okf/concepts/ab0e2528-c5ad-4abe-bc3a-3011f27b7d65.md
- labels: en = interior product；zh-CN = 内积运算
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.4 · `zh/chapter-08/04-interior-product#first-d519ee912b1b84a817f3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.4 · `en/chapter-08/04-interior-product#first-d519ee912b1b84a817f3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/04-interior-product.md

## 静止质量(rest mass)

- UUID: `abba4ed9-ce64-4c31-b396-9603c74a21d9` · okf_url: https://limbo137.top/okf/concepts/abba4ed9-ce64-4c31-b396-9603c74a21d9.md
- labels: en = rest mass；zh-CN = 静止质量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-aca6b970c0a074c62ef2-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-aca6b970c0a074c62ef2-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 度规张量(metric tensor)

- UUID: `ac724144-5a08-407a-a27a-796eb367a161` · okf_url: https://limbo137.top/okf/concepts/ac724144-5a08-407a-a27a-796eb367a161.md
- labels: en = metric tensor；zh-CN = 度规张量
- aliases: metric；度规；度量张量
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-52084e4e3766521c7bd5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-52084e4e3766521c7bd5-zh` · translation)
- 已审定义(definitions):
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `$$ g = g _ {\mu \nu} (x) d x ^ {\mu} \otimes d x ^ {\nu} $$`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `Definition: A metric g is a (0, 2) tensor field that is:`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `• Non-Degenerate: If, for any $p \in M , g ( X , Y ) { \big | } _ { p } = 0$ for all $Y \in T _ { p } ( M )$ then $X _ { p } = 0$ With a choice of coordinates, we can write the metric as`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `This is the form that we saw previously in (1.4). The metric components can be extracted by evaluating the metric on a pair of basis elements,`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `$$ d s ^ {2} = g _ {\mu \nu} (x) d x ^ {\mu} d x ^ {\nu} $$`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `• Symmetric: $g ( X , Y ) = g ( Y , X )$`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `$$ g _ {\mu \nu} (x) = g \left(\frac {\partial}{\partial x ^ {\mu}}, \frac {\partial}{\partial x ^ {\nu}}\right) $$`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `The object $g$ is often written as a line element $d s ^ { 2 }$ and this expression is abbreviated as`
  - david-tong-general-relativity/michaelmas-2019-source-20260923 §3.1 (pdf_page=1, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 光滑流形上的对称、非退化(0,2)张量场g，逐点作用于切向量；正定时为黎曼度规，允许洛伦兹等不定符号。Tong§3.1明确按此定义。与距离函数d:X×X→R分开，也不把单个向量空间上的内积自动当作已给定的张量场；度规/度量张量为该张量语境异名。 · 原文: `The metric $g _ { \mu \nu }$ is a symmetric matrix. We can always pick a basis $e _ { \mu }$ of each $T _ { p } ( M )$ so that this matrix is diagonal. The non-degeneracy condition above ensures that none of these diagonal elements vanish. Some are positive, some are negative. Sylvester’s law of inertia is a theorem in algebra which states that the number...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 李代数的结构常数(structure constants of a Lie algebra)

- UUID: `acfdabfa-4d2d-4076-b0b7-228126379ad7` · okf_url: https://limbo137.top/okf/concepts/acfdabfa-4d2d-4076-b0b7-228126379ad7.md
- labels: en = structure constants of a Lie algebra；zh-CN = 李代数的结构常数
- aliases: structure constants；结构常数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-structure-constants` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-f998b514210726440578-lie-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.1 · `books/woit-qtgr/2025-10-20/zh/chapter-05/01-lie-algebras#first-1501fe5e9e87c20269e1-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.1 · `books/woit-qtgr/2025-10-20/en/chapter-05/01-lie-algebras#first-1501fe5e9e87c20269e1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/01-lie-algebras.md

## 立方体(cube)

- UUID: `ada8bcbb-bf0e-4ae1-8b86-90da4e5a754c` · okf_url: https://limbo137.top/okf/concepts/ada8bcbb-bf0e-4ae1-8b86-90da4e5a754c.md
- labels: en = cube；zh-CN = 立方体
- aliases: 方体
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.3 · `en/chapter-17/03-stokes-theorem#first-cddbe9229816637ff88a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/03-stokes-theorem.md

## 狄拉克测度(Dirac measure)

- UUID: `adc2298d-a628-495d-aacf-599f09c84dd9` · okf_url: https://limbo137.top/okf/concepts/adc2298d-a628-495d-aacf-599f09c84dd9.md
- labels: en = Dirac measure；zh-CN = 狄拉克测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-b0f5a78d1efe2358093f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 维格纳定理(Wigner’s theorem)

- UUID: `adc722bc-8c6d-4761-904c-6d8926e72330` · okf_url: https://limbo137.top/okf/concepts/adc722bc-8c6d-4761-904c-6d8926e72330.md
- labels: en = Wigner’s theorem；zh-CN = 维格纳定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-c02633113fb56716faaf-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-c02633113fb56716faaf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 特殊酉群(special unitary group)

- UUID: `ae68704b-48d9-4342-ba60-9973c64e5093` · okf_url: https://limbo137.top/okf/concepts/ae68704b-48d9-4342-ba60-9973c64e5093.md
- labels: en = special unitary group；zh-CN = 特殊酉群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-special-unitary-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §3 · `books/woit-qtgr/2025-10-20/zh/chapter-03/index#first-2fa85f560ed99df11f7b-zh` · translation)；woit-qtgr/2025-10-20 (en · §3 · `books/woit-qtgr/2025-10-20/en/chapter-03/index#first-2fa85f560ed99df11f7b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/08-for-further-reading.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## Einstein求和约定(Einstein summation convention)

- UUID: `aebcd7da-6aaa-47db-831d-2be25a85038a` · okf_url: https://limbo137.top/okf/concepts/aebcd7da-6aaa-47db-831d-2be25a85038a.md
- labels: en = Einstein summation convention；zh-CN = Einstein求和约定
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.1 · `books/woit-qtgr/2025-10-20/zh/chapter-40/01-minkowski-space#first-85cf4063ce0a0f6f11cd-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §40.1 · `books/woit-qtgr/2025-10-20/en/chapter-40/01-minkowski-space#first-85cf4063ce0a0f6f11cd-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 特征方程(characteristic equation)

- UUID: `aec6e1be-5a6e-4322-9fb6-47824c0fb508` · okf_url: https://limbo137.top/okf/concepts/aec6e1be-5a6e-4322-9fb6-47824c0fb508.md
- labels: en = characteristic equation；zh-CN = 特征方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.1 · `zh/chapter-04/01-eigenspaces-and-characteristic-equations#concept-4-characteristic-equation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/01-eigenspaces-and-characteristic-equations.md

## 反对称张量(antisymmetric tensor)

- UUID: `af3d858c-3f7c-4b30-9ec1-451a71456675` · okf_url: https://limbo137.top/okf/concepts/af3d858c-3f7c-4b30-9ec1-451a71456675.md
- labels: en = antisymmetric tensor；zh-CN = 反对称张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-012c88f118329be2f3e1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/01-r-vectors-and-r-forms.md

## 相容可观测量(compatible observable)

- UUID: `af56883b-d7e8-4c1e-b9c0-a181f6c21c39` · okf_url: https://limbo137.top/okf/concepts/af56883b-d7e8-4c1e-b9c0-a181f6c21c39.md
- labels: en = compatible observable；zh-CN = 相容可观测量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-bc86768410549be871e8-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 瞬时静止系(instantaneous rest frame)

- UUID: `af612566-84dd-4584-9981-ae83c203343a` · okf_url: https://limbo137.top/okf/concepts/af612566-84dd-4584-9981-ae83c203343a.md
- labels: en = instantaneous rest frame；zh-CN = 瞬时静止系
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-d663ed3ab1955b715494-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-d663ed3ab1955b715494-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 完备正交标准集(complete orthonormal set)

- UUID: `af7548ea-bfb5-40fb-8155-60d865395208` · okf_url: https://limbo137.top/okf/concepts/af7548ea-bfb5-40fb-8155-60d865395208.md
- labels: en = complete orthonormal set；zh-CN = 完备正交标准集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-complete-orthonormal-set` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-ff5316519821afc51b79-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 量子场算符(quantum field operator)

- UUID: `afeba7ff-445b-48d6-9000-97af6ab5abfe` · okf_url: https://limbo137.top/okf/concepts/afeba7ff-445b-48d6-9000-97af6ab5abfe.md
- labels: en = quantum field operator；zh-CN = 量子场算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §37.1 · `books/woit-qtgr/2025-10-20/zh/chapter-37/01-quantum-field-operators#first-2ea8da5926d9d7793dbb-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/01-quantum-field-operators.md

## 联络的曲率、电磁场强度(Curvature of a connection, electromagnetic field strengths)

- UUID: `affd6976-ee5c-4679-bd92-106d209f88a7` · okf_url: https://limbo137.top/okf/concepts/affd6976-ee5c-4679-bd92-106d209f88a7.md
- labels: en = Curvature of a connection, electromagnetic field strengths；zh-CN = 联络的曲率、电磁场强度
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.2 · `books/woit-qtgr/2025-10-20/zh/chapter-45/02-curvature-electric-and-magnetic-fields#first-a6ddc4eb0f94c585e477-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.2 · `books/woit-qtgr/2025-10-20/en/chapter-45/02-curvature-electric-and-magnetic-fields#first-27df854a13a2b6066338-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `联络 $A_\mu$ 的曲率由下式给出 $$ F _ {\mu \nu} = \frac {i}{e} [ D _ {\mu} ^ {A}, D _ {\nu} ^ {A} ] $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/02-curvature-electric-and-magnetic-fields.md

## 共轭复向量空间(conjugate complex vector space)

- UUID: `b001334e-e824-4a5a-bc84-8b3ba855aebf` · okf_url: https://limbo137.top/okf/concepts/b001334e-e824-4a5a-bc84-8b3ba855aebf.md
- labels: en = conjugate complex vector space；zh-CN = 共轭复向量空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §26.4 · `books/woit-qtgr/2025-10-20/en/chapter-26/04-complex-vector-spaces-with-hermitian-inner-product-as-phase-spaces#first-02f4f475da2f9d50635e-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26.4 · `books/woit-qtgr/2025-10-20/zh/chapter-26/04-complex-vector-spaces-with-hermitian-inner-product-as-phase-spaces#first-02f4f475da2f9d50635e-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/04-complex-vector-spaces-with-hermitian-inner-product-as-phase-spaces.md

## 自旋–统计定理(spin-statistics theorem)

- UUID: `b087a1f3-81a9-4c70-a5b6-29950378bdd0` · okf_url: https://limbo137.top/okf/concepts/b087a1f3-81a9-4c70-a5b6-29950378bdd0.md
- labels: en = spin-statistics theorem；zh-CN = 自旋–统计定理
- 分书定义状态: schwartz-qft/local-experiment = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §38.4 · `books/woit-qtgr/2025-10-20/en/chapter-38/09-fermionic-fields#first-d0a75f032e011568a30c-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §38.4 · `books/woit-qtgr/2025-10-20/zh/chapter-38/09-fermionic-fields#first-d0a75f032e011568a30c-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): schwartz-qft/local-experiment (1 条, completeness=recorded-only, target_uuids=b087a1f3-81a9-4c70-a5b6-29950378bdd0) · 摘录: `The creation and annihilation operators for spinors must anticommute by the spin-statistics theorem:`
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/09-fermionic-fields.md

## 可缩空间(contractible space)

- UUID: `b0e05644-5634-527e-83d1-377c92d24648` · okf_url: https://limbo137.top/okf/concepts/b0e05644-5634-527e-83d1-377c92d24648.md
- labels: en = contractible space；zh-CN = 可缩空间
- aliases: contractible；可缩的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=150, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.9：可缩空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.9. If a point $a \in X$ is a deformation retract of $X$ , $X$ is said to be contractible.`
  - nakahara-geometry-topology-physics/second-edition-local §4.2.2 (pdf_page=150, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.9：可缩空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Let $c_{a}: X \to \{a\}$ be a constant map. If $X$ is contractible, there exists a homotopy $H: X \times I \to X$ such that $H(x, 0) = c_{a}(x) = a$ and $H(x, 1) = \mathrm{id}_{X}(x) = x$ for any $x \in X$ and, moreover, $H(a, t) = a$ for any $t \in I$ . The homotopy $H$ is called the contraction.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 二阶酉群(unitary group in dimension two)

- UUID: `b0f1cb61-5cdc-49df-9d91-612c9a805bc9` · okf_url: https://limbo137.top/okf/concepts/b0f1cb61-5cdc-49df-9d91-612c9a805bc9.md
- labels: en = unitary group in dimension two；zh-CN = 二阶酉群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-2e8b3d89d2c28f6a1e63-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-2e8b3d89d2c28f6a1e63-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 笛卡尔张量(cartesian tensor)

- UUID: `b1e90b67-3649-43c4-b3f2-a9a950456413` · okf_url: https://limbo137.top/okf/concepts/b1e90b67-3649-43c4-b3f2-a9a950456413.md
- labels: en = cartesian tensor；zh-CN = 笛卡尔张量
- aliases: cartesian tensors
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-6cdf0da76103f2df7850-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-8751a6a77b54504fea72-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 交换环(commutative ring)

- UUID: `b1e9ae23-1814-48c8-8a33-0b0337c674d8` · okf_url: https://limbo137.top/okf/concepts/b1e9ae23-1814-48c8-8a33-0b0337c674d8.md
- labels: en = commutative ring；zh-CN = 交换环
- aliases: ring；交换环（Szekeres约定）；环
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.1 · `zh/chapter-03/01-rings-and-fields#concept-3-ring` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/01-rings-and-fields.md

## 普朗克常数(Planck’s constant)

- UUID: `b24f4256-9aea-456c-bb41-3cd97e36a051` · okf_url: https://limbo137.top/okf/concepts/b24f4256-9aea-456c-bb41-3cd97e36a051.md
- labels: en = Planck’s constant；zh-CN = 普朗克常数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.2 · `books/woit-qtgr/2025-10-20/zh/chapter-01/02-basic-principles-of-quantum-mechanics#first-5f080eaeb8a7fec94863-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/02-basic-principles-of-quantum-mechanics.md

## 更强的拓扑(stronger topology)

- UUID: `b28960be-19c3-42ff-a50a-61fe827934e3` · okf_url: https://limbo137.top/okf/concepts/b28960be-19c3-42ff-a50a-61fe827934e3.md
- labels: en = stronger topology；zh-CN = 更强的拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-cf2639f491dfe774ba03-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 尘埃(dust)

- UUID: `b2cbc773-02cd-4a8c-982f-cec76e5ec795` · okf_url: https://limbo137.top/okf/concepts/b2cbc773-02cd-4a8c-982f-cec76e5ec795.md
- labels: en = dust；zh-CN = 尘埃
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-3467b4cc92a3ac960a74-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-3467b4cc92a3ac960a74-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 函数空间(function space)

- UUID: `b2da5ce9-fdbc-4d32-9495-60eba58b0cab` · okf_url: https://limbo137.top/okf/concepts/b2da5ce9-fdbc-4d32-9495-60eba58b0cab.md
- labels: en = function space；zh-CN = 函数空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-467fca2b7844dc4bfc62-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 路径相关的相位因子(Path-dependent phase factor)

- UUID: `b2e0217f-a007-4c41-9d08-b3c942a60422` · okf_url: https://limbo137.top/okf/concepts/b2e0217f-a007-4c41-9d08-b3c942a60422.md
- labels: en = Path-dependent phase factor；zh-CN = 路径相关的相位因子
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.4 · `books/woit-qtgr/2025-10-20/zh/chapter-45/04-the-geometric-significance-of-the-connection#first-4754b27cd45473b234ed-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.4 · `books/woit-qtgr/2025-10-20/en/chapter-45/04-the-geometric-significance-of-the-connection#first-17b648bce0665e965344-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `给定一个联络 $A _ { \mu } ( x )$，对于任意由 $\tau \in [ 0 , 1 ]$ 参数化的曲线 $\gamma$，其在时刻 $\tau$ 的位置由 $x(τ)$ 给出，可以定义路径相关的相位因子 $$ \int_ {\gamma} A \equiv \int_ {0} ^ {1} \sum_ {\mu = 1} ^ {4} A _ {\mu} (x) \frac {d x _ {\mu}}{d \tau} d \tau $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/04-the-geometric-significance-of-the-connection.md

## 积可测空间(product measurable space)

- UUID: `b2eadd99-3d29-457f-8e36-92406c08f412` · okf_url: https://limbo137.top/okf/concepts/b2eadd99-3d29-457f-8e36-92406c08f412.md
- labels: en = product measurable space；zh-CN = 积可测空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-649a23a7929243a8dd38-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 子空间(subspace)

- UUID: `b2f917c9-9ada-4911-8188-2aeccc652fa5` · okf_url: https://limbo137.top/okf/concepts/b2f917c9-9ada-4911-8188-2aeccc652fa5.md
- labels: en = subspace；zh-CN = 子空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-e5723211149818a218c1-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-e5723211149818a218c1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 最高权与最高权向量(Highest weights and highest weight vectors)

- UUID: `b2fa8d60-7325-4e2d-b009-3aea4de3c3c6` · okf_url: https://limbo137.top/okf/concepts/b2fa8d60-7325-4e2d-b009-3aea4de3c3c6.md
- labels: en = Highest weights and highest weight vectors；zh-CN = 最高权与最高权向量
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §8.1 · `books/woit-qtgr/2025-10-20/zh/chapter-08/01-representations-of-su2-classification#first-8d62ab003cac44d0fd8e-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §8.1 · `books/woit-qtgr/2025-10-20/en/chapter-08/01-representations-of-su2-classification#first-65668da92bde873c6565-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A non-zero vector $v \in V _ { n } \subset V$ such that $$ \pi^ {\prime} (S _ {+}) v = 0 $$`
  - woit-qtgr/2025-10-20 §8.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `一个非零向量 $v \in V _ { n } \subset V$，使得 $$ \pi^ {\prime} (S _ {+}) v = 0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/01-representations-of-su2-classification.md

## 局部有限的(locally finite)

- UUID: `b32fccab-0ef7-4fc4-bec9-9aee534e4afc` · okf_url: https://limbo137.top/okf/concepts/b32fccab-0ef7-4fc4-bec9-9aee534e4afc.md
- labels: en = locally finite；zh-CN = 局部有限的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.1 · `en/chapter-17/01-partitions-of-unity#first-cfb8584a54bc36367766-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.1 · `zh/chapter-17/01-partitions-of-unity#first-cfb8584a54bc36367766-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/01-partitions-of-unity.md

## 可积表示(integrable representation)

- UUID: `b3d9b0a6-b76b-4571-a20d-5ca7c7501ca2` · okf_url: https://limbo137.top/okf/concepts/b3d9b0a6-b76b-4571-a20d-5ca7c7501ca2.md
- labels: en = integrable representation；zh-CN = 可积表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-313cf2013b0dcefee830-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 自由阿贝尔群(free Abelian group)

- UUID: `b405e339-ed38-485b-be7b-b30aab2271a7` · okf_url: https://limbo137.top/okf/concepts/b405e339-ed38-485b-be7b-b30aab2271a7.md
- labels: en = free Abelian group；zh-CN = 自由阿贝尔群
- aliases: rank
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.2 (pdf_page=115, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.1：自由阿贝尔群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.1. If $G$ is finitely generated by $r$ linearly independent elements, $G$ is called a free Abelian group of rank $r$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 总粒子数算符(total number operator)

- UUID: `b44450c5-d0c8-4518-b1d0-c5f0941c5f71` · okf_url: https://limbo137.top/okf/concepts/b44450c5-d0c8-4518-b1d0-c5f0941c5f71.md
- labels: en = total number operator；zh-CN = 总粒子数算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §36.2 · `books/woit-qtgr/2025-10-20/en/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-a277d131a7ccf32bdcb1-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §36.2 · `books/woit-qtgr/2025-10-20/zh/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism#first-a277d131a7ccf32bdcb1-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/04-multi-particle-quantum-systems-of-free-particles-finite-cutof-formalism.md

## 作用量泛函(action functional)

- UUID: `b48afbf0-a19f-42a5-8dfc-5b0dc4ba7ed2` · okf_url: https://limbo137.top/okf/concepts/b48afbf0-a19f-42a5-8dfc-5b0dc4ba7ed2.md
- labels: en = action functional；zh-CN = 作用量泛函
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-c7f57f0aa3666ecbdb86-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-2dc4536000fe14e62eea-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The [action](/concepts/mechanics/action-functional) $S$ of a path $\gamma$ is $$ S[\gamma]=\int_{t_1}^{t_2}L(\mathbf q(t),\dot{\mathbf q}(t))\,dt. $$`
  - woit-qtgr/2025-10-20 §35.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `路径 $\gamma$ 的[作用量(action)](/concepts/mechanics/action-functional)为 $$ S[\gamma]=\int_{t_1}^{t_2}L(\mathbf q(t),\dot{\mathbf q}(t))\,dt. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 普朗歇尔(Plancherel)

- UUID: `b4b2ccab-421f-4d85-afd2-a2f92d03737e` · okf_url: https://limbo137.top/okf/concepts/b4b2ccab-421f-4d85-afd2-a2f92d03737e.md
- labels: en = Plancherel；zh-CN = 普朗歇尔
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §11.2 · `books/woit-qtgr/2025-10-20/en/chapter-11/02-the-group-r-and-the-fourier-transform#first-de9c0c7ed31cb268bea0-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §11.2 · `books/woit-qtgr/2025-10-20/zh/chapter-11/02-the-group-r-and-the-fourier-transform#first-debb81c2102d9c08209c-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-11/02-the-group-r-and-the-fourier-transform.md

## 斜伴随(skew-adjoint)

- UUID: `b4e48583-4b04-48fc-83d8-e77fc248bcb3` · okf_url: https://limbo137.top/okf/concepts/b4e48583-4b04-48fc-83d8-e77fc248bcb3.md
- labels: en = skew-adjoint；zh-CN = 斜伴随
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §1.4 · `books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics#concept-1-skew-adjoint` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §1.4 · `books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics#concept-1-skew-adjoint` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/05-adjoint-operators.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md

## 德拉姆上同调群(de Rham cohomology group)

- UUID: `b536c648-0767-49e9-b712-1849d0825566` · okf_url: https://limbo137.top/okf/concepts/b536c648-0767-49e9-b712-1849d0825566.md
- labels: en = de Rham cohomology group；zh-CN = 德拉姆上同调群
- aliases: de Rham cohomology；de Rham cohomology groups；德拉姆上同调
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-57c2dbd73bffb785e74e-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-57c2dbd73bffb785e74e-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §17.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `The $r$th de Rham cohomology space (group) is defined as the factor space $H ^ { r } ( M ) = Z ^ { r } ( M ) / B ^ { r } ( M )$ , and any two $r$-cocycles $\alpha$ and $\beta$ are said to be cohomologous if they belong to the same coset, ${ \alpha - \beta = { \mathrm { d } } \gamma }$ for some $(r-1)$-cochain $\gamma$ . The dimensions of the vector spaces...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 哈密顿对称性(Hamiltonian symmetry)

- UUID: `b57335c3-b291-4b5c-9ce9-0c7d580b664e` · okf_url: https://limbo137.top/okf/concepts/b57335c3-b291-4b5c-9ce9-0c7d580b664e.md
- labels: en = Hamiltonian symmetry；zh-CN = 哈密顿对称性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-19dcb5324926d69535b4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-19dcb5324926d69535b4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 双线性(bilinear)

- UUID: `b5a8ec0e-a1f9-44eb-9ffc-7094a35eae32` · okf_url: https://limbo137.top/okf/concepts/b5a8ec0e-a1f9-44eb-9ffc-7094a35eae32.md
- labels: en = bilinear；zh-CN = 双线性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-bilinear` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 麦克斯韦2-形式(Maxwell 2-form)

- UUID: `b5f9a010-d584-41c0-b677-d34ea1318faf` · okf_url: https://limbo137.top/okf/concepts/b5f9a010-d584-41c0-b677-d34ea1318faf.md
- labels: en = Maxwell 2-form；zh-CN = 麦克斯韦2-形式
- aliases: 麦克斯韦二形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.5 · `zh/chapter-17/05-the-poincar-e-lemma#first-64cd65c5c94f7db73019-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.5 · `en/chapter-17/05-the-poincar-e-lemma#first-64cd65c5c94f7db73019-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/05-the-poincar-e-lemma.md

## 指向过去的(past-pointing)

- UUID: `b642e129-ab4e-43a2-aff0-96debafdeb6e` · okf_url: https://limbo137.top/okf/concepts/b642e129-ab4e-43a2-aff0-96debafdeb6e.md
- labels: en = past-pointing；zh-CN = 指向过去的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-3e14d5fe3bfa1870e1aa-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-3e14d5fe3bfa1870e1aa-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 四维余向量(4-covector)

- UUID: `b6446ff8-b20d-4aef-b0ae-fccd4af25b48` · okf_url: https://limbo137.top/okf/concepts/b6446ff8-b20d-4aef-b0ae-fccd4af25b48.md
- labels: en = 4-covector；zh-CN = 四维余向量
- aliases: 4-covectors
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-0e5eff462969f4ad4cea-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-5b128fd930d05c2229b2-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 基变换(change of basis)

- UUID: `b66bf048-7f86-4318-8350-fbb64491a8b4` · okf_url: https://limbo137.top/okf/concepts/b66bf048-7f86-4318-8350-fbb64491a8b4.md
- labels: en = change of basis；zh-CN = 基变换
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.3 · `books/woit-qtgr/2025-10-20/zh/chapter-04/03-change-of-basis#first-fdef6b203b0a71d6f29d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/03-change-of-basis.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/06-orthogonal-and-unitary-transformations.md

## 有序单纯形(ordered simplex)

- UUID: `b67e7bbb-81b0-4044-bf43-bd2dcd1e0eb0` · okf_url: https://limbo137.top/okf/concepts/b67e7bbb-81b0-4044-bf43-bd2dcd1e0eb0.md
- labels: en = ordered simplex；zh-CN = 有序单纯形
- aliases: ordered
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-8d35a6f6df64572c236c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-4c1c4c7074ce818364b5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## Hopf 纤维化(Hopf fibration)

- UUID: `b6e6358a-67f2-4f17-abe3-6047c1590084` · okf_url: https://limbo137.top/okf/concepts/b6e6358a-67f2-4f17-abe3-6047c1590084.md
- labels: en = Hopf fibration；zh-CN = Hopf 纤维化
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-7dcdf66d410d25744005-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-7dcdf66d410d25744005-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 控制收敛(dominated convergence)

- UUID: `b6f607ec-8378-466a-8a92-5b116928a84f` · okf_url: https://limbo137.top/okf/concepts/b6f607ec-8378-466a-8a92-5b116928a84f.md
- labels: en = dominated convergence；zh-CN = 控制收敛
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-35af47d7452a03063fb0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 傅里叶级数(Fourier series)

- UUID: `b731fa52-a18c-41d3-b817-67eb77e84ee3` · okf_url: https://limbo137.top/okf/concepts/b731fa52-a18c-41d3-b817-67eb77e84ee3.md
- labels: en = Fourier series；zh-CN = 傅里叶级数
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §11.1 · `books/woit-qtgr/2025-10-20/zh/chapter-11/01-periodic-boundary-conditions-and-the-group-u-1#first-12a910793de1a38ee826-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §11.1 · `books/woit-qtgr/2025-10-20/en/chapter-11/01-periodic-boundary-conditions-and-the-group-u-1#first-993b36dd8b0dd2510fd0-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-11/01-periodic-boundary-conditions-and-the-group-u-1.md

## 包含映射(inclusion mapping)

- UUID: `b74f81a5-eb04-4d44-8595-eaa96ef287ed` · okf_url: https://limbo137.top/okf/concepts/b74f81a5-eb04-4d44-8595-eaa96ef287ed.md
- labels: en = inclusion mapping；zh-CN = 包含映射
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-b5ae2a5b25b7042a03d8-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 概率空间(probability space)

- UUID: `b7c5da7f-0ff9-4a9d-814c-974d791729c9` · okf_url: https://limbo137.top/okf/concepts/b7c5da7f-0ff9-4a9d-814c-974d791729c9.md
- labels: en = probability space；zh-CN = 概率空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-570b984461255e9c0139-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 德西特宇宙(de Sitter universe)

- UUID: `b808cb77-61b9-448a-9a27-f02d25fcacd0` · okf_url: https://limbo137.top/okf/concepts/b808cb77-61b9-448a-9a27-f02d25fcacd0.md
- labels: en = de Sitter universe；zh-CN = 德西特宇宙
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-50b27f3d9723852373f3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 黑洞(black hole)

- UUID: `b8325660-7a21-4972-969c-e88b5bf54aac` · okf_url: https://limbo137.top/okf/concepts/b8325660-7a21-4972-969c-e88b5bf54aac.md
- labels: en = black hole；zh-CN = 黑洞
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-a6eac3f03f429b257e85-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-a6eac3f03f429b257e85-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 定向3-单纯形(oriented 3-simplex)

- UUID: `b8acb2d9-0f05-4c7a-9a71-e115faebd5a9` · okf_url: https://limbo137.top/okf/concepts/b8acb2d9-0f05-4c7a-9a71-e115faebd5a9.md
- labels: en = oriented 3-simplex；zh-CN = 定向3-单纯形
- aliases: $\sigma_3 = (p_0p_1p_2p_3)$；定向 3-单形；定向 3-单纯形
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 四顶点四面体的序列及排列符号约定；保留维数3特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ P = \left( \begin{array}{c c c c} 0 & 1 & 2 & 3 \\ i & j & k & l \end{array} \right) $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 四顶点四面体的序列及排列符号约定；保留维数3特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `An oriented 3-simplex $\sigma_3 = (p_0p_1p_2p_3)$ is an ordered sequence of four vertices of a tetrahedron. Let`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 四顶点四面体的序列及排列符号约定；保留维数3特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ \left(p _ {i} p _ {j} p _ {k} p _ {l}\right) = \operatorname {s g n} (P) \left(p _ {0} p _ {1} p _ {2} p _ {3}\right). $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.1 (pdf_page=121, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 四顶点四面体的序列及排列符号约定；保留维数3特化。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `be a permutation. We define`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 克里斯托费尔符号(Christoffel symbols)

- UUID: `b902db54-268c-45d6-b3fa-ea53f35cb630` · okf_url: https://limbo137.top/okf/concepts/b902db54-268c-45d6-b3fa-ea53f35cb630.md
- labels: en = Christoffel symbols；zh-CN = 克里斯托费尔符号
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-8793835674118afa92ef-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-8793835674118afa92ef-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 上确界范数(supremum norm)

- UUID: `b91ff137-b70e-4b45-b5a8-d7d8478cc0f2` · okf_url: https://limbo137.top/okf/concepts/b91ff137-b70e-4b45-b5a8-d7d8478cc0f2.md
- labels: en = supremum norm；zh-CN = 上确界范数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.9 · `zh/chapter-10/09-topological-vector-spaces#first-1305a3e9156dc317c975-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md

## 庞加莱群(Poincaré group)

- UUID: `b9250f96-c06a-4ae1-ab1f-8035202bea26` · okf_url: https://limbo137.top/okf/concepts/b9250f96-c06a-4ae1-ab1f-8035202bea26.md
- labels: en = Poincaré group；zh-CN = 庞加莱群
- aliases: Poincaré群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-poincare-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §40 · `books/woit-qtgr/2025-10-20/zh/chapter-40/index#first-4c72e0fabfe1ab841067-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §40 · `books/woit-qtgr/2025-10-20/en/chapter-40/index#first-4c72e0fabfe1ab841067-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §42.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `庞加莱群是半直积 $$ \mathcal {P} = \mathbf {R} ^ {4} \rtimes S O (3, 1) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-42/01-the-poincar-e-group-and-its-lie-algebra.md

## 联络(connection)

- UUID: `b94af6bd-ce81-4af1-b0c0-3c1c8b55401d` · okf_url: https://limbo137.top/okf/concepts/b94af6bd-ce81-4af1-b0c0-3c1c8b55401d.md
- labels: en = connection；zh-CN = 联络
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-52390a9f13329d43d684-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-52390a9f13329d43d684-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 边缘群(boundary group)

- UUID: `b997f55f-72cd-41dd-a707-f2a61744a707` · okf_url: https://limbo137.top/okf/concepts/b997f55f-72cd-41dd-a707-f2a61744a707.md
- labels: en = boundary group；zh-CN = 边缘群
- aliases: p-boundary group；r-boundary group；$p$-边界群；$p$-边缘群；$r$-边界群；$r$-边缘群；边界群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-14f4d6dbed8b0089f13d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-14f4d6dbed8b0089f13d-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `then $c$ is called an $r$ -boundary. The set of $r$ -boundaries $B_r(K)$ is a subgroup of $C_r(K)$ and is called the $r$ -boundary group. Note that $B_r(K) = \operatorname{im} \partial_{r+1}$ . [Remark: $B_n(K)$ is defined to be 0.]`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , which is crucial in the definition of homology groups.`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 3.4. Let $K$ be an $n$ -dimensional simplicial complex and let $c \in C_r(K)$ . If there exists an element $d \in C_{r + 1}(K)$ such that`
  - nakahara-geometry-topology-physics/second-edition-local §3.3.2 (pdf_page=124, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 3.4：边界群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ c = \partial_ {r + 1} d \tag {3.25} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 产生算符(creation operator)

- UUID: `ba05cd48-5659-4250-8261-dc31a2e129be` · okf_url: https://limbo137.top/okf/concepts/ba05cd48-5659-4250-8261-dc31a2e129be.md
- labels: en = creation operator；zh-CN = 产生算符
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-f22eab78283659614d38-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-creation-operator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 缩并(contraction)

- UUID: `ba852562-0710-4e32-9327-b03efa03811d` · okf_url: https://limbo137.top/okf/concepts/ba852562-0710-4e32-9327-b03efa03811d.md
- labels: en = contraction；zh-CN = 缩并
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-cfc051680d128a6d1910-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-cfc051680d128a6d1910-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 左不变向量场(left-invariant vector field)

- UUID: `bb3b541c-b4ed-45bc-a26e-9365d65888f6` · okf_url: https://limbo137.top/okf/concepts/bb3b541c-b4ed-45bc-a26e-9365d65888f6.md
- labels: en = left-invariant vector field；zh-CN = 左不变向量场
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-left-invariant-vector-field` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.10：左不变向量场。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.10. Let $X$ be a vector field on a Lie group $G$ . $X$ is said to be a left-invariant vector field if $L_{a*}X|_g = X|_{ag}$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 李代数的酉表示(unitary Lie algebra representation)

- UUID: `bb49c0ac-9eb0-4734-901d-f537f22d4855` · okf_url: https://limbo137.top/okf/concepts/bb49c0ac-9eb0-4734-901d-f537f22d4855.md
- labels: en = unitary Lie algebra representation；zh-CN = 李代数的酉表示
- aliases: unitary representation；酉表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-851e0548d24298e19cdf-lie-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 完备测度(complete measure)

- UUID: `bbaa06a0-b3ff-4f57-818c-60cf45c8c9e7` · okf_url: https://limbo137.top/okf/concepts/bbaa06a0-b3ff-4f57-818c-60cf45c8c9e7.md
- labels: en = complete measure；zh-CN = 完备测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-f148cf5b376bb77567df-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 生成的子空间(subspace generated by)

- UUID: `bbb8c33f-05dd-4f24-a5c7-cc7a839194f8` · okf_url: https://limbo137.top/okf/concepts/bbb8c33f-05dd-4f24-a5c7-cc7a839194f8.md
- labels: en = subspace generated by；zh-CN = 生成的子空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.2 · `zh/chapter-13/02-expansion-theorems#concept-13-subspace-generated-by` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.2 · `en/chapter-13/02-expansion-theorems#first-b6544f07eac0124ae411-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/02-expansion-theorems.md

## 值域(range)

- UUID: `bc816847-9a37-4e92-840e-b13a6380a072` · okf_url: https://limbo137.top/okf/concepts/bc816847-9a37-4e92-840e-b13a6380a072.md
- labels: en = range；zh-CN = 值域
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-4d610d239e935fc70ba3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-range` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §13.6 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: R_A=A(D_A)，实际像；不等于任意预先指定的目标集。 · 原文: `Its range is defined as the set of vectors that are mapped onto, $R _ { A } = A ( D _ { A } )$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 哈密顿向量场(Hamiltonian vector field)

- UUID: `bd322781-a622-431f-a000-1be56b3c37e1` · okf_url: https://limbo137.top/okf/concepts/bd322781-a622-431f-a000-1be56b3c37e1.md
- labels: en = Hamiltonian vector field；zh-CN = 哈密顿向量场
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-0acbbd0ad314962e6deb-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-0acbbd0ad314962e6deb-en` · original_transcription)；woit-qtgr/2025-10-20 (en · §15.2 · `books/woit-qtgr/2025-10-20/en/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations#first-3fcaaf2a4cf66d90d2dd-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §15.2 · `books/woit-qtgr/2025-10-20/zh/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations#first-32812f1047753eb9505d-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §15.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若 $M=\mathbf R^2$ 上的向量场具有下列形式 $$ X_f=\frac{\partial f}{\partial p}\frac{\partial}{\partial q}-\frac{\partial f}{\partial q}\frac{\partial}{\partial p}=-\{f,\cdot\}, $$ 其中 $f$ 是 $M$ 上的函数，则称它为哈密顿向量场，记作 $X_f$。在更高维的情形下，$M=\mathbf R^{2d}$ 上的哈密顿向量场具有如下形式： $$ X_f=\sum_{j=1}^{d}\left(\frac{\partial f}{\partial p_j}\frac{\partial}{\partial q_j}-\frac{\partial f}{\...`
  - woit-qtgr/2025-10-20 §15.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A vector field on $M=\mathbf R^2$ of the form $$ X_f=\frac{\partial f}{\partial p}\frac{\partial}{\partial q}-\frac{\partial f}{\partial q}\frac{\partial}{\partial p}=-\{f,\cdot\} $$ for a function $f$ on $M$ is called a Hamiltonian vector field and is denoted $X_f$. In higher dimensions, Hamiltonian vector fields on $M=\mathbf R^{2d}$ have the form $$ X_...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-15/02-hamiltonian-vector-fields-and-canonical-transformations.md

## 自伴矩阵(self-adjoint matrix)

- UUID: `bd347ccc-dc4f-4ee4-8f5c-b6a79f4d41de` · okf_url: https://limbo137.top/okf/concepts/bd347ccc-dc4f-4ee4-8f5c-b6a79f4d41de.md
- labels: en = self-adjoint matrix；zh-CN = 自伴矩阵
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.7 · `books/woit-qtgr/2025-10-20/zh/chapter-04/07-eigenvalues-and-eigenvectors#first-486861a6c147c48caf24-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 可数可加(countably additive)

- UUID: `bd8e72eb-158e-4dd0-a213-78b06d6ba9d5` · okf_url: https://limbo137.top/okf/concepts/bd8e72eb-158e-4dd0-a213-78b06d6ba9d5.md
- labels: en = countably additive；zh-CN = 可数可加
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-90a50142ca051d7f3745-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 几乎处处相等(equal almost everywhere)

- UUID: `bd926011-fb15-41ba-8200-7005cf33325f` · okf_url: https://limbo137.top/okf/concepts/bd926011-fb15-41ba-8200-7005cf33325f.md
- labels: en = equal almost everywhere；zh-CN = 几乎处处相等
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-equal-almost-everywhere` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 有界算符的伴随(adjoint of a bounded operator)

- UUID: `bdb7f779-f3cc-47cb-8adf-77956a39f0d4` · okf_url: https://limbo137.top/okf/concepts/bdb7f779-f3cc-47cb-8adf-77956a39f0d4.md
- labels: en = adjoint of a bounded operator；zh-CN = 有界算符的伴随
- aliases: adjoint；伴随
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-2e0a164d8dd3ffb235a5-operator-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-adjoint` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 群在集合上的作用(Group action on a set)

- UUID: `be2d8436-d3dc-4871-a029-122366c0f690` · okf_url: https://limbo137.top/okf/concepts/be2d8436-d3dc-4871-a029-122366c0f690.md
- labels: en = Group action on a set；zh-CN = 群在集合上的作用
- aliases: group action on a set
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-5dc3c5d27aaa22db3338-zh` · translation)；woit-qtgr/2025-10-20 (en · §1.3 · `books/woit-qtgr/2025-10-20/en/chapter-01/03-unitary-group-representations#first-5dc3c5d27aaa22db3338-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §1.3 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Group action on a set). An action of a group $G$ on a set $M$ is given by a map $$ (g, x) \in G \times M \to g \cdot x \in M $$ that takes a pair $( g , x )$ of a group element $g \in G$ and an element $x \in M$ to another element $g \cdot x \in M$ such that $$ g _ {1} \cdot (g _ {2} \cdot x) = (g _ {1} g _ {2}) \cdot x\tag{1.2} $$ and $$ e \c...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 常曲率(constant curvature)

- UUID: `be45079f-5d67-4742-8671-a499a61b9ade` · okf_url: https://limbo137.top/okf/concepts/be45079f-5d67-4742-8671-a499a61b9ade.md
- labels: en = constant curvature；zh-CN = 常曲率
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-5c7d34002c4e22812ef0-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-constant-curvature` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 幂集(power set)

- UUID: `be638422-462d-4f5d-a903-6026472a0465` · okf_url: https://limbo137.top/okf/concepts/be638422-462d-4f5d-a903-6026472a0465.md
- labels: en = power set；zh-CN = 幂集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.2 · `zh/chapter-01/02-subsets-and-operations#first-4e4da7d9df37a8f1cc07-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/02-subsets-and-operations.md

## 事件(event)

- UUID: `be792ca7-e48a-456a-b71a-d2e7ce013ee1` · okf_url: https://limbo137.top/okf/concepts/be792ca7-e48a-456a-b71a-d2e7ce013ee1.md
- labels: en = event；zh-CN = 事件
- aliases: events
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-720add771c189e466439-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-event` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 受限正交群(restricted orthogonal group)

- UUID: `be89d96d-f2fe-4fd0-b474-8739def4a433` · okf_url: https://limbo137.top/okf/concepts/be89d96d-f2fe-4fd0-b474-8739def4a433.md
- labels: en = restricted orthogonal group；zh-CN = 受限正交群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §39.2 · `books/woit-qtgr/2025-10-20/en/chapter-39/02-the-restricted-symplectic-group#first-8f341fbb416145e08efb-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §39.2 · `books/woit-qtgr/2025-10-20/zh/chapter-39/02-the-restricted-symplectic-group#first-8f341fbb416145e08efb-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/02-the-restricted-symplectic-group.md

## 无处稠密(nowhere dense)

- UUID: `be9d7d7b-195a-4a88-864a-6fd1e6eaf2f7` · okf_url: https://limbo137.top/okf/concepts/be9d7d7b-195a-4a88-864a-6fd1e6eaf2f7.md
- labels: en = nowhere dense；zh-CN = 无处稠密
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-1573c974299f73854745-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 基本群(fundamental group)

- UUID: `bed044e3-0bd6-48d8-8aab-253bd7096faa` · okf_url: https://limbo137.top/okf/concepts/bed044e3-0bd6-48d8-8aab-253bd7096faa.md
- labels: en = fundamental group；zh-CN = 基本群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-fund` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-fund` · machine_draft)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.1.4 (pdf_page=144, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.5：基本群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.5. Let $X$ be a topological space. The set of homotopy classes of loops at $x_0 \in X$ is denoted by $\pi_1(X, x_0)$ and is called the fundamental group (or the first homotopy group) of $X$ at $x_0$ . The product of homotopy classes $[\alpha]$ and $[\beta]$ is defined by`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.4 (pdf_page=144, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.5：基本群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ [ \alpha ] * [ \beta ] = [ \alpha * \beta ]. \tag {4.4} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 恒等态射(identity morphism)

- UUID: `bf12827c-4e18-4a4b-b0cd-beaf17f4c035` · okf_url: https://limbo137.top/okf/concepts/bf12827c-4e18-4a4b-b0cd-beaf17f4c035.md
- labels: en = identity morphism；zh-CN = 恒等态射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-5e0f22120a3c6f2ec2ec-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 拓扑商空间(topological quotient space)

- UUID: `bf25e16b-0efc-4514-88b9-3e7aa54578a2` · okf_url: https://limbo137.top/okf/concepts/bf25e16b-0efc-4514-88b9-3e7aa54578a2.md
- labels: en = topological quotient space；zh-CN = 拓扑商空间
- aliases: quotient space；商空间；拓扑商空间（射影空间构造）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-abc346eea4153b0f12a9-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-abc346eea4153b0f12a9-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 欧氏变换(Euclidean transformation)

- UUID: `bf847762-9cf4-4164-962c-97558cdc6762` · okf_url: https://limbo137.top/okf/concepts/bf847762-9cf4-4164-962c-97558cdc6762.md
- labels: en = Euclidean transformation；zh-CN = 欧氏变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-euclidean-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 向量空间的内直和(internal direct sum of vector spaces)

- UUID: `c03015ff-9150-4204-b5f2-9fbc19660ab2` · okf_url: https://limbo137.top/okf/concepts/c03015ff-9150-4204-b5f2-9fbc19660ab2.md
- labels: en = internal direct sum of vector spaces；zh-CN = 向量空间的内直和
- aliases: direct sum；直和
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.4 · `zh/chapter-03/04-vector-subspaces-and-quotient-spaces#concept-3-direct-sum` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/04-vector-subspaces-and-quotient-spaces.md

## 稠定的(densely defined)

- UUID: `c0c52ad6-989e-4d1b-8ed9-9f4d55ba0d40` · okf_url: https://limbo137.top/okf/concepts/c0c52ad6-989e-4d1b-8ed9-9f4d55ba0d40.md
- labels: en = densely defined；zh-CN = 稠定的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.6 · `zh/chapter-13/06-unbounded-operators#concept-13-densely-defined` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-c399a33322b23377e070-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md

## 同调的(homologous)

- UUID: `c15b2f02-2ed4-430f-b86c-876c372f938e` · okf_url: https://limbo137.top/okf/concepts/c15b2f02-2ed4-430f-b86c-876c372f938e.md
- labels: en = homologous；zh-CN = 同调的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-5fa48f22e3b9d9c0bad1-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-5fa48f22e3b9d9c0bad1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 超势(superpotential)

- UUID: `c15c5aa4-a64e-4d12-9ded-57d386a5a99d` · okf_url: https://limbo137.top/okf/concepts/c15c5aa4-a64e-4d12-9ded-57d386a5a99d.md
- labels: en = superpotential；zh-CN = 超势
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33.2 · `books/woit-qtgr/2025-10-20/en/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential#first-e24fd89c4c61cc5d188d-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §33.2 · `books/woit-qtgr/2025-10-20/zh/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential#first-e24fd89c4c61cc5d188d-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/02-supersymmetric-quantum-mechanics-with-a-superpotential.md

## 域(field)

- UUID: `c16f8888-996a-428f-a6b8-a4de7e108868` · okf_url: https://limbo137.top/okf/concepts/c16f8888-996a-428f-a6b8-a4de7e108868.md
- labels: en = field；zh-CN = 域
- aliases: field (algebra)；域（代数）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.1 · `zh/chapter-03/01-rings-and-fields#concept-3-field` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/01-rings-and-fields.md

## 切空间(tangent space)

- UUID: `c1d46382-437d-4926-a7fd-d9d0cd36a531` · okf_url: https://limbo137.top/okf/concepts/c1d46382-437d-4926-a7fd-d9d0cd36a531.md
- labels: en = tangent space；zh-CN = 切空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-1fd2c442e5431ea73af8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-tangent-space` · translation)；woit-qtgr/2025-10-20 (zh-CN · §3.2 · `books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices#first-c1cbeb5580362deafa51-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.2 · `books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices#first-c1cbeb5580362deafa51-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/02-commutation-relations-for-pauli-matrices.md

## 忠实表示(faithful representation)

- UUID: `c1e17053-8805-4fd9-93e0-eb9997fdaf8f` · okf_url: https://limbo137.top/okf/concepts/c1e17053-8805-4fd9-93e0-eb9997fdaf8f.md
- labels: en = faithful representation；zh-CN = 忠实表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.4 · `zh/chapter-04/04-introduction-to-group-representation-theory#concept-4-faithful-representation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/04-introduction-to-group-representation-theory.md

## 复射影直线(complex projective line)

- UUID: `c2119140-62f5-42ca-b162-fd313d43319e` · okf_url: https://limbo137.top/okf/concepts/c2119140-62f5-42ca-b162-fd313d43319e.md
- labels: en = complex projective line；zh-CN = 复射影直线
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-de54afbc6b9c88ca980a-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-de54afbc6b9c88ca980a-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 闵可夫斯基向量空间(Minkowski vector space)

- UUID: `c220fb5a-3d41-491f-865c-8b2a4c28fa3f` · okf_url: https://limbo137.top/okf/concepts/c220fb5a-3d41-491f-865c-8b2a4c28fa3f.md
- labels: en = Minkowski vector space；zh-CN = 闵可夫斯基向量空间
- aliases: Minkowski space；闵可夫斯基空间
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §40.1 · `books/woit-qtgr/2025-10-20/en/chapter-40/01-minkowski-space#first-d231792fa1ef91da6bdc-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §40.1 · `books/woit-qtgr/2025-10-20/zh/chapter-40/01-minkowski-space#first-5f65d3b7cad0ac9ea86e-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §40.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[闵可夫斯基空间(Minkowski space)](/concepts/geometry/minkowski-space) $M^4$ 是带下列不定内积的向量空间 $\mathbb R^4$： $$ (x,y)\equiv x\cdot y=-x_0y_0+x_1y_1+x_2y_2+x_3y_3. $$ 其中 $(x_0,x_1,x_2,x_3)$ 与 $(y_0,y_1,y_2,y_3)$ 分别是 $x,y\in\mathbb R^4$ 的坐标。 *采用 $(-,+,+,+)$ 符号约定，理由如下：* - *把时间变量 $x_0$ 解析延拓为 $ix_0$ 后，内积变为正定。* - *限制到空间分量时，此前关于欧氏空间对称群 $E(3)$ 的公式保持原样。* - *这一选...`
  - woit-qtgr/2025-10-20 §40.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[Minkowski space](/concepts/geometry/minkowski-space) $M^4$ is $\mathbb R^4$ with the indefinite inner product $$ (x,y)\equiv x\cdot y=-x_0y_0+x_1y_1+x_2y_2+x_3y_3. $$ Here $(x_0,x_1,x_2,x_3)$ and $(y_0,y_1,y_2,y_3)$ are the coordinates of $x,y\in\mathbb R^4$. *We use the $(-,+,+,+)$ rather than $(+,-,-,-)$ convention for these reasons:* - *...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/01-minkowski-space.md

## 上闭链(cocycle)

- UUID: `c26a11a5-b8bb-4cd7-8920-cd092b71eb06` · okf_url: https://limbo137.top/okf/concepts/c26a11a5-b8bb-4cd7-8920-cd092b71eb06.md
- labels: en = cocycle；zh-CN = 上闭链
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-6a9d37016a96e22681a4-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-6a9d37016a96e22681a4-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 欧拉角(Euler angles)

- UUID: `c2c3d648-ab4b-4df7-9d8d-de852327c911` · okf_url: https://limbo137.top/okf/concepts/c2c3d648-ab4b-4df7-9d8d-de852327c911.md
- labels: en = Euler angles；zh-CN = 欧拉角
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §6.1 · `books/woit-qtgr/2025-10-20/zh/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-euler-angles` · machine_draft)；woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-euler-angles` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md

## 电磁场(electromagnetic field)

- UUID: `c2cf2f8f-2a79-4942-863c-c7fc5e8bd40f` · okf_url: https://limbo137.top/okf/concepts/c2cf2f8f-2a79-4942-863c-c7fc5e8bd40f.md
- labels: en = electromagnetic field；zh-CN = 电磁场
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-f9754db4fef33a82eb6d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 薛定谔表示，李群版本(Schrödinger representation, Lie group version)

- UUID: `c2dfd358-353e-4ba7-8774-c693c50cba5f` · okf_url: https://limbo137.top/okf/concepts/c2dfd358-353e-4ba7-8774-c693c50cba5f.md
- labels: en = Schrödinger representation, Lie group version；zh-CN = 薛定谔表示，李群版本
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §13.3 · `books/woit-qtgr/2025-10-20/zh/chapter-13/03-the-schr-odinger-representation#first-022e33e5f115ab4de080-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §13.3 · `books/woit-qtgr/2025-10-20/en/chapter-13/03-the-schr-odinger-representation#first-aca811e9ff50bc4c19ca-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡李群 $H_3$ 的薛定谔表示是 $(\Gamma_S,L^2(\mathbf R))$，由下式定义： $$ \Gamma_S\!\left(\left(\binom{x}{y},z\right)\right)\psi(q)=e^{-iz}e^{i\frac{xy}{2}}e^{-ixq}\psi(q-y)\tag{13.3} $$`
  - woit-qtgr/2025-10-20 §13.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Schrödinger representation of the Heisenberg Lie group $H_3$ is $(\Gamma_S,L^2(\mathbf R))$, defined by $$ \Gamma_S\!\left(\left(\binom{x}{y},z\right)\right)\psi(q)=e^{-iz}e^{i\frac{xy}{2}}e^{-ixq}\psi(q-y)\tag{13.3} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-13/03-the-schr-odinger-representation.md

## 阿贝尔李代数(abelian Lie algebra)

- UUID: `c30bfb5d-dc0a-47ce-8225-a08fe05164ac` · okf_url: https://limbo137.top/okf/concepts/c30bfb5d-dc0a-47ce-8225-a08fe05164ac.md
- labels: en = abelian Lie algebra；zh-CN = 阿贝尔李代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-abelian-lie-algebra` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 厄米的(hermitian)

- UUID: `c353238d-f4ab-4fe1-9ac1-1a9f4e0317a9` · okf_url: https://limbo137.top/okf/concepts/c353238d-f4ab-4fe1-9ac1-1a9f4e0317a9.md
- labels: en = hermitian；zh-CN = 厄米的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-b2e898ecd6968a4f67a9-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-hermitian` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## $\mathfrak{sl}(2,\mathbb R)$ 的元辛表示(Metaplectic representation of $\mathfrak{sl}(2,\mathbb R)$)

- UUID: `c38865ab-fa4a-4617-809e-1d99d66a3793` · okf_url: https://limbo137.top/okf/concepts/c38865ab-fa4a-4617-809e-1d99d66a3793.md
- labels: en = Metaplectic representation of $\mathfrak{sl}(2,\mathbb R)$；zh-CN = $\mathfrak{sl}(2,\mathbb R)$ 的元辛表示
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §24.1 · `books/woit-qtgr/2025-10-20/zh/chapter-24/01-the-metaplectic-representation-for-d-1-in-terms-of-a-and-a#first-ebf8ac7cc616453eec4b-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §24.1 · `books/woit-qtgr/2025-10-20/en/chapter-24/01-the-metaplectic-representation-for-d-1-in-terms-of-a-and-a#first-efc7739eee72cb7c382d-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §24.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The representation $\Gamma_{BF}^{\prime}$ on $\mathcal F$ is given by $$ \Gamma_ {B F} ^ {\prime} (G) = \Gamma_ {B F} ^ {\prime} (X _ {+} + X _ {-}) = - \frac {i}{2} ((a ^ {\dagger}) ^ {2} + a ^ {2}) $$ $$ \Gamma_ {B F} ^ {\prime} (E + F) = \Gamma_ {B F} ^ {\prime} (- i (X _ {+} - X _ {-})) = - \frac {1}{2} ((a ^ {\dagger}) ^ {2} - a ^ {2})\tag{24.2} $$ $...`
  - woit-qtgr/2025-10-20 §24.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `$\mathcal F$ 上的表示 $\Gamma_{BF}^{\prime}$ 定义如下： $$ \Gamma_ {B F} ^ {\prime} (G) = \Gamma_ {B F} ^ {\prime} (X _ {+} + X _ {-}) = - \frac {i}{2} ((a ^ {\dagger}) ^ {2} + a ^ {2}) $$ $$ \Gamma_ {B F} ^ {\prime} (E + F) = \Gamma_ {B F} ^ {\prime} (- i (X _ {+} - X _ {-})) = - \frac {1}{2} ((a ^ {\dagger}) ^ {2} - a ^ {2})\tag{24.2} $$ $$ \Gamma_ {B F} ^ {\pri...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-24/01-the-metaplectic-representation-for-d-1-in-terms-of-a-and-a.md

## 开集(open set)

- UUID: `c3ac7e5b-190b-46ab-86f5-5288c831eecc` · okf_url: https://limbo137.top/okf/concepts/c3ac7e5b-190b-46ab-86f5-5288c831eecc.md
- labels: en = open set；zh-CN = 开集
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-6c8a28b7ca0c9a5df668-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## 四维动量(4-momentum)

- UUID: `c3b42b3b-ad25-42f3-96f7-5642573c8462` · okf_url: https://limbo137.top/okf/concepts/c3b42b3b-ad25-42f3-96f7-5642573c8462.md
- labels: en = 4-momentum；zh-CN = 四维动量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.3 · `en/chapter-09/03-particle-dynamics#first-bcf9c0b6ea6f972159e5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.3 · `zh/chapter-09/03-particle-dynamics#first-bcf9c0b6ea6f972159e5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/03-particle-dynamics.md

## 李超代数表示(Representation of a Lie superalgebra)

- UUID: `c45f5876-ea6e-409e-b221-625213a50b9f` · okf_url: https://limbo137.top/okf/concepts/c45f5876-ea6e-409e-b221-625213a50b9f.md
- labels: en = Representation of a Lie superalgebra；zh-CN = 李超代数表示
- aliases: representation of a Lie superalgebra
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §31.1 · `books/woit-qtgr/2025-10-20/en/chapter-31/01-quantization-of-pseudo-classical-systems#first-b64fd7ba5ab4854767e5-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §31.1 · `books/woit-qtgr/2025-10-20/zh/chapter-31/01-quantization-of-pseudo-classical-systems#first-9fdbeb8fa684a62e53aa-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §31.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A [representation of a Lie superalgebra](/concepts/representation-theory/lie-superalgebra-representation) is a homomorphism $\Phi$ preserving the superbracket: $$ [\Phi(X),\Phi(Y)]_\pm=\Phi([X,Y]_\pm). $$ It takes values in a Lie superalgebra of linear operators, with $|\Phi(X)|=|X|$ and $$ [\Phi(X),\Phi(Y)]_\pm=\Phi(X)\Phi(Y)-(-)^{|X||Y|}\P...`
  - woit-qtgr/2025-10-20 §31.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `[李超代数表示(representation of a Lie superalgebra)](/concepts/representation-theory/lie-superalgebra-representation)是保持超括号的同态 $\Phi$： $$ [\Phi(X),\Phi(Y)]_\pm=\Phi([X,Y]_\pm). $$ 它取值于线性算符的李超代数，满足 $|\Phi(X)|=|X|$，其中 $$ [\Phi(X),\Phi(Y)]_\pm=\Phi(X)\Phi(Y)-(-)^{|X||Y|}\Phi(Y)\Phi(X). $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-31/01-quantization-of-pseudo-classical-systems.md

## 条件概率(conditional probability)

- UUID: `c4872e54-8e98-4b2d-803d-b1a24bfb06ce` · okf_url: https://limbo137.top/okf/concepts/c4872e54-8e98-4b2d-803d-b1a24bfb06ce.md
- labels: en = conditional probability；zh-CN = 条件概率
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-c8a136e13e424f876c4d-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 库仑规范(Coulomb gauge)

- UUID: `c4aefa07-365f-49a3-899c-1f5c16dda09d` · okf_url: https://limbo137.top/okf/concepts/c4aefa07-365f-49a3-899c-1f5c16dda09d.md
- labels: en = Coulomb gauge；zh-CN = 库仑规范
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §46.4 · `books/woit-qtgr/2025-10-20/zh/chapter-46/04-quantization-in-coulomb-gauge#first-a1277e5d81e65a590971-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §46.4 · `books/woit-qtgr/2025-10-20/en/chapter-46/04-quantization-in-coulomb-gauge#first-5e4b7b3850019e000df1-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §46.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `若 $\nabla \cdot \mathbf { A } = 0$，则称矢势 $A _ { \mu }$ 处于库仑规范。`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-46/04-quantization-in-coulomb-gauge.md

## de Rham微分(de Rham differential)

- UUID: `c4df2f3d-e41b-4bde-949e-8989739a863d` · okf_url: https://limbo137.top/okf/concepts/c4df2f3d-e41b-4bde-949e-8989739a863d.md
- labels: en = de Rham differential；zh-CN = de Rham微分
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §33.3 · `books/woit-qtgr/2025-10-20/en/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-f2d1370f98e339cb27f5-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §33.3 · `books/woit-qtgr/2025-10-20/zh/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-f2d1370f98e339cb27f5-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms.md

## 庞加莱引理(Poincaré lemma)

- UUID: `c5c6f787-2192-4331-ba80-8bede20d45b7` · okf_url: https://limbo137.top/okf/concepts/c5c6f787-2192-4331-ba80-8bede20d45b7.md
- labels: en = Poincaré lemma；zh-CN = 庞加莱引理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.5 · `en/chapter-17/05-the-poincar-e-lemma#first-082959daf840e444a6f3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.5 · `zh/chapter-17/05-the-poincar-e-lemma#first-082959daf840e444a6f3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/05-the-poincar-e-lemma.md

## 坡印廷矢量(Poynting vector)

- UUID: `c60eeabd-be06-4756-8ba6-12e3fdc72872` · okf_url: https://limbo137.top/okf/concepts/c60eeabd-be06-4756-8ba6-12e3fdc72872.md
- labels: en = Poynting vector；zh-CN = 坡印廷矢量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.5 · `zh/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-f02720824f861d993cad-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.5 · `en/chapter-09/05-conservation-laws-and-energy-stress-tensors#first-f02720824f861d993cad-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/05-conservation-laws-and-energy-stress-tensors.md

## 狄拉克方程(Dirac equation)

- UUID: `c6117e48-9e8f-4f0a-88b6-20779e5159d4` · okf_url: https://limbo137.top/okf/concepts/c6117e48-9e8f-4f0a-88b6-20779e5159d4.md
- labels: en = Dirac equation；zh-CN = 狄拉克方程
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §47.1 · `books/woit-qtgr/2025-10-20/en/chapter-47/01-the-dirac-equation-in-minkowski-space#first-d0bd80525eaa32470ada-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §47.1 · `books/woit-qtgr/2025-10-20/zh/chapter-47/01-the-dirac-equation-in-minkowski-space#first-3d93a11157db450d916e-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §47.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `狄拉克方程是微分方程 $$ (\not\partial-m)\psi(x)=0\tag{47.2} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-47/01-the-dirac-equation-in-minkowski-space.md

## 阿贝尔李代数(abelian Lie algebra)

- UUID: `c6303f8a-9c2e-414b-b285-c5671024e962` · okf_url: https://limbo137.top/okf/concepts/c6303f8a-9c2e-414b-b285-c5671024e962.md
- labels: en = abelian Lie algebra；zh-CN = 阿贝尔李代数
- aliases: abelian
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-882b659f526d0c278d26-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 半直积(semi-direct product)

- UUID: `c65e599c-ab76-4124-9f5a-0f1f9997e585` · okf_url: https://limbo137.top/okf/concepts/c65e599c-ab76-4124-9f5a-0f1f9997e585.md
- labels: en = semi-direct product；zh-CN = 半直积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.7 · `zh/chapter-02/07-symmetry-groups#concept-2-semi-direct-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/07-symmetry-groups.md

## 反线性(antilinear)

- UUID: `c68e97f6-2eed-4d37-be47-ad318e6c9ba5` · okf_url: https://limbo137.top/okf/concepts/c68e97f6-2eed-4d37-be47-ad318e6c9ba5.md
- labels: en = antilinear；zh-CN = 反线性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-antilinear` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-b24bdbf04b0ca489ab48-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 张成或生成的子空间(span)

- UUID: `c721f2ec-d052-4cd6-a1b4-e27216b017b1` · okf_url: https://limbo137.top/okf/concepts/c721f2ec-d052-4cd6-a1b4-e27216b017b1.md
- labels: en = span；zh-CN = 张成或生成的子空间
- aliases: spanned；张成
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-span` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 马施克定理(Maschke’s theorem)

- UUID: `c75b3404-beda-4f83-b6ce-b1d83e5669ad` · okf_url: https://limbo137.top/okf/concepts/c75b3404-beda-4f83-b6ce-b1d83e5669ad.md
- labels: en = Maschke’s theorem；zh-CN = 马施克定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#first-50699cbe14af28439bed-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §5.3 · `en/chapter-05/03-representations-of-finite-groups#first-50699cbe14af28439bed-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md

## 点谱(point spectrum)

- UUID: `c768f687-a6f9-4e14-977a-488822437d2a` · okf_url: https://limbo137.top/okf/concepts/c768f687-a6f9-4e14-977a-488822437d2a.md
- labels: en = point spectrum；zh-CN = 点谱
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §13.5 · `en/chapter-13/05-spectral-theory#first-db910a0bdb87906d73dc-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.5 · `zh/chapter-13/05-spectral-theory#concept-13-point-spectrum` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/05-spectral-theory.md

## 亥维赛德阶跃函数(Heaviside step function)

- UUID: `c7a623f9-79cd-45e9-ad58-94b359b2bff2` · okf_url: https://limbo137.top/okf/concepts/c7a623f9-79cd-45e9-ad58-94b359b2bff2.md
- labels: en = Heaviside step function；zh-CN = 亥维赛德阶跃函数
- aliases: Heaviside function；Heaviside函数
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.2 · `zh/chapter-12/02-operations-on-distributions#first-e32fef5c881d81e4aca4-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §1.3.3 (pdf_page=47, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定义条件与符号保留在引文中；不推断跨书等价。 · 原文: `13 The Heaviside function is defined by`
  - szekeres/source-snapshot-bc3404156f6e §12.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: x<0为0，x≥0为1；两书在零点约定一致。 · 原文: `Set $\theta ( x )$ to be the Heaviside step function $$ \theta (x) = \left\{ \begin{array}{l l} 1 & \text { if } x \geq 0, \\ 0 & \text { if } x $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/02-operations-on-distributions.md

## 内积空间的自伴条件(self-adjoint condition in an inner product space)

- UUID: `c7a78756-912d-4e4f-a31b-762d35ae477b` · okf_url: https://limbo137.top/okf/concepts/c7a78756-912d-4e4f-a31b-762d35ae477b.md
- labels: en = self-adjoint condition in an inner product space；zh-CN = 内积空间的自伴条件
- aliases: self-adjoint operator；内积空间的自伴条件（练习）；自伴算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-self-adjoint-operator` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 恰当形式(exact form)

- UUID: `c7b763c7-2046-4023-85d1-954aaaf12c0d` · okf_url: https://limbo137.top/okf/concepts/c7b763c7-2046-4023-85d1-954aaaf12c0d.md
- labels: en = exact form；zh-CN = 恰当形式
- aliases: exact
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-b62ff826920c0e8eb620-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-21de2db070b1774c8490-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 截面曲率(sectional curvature)

- UUID: `c7c5b229-3b71-47b1-94df-4c6f7c6b7828` · okf_url: https://limbo137.top/okf/concepts/c7c5b229-3b71-47b1-94df-4c6f7c6b7828.md
- labels: en = sectional curvature；zh-CN = 截面曲率
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.7 · `en/chapter-18/07-cartan-formalism#first-f5d79a6c86fd49aa3042-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.7 · `zh/chapter-18/07-cartan-formalism#first-f5d79a6c86fd49aa3042-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/07-cartan-formalism.md

## 诺特定理(Noether’s theorem)

- UUID: `c84c4811-3c5d-480b-b7df-17a2505e95b0` · okf_url: https://limbo137.top/okf/concepts/c84c4811-3c5d-480b-b7df-17a2505e95b0.md
- labels: en = Noether’s theorem；zh-CN = 诺特定理
- aliases: Noether定理
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = not-recorded；nakahara-geometry-topology-physics/second-edition-local = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.2 · `books/woit-qtgr/2025-10-20/en/chapter-35/02-noether-s-theorem-and-symmetries-in-the-lagrangian-formalism#first-f2721f2f42e84914421b-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.2 · `books/woit-qtgr/2025-10-20/zh/chapter-35/02-noether-s-theorem-and-symmetries-in-the-lagrangian-formalism#first-f2721f2f42e84914421b-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/02-noether-s-theorem-and-symmetries-in-the-lagrangian-formalism.md

## 集合族(family of sets)

- UUID: `c8c1d8df-16cb-4f99-a42a-17e387eef280` · okf_url: https://limbo137.top/okf/concepts/c8c1d8df-16cb-4f99-a42a-17e387eef280.md
- labels: en = family of sets；zh-CN = 集合族
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-d459549609296007e3aa-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 三维自旋旋量(two-component spinor)

- UUID: `c8c7bd36-41d4-4cd0-9c52-68d1e458bf59` · okf_url: https://limbo137.top/okf/concepts/c8c7bd36-41d4-4cd0-9c52-68d1e458bf59.md
- labels: en = two-component spinor；zh-CN = 三维自旋旋量
- aliases: 三维自旋旋量（二分量）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.1 · `books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation#first-6eb127ba47451a125e00-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.1 · `books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation#first-ca37a43c4461c705ee15-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 粒子数守恒(particle number conservation)

- UUID: `c922485e-38de-4c71-8b31-2cf8e45ac65a` · okf_url: https://limbo137.top/okf/concepts/c922485e-38de-4c71-8b31-2cf8e45ac65a.md
- labels: en = particle number conservation；zh-CN = 粒子数守恒
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §38.2 · `books/woit-qtgr/2025-10-20/zh/chapter-38/02-internal-symmetries#first-955c0514be01bdc82953-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §38.2 · `books/woit-qtgr/2025-10-20/en/chapter-38/02-internal-symmetries#first-955c0514be01bdc82953-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/02-internal-symmetries.md

## 伪欧氏内积(pseudo-Euclidean inner product)

- UUID: `c9477834-26c4-4a2b-85fb-2253be1709bf` · okf_url: https://limbo137.top/okf/concepts/c9477834-26c4-4a2b-85fb-2253be1709bf.md
- labels: en = pseudo-Euclidean inner product；zh-CN = 伪欧氏内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-pseudo-euclidean-inner-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 正交复结构(orthogonal complex structure)

- UUID: `c985d06a-9bb8-4419-8598-30d94e1b91f8` · okf_url: https://limbo137.top/okf/concepts/c985d06a-9bb8-4419-8598-30d94e1b91f8.md
- labels: en = orthogonal complex structure；zh-CN = 正交复结构
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §31.4 (pdf_page=361, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。 · 原文: `$$ (J (\cdot), J (\cdot)) = (\cdot , \cdot) $$`
  - woit-qtgr/2025-10-20 §31.4 (pdf_page=361, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。 · 原文: `The construction of the spinor representation given here has used a specific choice of the $\theta _ { j } , \overline { { \theta } } _ { j }$ (see equations 31.2) and the fermionic annihilation and creation operators. This corresponds to a standard choice of complex structure $J _ { 0 }$ , which appears in a manner closely parallel to that of the Bargman...`
  - woit-qtgr/2025-10-20 §31.4 (pdf_page=361, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。 · 原文: `We will here restrict to the case of $( \cdot , \cdot )$ positive definite, and unlike in the bosonic case, no additional positivity condition on $J$ will then be required.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 有序 n 元组(ordered n-tuple)

- UUID: `c9f10499-c604-49be-9205-1e80391938af` · okf_url: https://limbo137.top/okf/concepts/c9f10499-c604-49be-9205-1e80391938af.md
- labels: en = ordered n-tuple；zh-CN = 有序 n 元组
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-e589a0dd45597382660c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 洛伦兹–菲茨杰拉德收缩(Lorentz–Fitzgerald contraction)

- UUID: `ca0c8a69-312f-4ac8-8b2b-365181b6cf02` · okf_url: https://limbo137.top/okf/concepts/ca0c8a69-312f-4ac8-8b2b-365181b6cf02.md
- labels: en = Lorentz–Fitzgerald contraction；zh-CN = 洛伦兹–菲茨杰拉德收缩
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.2 · `zh/chapter-09/02-relativistic-kinematics#first-9abf0457df0d9b326558-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.2 · `en/chapter-09/02-relativistic-kinematics#first-9abf0457df0d9b326558-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/02-relativistic-kinematics.md

## 自发对称破缺(spontaneous symmetry breaking)

- UUID: `ca209073-6b0b-4faa-b78f-0d00d3d11d24` · okf_url: https://limbo137.top/okf/concepts/ca209073-6b0b-4faa-b78f-0d00d3d11d24.md
- labels: en = spontaneous symmetry breaking；zh-CN = 自发对称破缺
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §39.4 · `books/woit-qtgr/2025-10-20/zh/chapter-39/04-spontaneous-symmetry-breaking#first-604a9d7ff08d53f734ab-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §39.4 · `books/woit-qtgr/2025-10-20/en/chapter-39/04-spontaneous-symmetry-breaking#first-604a9d7ff08d53f734ab-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/04-spontaneous-symmetry-breaking.md

## 内部(interior)

- UUID: `ca484c0c-e258-4cd3-8802-13fd736f4a07` · okf_url: https://limbo137.top/okf/concepts/ca484c0c-e258-4cd3-8802-13fd736f4a07.md
- labels: en = interior；zh-CN = 内部
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-8560acd76c736adab17b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 可逆的(invertible)

- UUID: `cb1aa3aa-d213-4d5c-9061-9e805aea073a` · okf_url: https://limbo137.top/okf/concepts/cb1aa3aa-d213-4d5c-9061-9e805aea073a.md
- labels: en = invertible；zh-CN = 可逆的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-invertible` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-1b9652eb4e8887748082-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 无挠的(torsion-free)

- UUID: `cb71a150-9540-4253-a1de-4fcb50907875` · okf_url: https://limbo137.top/okf/concepts/cb71a150-9540-4253-a1de-4fcb50907875.md
- labels: en = torsion-free；zh-CN = 无挠的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.3 · `zh/chapter-18/03-curvature-and-torsion#first-742245dfca473015595d-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.3 · `en/chapter-18/03-curvature-and-torsion#first-742245dfca473015595d-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/03-curvature-and-torsion.md

## 反对称矩阵(skew-symmetric matrix)

- UUID: `cbc9ebe8-e9e8-4f3d-a919-cb3725b7f68c` · okf_url: https://limbo137.top/okf/concepts/cbc9ebe8-e9e8-4f3d-a919-cb3725b7f68c.md
- labels: en = skew-symmetric matrix；zh-CN = 反对称矩阵
- aliases: skew-symmetric matrices
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.1 · `books/woit-qtgr/2025-10-20/en/chapter-06/01-the-rotation-group-in-three-dimensions#concept-6-skew-symmetric-matrix` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §5.2 · `books/woit-qtgr/2025-10-20/zh/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups#first-d489988c101c19cb1a72-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/02-lie-algebras-of-the-orthogonal-and-unitary-groups.md

## 实形式(real forms)

- UUID: `cbd1de1f-2a0b-436c-9be6-94fea9beb031` · okf_url: https://limbo137.top/okf/concepts/cbd1de1f-2a0b-436c-9be6-94fea9beb031.md
- labels: en = real forms；zh-CN = 实形式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §40.4 · `books/woit-qtgr/2025-10-20/zh/chapter-40/04-spin-and-the-lorentz-group#first-7c5708c86fe2f4d7b10f-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §40.4 · `books/woit-qtgr/2025-10-20/en/chapter-40/04-spin-and-the-lorentz-group#first-7c5708c86fe2f4d7b10f-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-40/04-spin-and-the-lorentz-group.md

## 反线性变换(antilinear transformation)

- UUID: `cc06cd52-4f36-4489-a96f-da283bf71562` · okf_url: https://limbo137.top/okf/concepts/cc06cd52-4f36-4489-a96f-da283bf71562.md
- labels: en = antilinear transformation；zh-CN = 反线性变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-3456599059b5eb5cb3e7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 调和形式(harmonic forms)

- UUID: `cc12d9b1-2846-4c8c-a8f5-6daa888b376e` · okf_url: https://limbo137.top/okf/concepts/cc12d9b1-2846-4c8c-a8f5-6daa888b376e.md
- labels: en = harmonic forms；zh-CN = 调和形式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §33.3 · `books/woit-qtgr/2025-10-20/zh/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-7e46059e165f3f894b8d-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §33.3 · `books/woit-qtgr/2025-10-20/en/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms#first-7e46059e165f3f894b8d-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/03-supersymmetric-quantum-mechanics-and-diferential-forms.md

## 真面(proper face)

- UUID: `cc566ee3-c318-49a8-9e0f-353b0ab878d5` · okf_url: https://limbo137.top/okf/concepts/cc566ee3-c318-49a8-9e0f-353b0ab878d5.md
- labels: en = proper face；zh-CN = 真面
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=117, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 相对于给定几何单纯形σ_r的面σ_q，且σ_q≠σ_r，记σ_q<σ_r；Nakahara约定0-单纯形没有真面。真面是面的严格子类，不是face的无条件异名；不声称首次出现。 · 原文: `Let $q$ be an integer such that $0 \leq q \leq r$ . If we choose $q + 1$ points $p_{i_0}, \ldots, p_{i_q}$ out of $p_0, \ldots, p_r$ , these $q + 1$ points define a $q$ -simplex $\sigma_q = \langle p_{i_0}, \ldots, p_{i_q} \rangle$ , which is called a $q$ -face of $\sigma_r$ . We write $\sigma_q \leq \sigma_r$ if $\sigma_q$ is a face of`
  - nakahara-geometry-topology-physics/second-edition-local §3.2.1 (pdf_page=118, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 相对于给定几何单纯形σ_r的面σ_q，且σ_q≠σ_r，记σ_q<σ_r；Nakahara约定0-单纯形没有真面。真面是面的严格子类，不是face的无条件异名；不声称首次出现。 · 原文: `$\sigma_r$ . If $\sigma_q \neq \sigma_r$ , we say $\sigma_q$ is a proper face of $\sigma_r$ , denoted as $\sigma_q < \sigma_r$ . Figure 3.3 shows a 0-face $p_0$ and a 2-face $\langle p_1p_2p_3\rangle$ of a 3-simplex $\langle p_0p_1p_2p_3\rangle$ . There are one 3-face, four 2-faces, six 1-faces and four 0-faces. The reader should verify that the number of...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 费米子数算符(fermionic number operator)

- UUID: `cccda163-e46e-4720-b300-e3ed738bb1e8` · okf_url: https://limbo137.top/okf/concepts/cccda163-e46e-4720-b300-e3ed738bb1e8.md
- labels: en = fermionic number operator；zh-CN = 费米子数算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §27.1 · `books/woit-qtgr/2025-10-20/en/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator#first-ccdd6b5da0d9f40a16da-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator.md

## 矩阵的相似变换(matrix similarity transformation)

- UUID: `cd3b5031-565b-48d6-9ef1-b20a427e78f5` · okf_url: https://limbo137.top/okf/concepts/cd3b5031-565b-48d6-9ef1-b20a427e78f5.md
- labels: en = matrix similarity transformation；zh-CN = 矩阵的相似变换
- aliases: conjugation；共轭变换
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.3 · `books/woit-qtgr/2025-10-20/zh/chapter-04/03-change-of-basis#first-85d6f47fc66bc72f8c25-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/03-change-of-basis.md

## 不定内积(indefinite inner product)

- UUID: `cd664333-1e91-40f6-879d-2b48d97fc6d5` · okf_url: https://limbo137.top/okf/concepts/cd664333-1e91-40f6-879d-2b48d97fc6d5.md
- labels: en = indefinite inner product；zh-CN = 不定内积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#concept-5-indefinite-inner-product` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 态射(morphism)

- UUID: `cd7c1e52-5581-4c58-a6a9-96f8fbe4fcfe` · okf_url: https://limbo137.top/okf/concepts/cd7c1e52-5581-4c58-a6a9-96f8fbe4fcfe.md
- labels: en = morphism；zh-CN = 态射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-9d18b9bce96c8bc75704-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 开区间(open interval)

- UUID: `cd84b279-ed33-43fd-be4d-3bf0bfb28da5` · okf_url: https://limbo137.top/okf/concepts/cd84b279-ed33-43fd-be4d-3bf0bfb28da5.md
- labels: en = open interval；zh-CN = 开区间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.1 · `zh/chapter-10/01-euclidean-topology#first-8686d43e8087afd9ddfc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## 商空间(quotient space)

- UUID: `ce472884-6d77-4065-8052-02106bdf8345` · okf_url: https://limbo137.top/okf/concepts/ce472884-6d77-4065-8052-02106bdf8345.md
- labels: en = quotient space；zh-CN = 商空间
- aliases: factor space；quotient set；商集；商集（等价类集合）
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-aa0ca7663dc0cd7c7f1f-set-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=89, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §2.1.2以等价类集合定义quotient space，并立即给圆、环面、球面、G/H等几何实现；G/H未必为群。 · 原文: `Let $X$ be a space in our usual sense. (To be more precise, we need the notion of topological space, which will be defined in section 2.3. For the time being we depend on our intuitive notion of 'space'.) Then quotient spaces may be realized as geometrical figures. For example, let $x$ and $y$ be two points in $\mathbb{R}$ .`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=94, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §2.1.2以等价类集合定义quotient space，并立即给圆、环面、球面、G/H等几何实现；G/H未必为群。 · 原文: `Example 2.6. Let $G$ be a group and $H$ a subgroup of $G$ . Let $g, g' \in G$ and introduce an equivalence relation $\sim$ by $g \sim g'$ if there exists $h \in H$ such that $g' = gh$ . We denote the equivalence class $[g] = \{gh | h \in H\}$ by $gH$ . The class $gH$ is called a (left) coset. $gH$ satisfies either $gH \cap g'H = \emptyset$ or $gH = g'H$ ....`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=90, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara §2.1.2以等价类集合定义quotient space，并立即给圆、环面、球面、G/H等几何实现；G/H未必为群。 · 原文: `Introduce a relation $\sim$ by: $x\sim y$ if there exists $n\in \mathbb{Z}$ such that $y = x + 2\pi n$ . It is easily shown that $\sim$ is an equivalence relation. The class $[x]$ is the set $\{\ldots ,x - 2\pi ,x,x + 2\pi ,\ldots \}$ . A number $x\in [0,2\pi)$ serves as a representative of an equivalence class $[x]$ , see figure $2.2(a)$ . Note that $0$ ...`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=89, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书定义条件与符号保留在引文中；不推断跨书等价。 · 原文: `$[a]$ cannot be empty since $a \sim a$ . We now prove that if $[a] \cap [b] \neq \emptyset$ then $[a] = [b]$ . First note that $a \sim b$ . (Since $[a] \cap [b] \neq \emptyset$ there is at least one element in $[a] \cap [b]$ that satisfies $c \sim a$ and $c \sim b$ . From the transitivity, we have $a \sim b$ .) Next we show that $[a] \subset [b]$ . Take a...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 基灵向量(Killing vector)

- UUID: `ce4e2dec-50cc-4729-bb8e-6003fcf99e35` · okf_url: https://limbo137.top/okf/concepts/ce4e2dec-50cc-4729-bb8e-6003fcf99e35.md
- labels: en = Killing vector；zh-CN = 基灵向量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-killing-vector` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-4a92be17c89ec37308eb-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 实射影直线(real projective line)

- UUID: `ce618650-9f53-49e2-ab6f-718ffe06a89e` · okf_url: https://limbo137.top/okf/concepts/ce618650-9f53-49e2-ab6f-718ffe06a89e.md
- labels: en = real projective line；zh-CN = 实射影直线
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-cdbf26767d2e19aff4dc-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-cdbf26767d2e19aff4dc-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 量子哈密顿算符(quantum Hamiltonian operator)

- UUID: `ce9d1194-5bd1-4a10-bd18-3aa212bbd0c7` · okf_url: https://limbo137.top/okf/concepts/ce9d1194-5bd1-4a10-bd18-3aa212bbd0c7.md
- labels: en = quantum Hamiltonian operator；zh-CN = 量子哈密顿算符
- aliases: Hamiltonian；Hamiltonian operator；哈密顿算子；哈密顿算符；哈密顿量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-07aec2673237abc460d2-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-07aec2673237abc460d2-zh` · translation)；woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-14c43d8a9215edd8df7d-quantum-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §3.3 · `books/woit-qtgr/2025-10-20/zh/chapter-03/03-dynamics-of-a-two-state-system#first-7691a858bfeb3ed42625-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 蕴含(implication)

- UUID: `cea74f88-ca24-4874-89ca-5f0cb65d2c84` · okf_url: https://limbo137.top/okf/concepts/cea74f88-ca24-4874-89ca-5f0cb65d2c84.md
- labels: en = implication；zh-CN = 蕴含
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-99687605a0c32d40fc19-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 子代数(subalgebra)

- UUID: `cf37a212-b028-47fc-9040-136c5bb702d7` · okf_url: https://limbo137.top/okf/concepts/cf37a212-b028-47fc-9040-136c5bb702d7.md
- labels: en = subalgebra；zh-CN = 子代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-subalgebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-fa8a506377952853c56e-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 嵌入(embedding)

- UUID: `cfb63051-fb3e-5619-9cb0-a6f0afccfb9c` · okf_url: https://limbo137.top/okf/concepts/cfb63051-fb3e-5619-9cb0-a6f0afccfb9c.md
- labels: en = embedding；zh-CN = 嵌入
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：嵌入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。本书只要求单射浸入，比通常另要求到像的拓扑嵌入更弱；本次保留该书定义。 · 原文: `If $f$ is an immersion, $f^{*}$ maps $T_{p}M$ isomorphically to an $m$ -dimensional vector subspace of $T_{f(p)}N$ since $\mathrm{rank}f_{*} = \dim M$ . From theorem 2.1, we also find $\ker f_{*} = \{0\}$ . If $f$ is an embedding, $M$ is diffeomorphic to $f(M)$ . Examples will clarify these rather technical points. Consider a map $f: S^1 \to \mathbb{R}^2$...`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：嵌入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。本书只要求单射浸入，比通常另要求到像的拓扑嵌入更弱；本次保留该书定义。 · 原文: `Definition 5.3. (Immersion, submanifold, embedding) Let $f: M \to N$ be a smooth map and let $\dim M \leq \dim N$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.2.7 (pdf_page=207, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.3：嵌入。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。本书只要求单射浸入，比通常另要求到像的拓扑嵌入更弱；本次保留该书定义。 · 原文: `(a) The map $f$ is called an immersion of $M$ into $N$ if $f_{*}: T_{p}M \to T_{f(p)}N$ is an injection (one to one), that is $\operatorname{rank} f_{*} = \dim M$ . (b) The map $f$ is called an **embedding** if $f$ is an injection and an immersion. The image $f(M)$ is called a **submanifold** of $N$ . [In practice, $f(M)$ thus defined is diffeomorphic to ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 复向量空间(complex vector space)

- UUID: `cfc2af7a-6467-42cb-8699-52bf30033870` · okf_url: https://limbo137.top/okf/concepts/cfc2af7a-6467-42cb-8699-52bf30033870.md
- labels: en = complex vector space；zh-CN = 复向量空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-f4a8739c294321ab2ef9-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §1.1 · `books/woit-qtgr/2025-10-20/zh/chapter-01/01-introduction#first-f4a8739c294321ab2ef9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/01-introduction.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 表示的酉性(unitarity of a representation)

- UUID: `d06a0f44-cd99-4902-bac1-36de0284c944` · okf_url: https://limbo137.top/okf/concepts/d06a0f44-cd99-4902-bac1-36de0284c944.md
- labels: en = unitarity of a representation；zh-CN = 表示的酉性
- aliases: unitary
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-5c3c715d0d4f0fddcdab-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/01-the-symplectic-group.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 可观测量(observable)

- UUID: `d07b8252-8aa0-41c2-a5b3-a8bec45db852` · okf_url: https://limbo137.top/okf/concepts/d07b8252-8aa0-41c2-a5b3-a8bec45db852.md
- labels: en = observable；zh-CN = 可观测量
- aliases: observables
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-c5a94a7805820963224f-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-4f6956ccc70e62d5b6f9-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.7 · `books/woit-qtgr/2025-10-20/zh/chapter-04/07-eigenvalues-and-eigenvectors#first-c7877353e297ad512e88-zh` · translation)；woit-qtgr/2025-10-20 (en · §7.2 · `books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field#first-2208104e53ef5c4b2e7d-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 定义域(domain)

- UUID: `d091fdbd-b578-4a81-85b8-b28b54ceaf3d` · okf_url: https://limbo137.top/okf/concepts/d091fdbd-b578-4a81-85b8-b28b54ceaf3d.md
- labels: en = domain；zh-CN = 定义域
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-1a4e8f2ed8c7a5443668-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.6 · `en/chapter-13/06-unbounded-operators#first-1a4e8f2ed8c7a5443668-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/06-unbounded-operators.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## 偏序(partial order)

- UUID: `d092e15b-fc6b-4a6f-9f4b-b3a420ae66fc` · okf_url: https://limbo137.top/okf/concepts/d092e15b-fc6b-4a6f-9f4b-b3a420ae66fc.md
- labels: en = partial order；zh-CN = 偏序
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-3bbd79c9e48440583768-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 导子(derivation)

- UUID: `d15da6db-7ba5-435d-9eec-3740a703c137` · okf_url: https://limbo137.top/okf/concepts/d15da6db-7ba5-435d-9eec-3740a703c137.md
- labels: en = derivation；zh-CN = 导子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-8b5c0417fc94209a34df-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.4 · `zh/chapter-19/04-lie-groups-of-transformations#concept-19-derivation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md

## 选择函数(choice function)

- UUID: `d174ff8b-b83a-45b5-9080-8dfedd238cb1` · okf_url: https://limbo137.top/okf/concepts/d174ff8b-b83a-45b5-9080-8dfedd238cb1.md
- labels: en = choice function；zh-CN = 选择函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.5 · `zh/chapter-01/05-infinite-sets#first-17a5b1f09970c5451069-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/05-infinite-sets.md

## 电场与磁场(Electric and magnetic fields)

- UUID: `d18866dc-065d-461f-b2c0-269b4e51f2a9` · okf_url: https://limbo137.top/okf/concepts/d18866dc-065d-461f-b2c0-269b4e51f2a9.md
- labels: en = Electric and magnetic fields；zh-CN = 电场与磁场
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.2 · `books/woit-qtgr/2025-10-20/zh/chapter-45/02-curvature-electric-and-magnetic-fields#first-e0fc6567bd26a488c8b0-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.2 · `books/woit-qtgr/2025-10-20/en/chapter-45/02-curvature-electric-and-magnetic-fields#first-795b7e65a906a8446ee6-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `电场和磁场是从 $\mathbf R^4$ 到 $\mathbf R^3$ 的两个函数，其分量定义如下： $$ E _ {j} = F _ {j 0} = - \frac {\partial A _ {j}}{\partial t} + \frac {\partial A _ {0}}{\partial x _ {j}} $$ $$ B _ {j} = \frac {1}{2} \epsilon_ {j k l} F _ {k l} = \epsilon_ {j k l} \frac {\partial A _ {l}}{\partial x _ {k}} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/02-curvature-electric-and-magnetic-fields.md

## 运动常量(constant of the motion)

- UUID: `d18ab3a8-acf6-4147-afc2-e59c5c4630dd` · okf_url: https://limbo137.top/okf/concepts/d18ab3a8-acf6-4147-afc2-e59c5c4630dd.md
- labels: en = constant of the motion；zh-CN = 运动常量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-d8af141658dc540dd209-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-d8af141658dc540dd209-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 正规空间(normal space)

- UUID: `d1c09844-e7aa-40eb-9ee7-c1c3ddf68490` · okf_url: https://limbo137.top/okf/concepts/d1c09844-e7aa-40eb-9ee7-c1c3ddf68490.md
- labels: en = normal space；zh-CN = 正规空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-1a67b10220e654865d60-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 实标量量子场(Real scalar quantum field)

- UUID: `d2085240-e280-4ce3-9875-2f28efcca408` · okf_url: https://limbo137.top/okf/concepts/d2085240-e280-4ce3-9875-2f28efcca408.md
- labels: en = Real scalar quantum field；zh-CN = 实标量量子场
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §43.4 · `books/woit-qtgr/2025-10-20/en/chapter-43/04-quantization-of-the-klein-gordon-theory#first-2b2d2a333829d6a2cd32-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §43.4 · `books/woit-qtgr/2025-10-20/zh/chapter-43/04-quantization-of-the-klein-gordon-theory#first-a11b4a19941a57ff0903-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §43.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `实标量量子场算符是由下式定义的算符值分布 $$ \widehat {\Phi} (\mathbf {x}) = \frac {1}{(2 \pi) ^ {3 / 2}} \int_ {\mathbf {R} ^ {3}} (a (\mathbf {p}) e ^ {i \mathbf {p} \cdot \mathbf {x}} + a ^ {\dagger} (\mathbf {p}) e ^ {- i \mathbf {p} \cdot \mathbf {x}}) \frac {d ^ {3} \mathbf {p}}{\sqrt {2 \omega_ {\mathbf {p}}}}\tag{43.19} $$ $$ \widehat {\Pi} (\mathbf {x}) = \frac {1}{(...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-43/04-quantization-of-the-klein-gordon-theory.md

## 坐标映射(coordinate map)

- UUID: `d2192d11-f0f8-4c35-a585-2094b21f2a11` · okf_url: https://limbo137.top/okf/concepts/d2192d11-f0f8-4c35-a585-2094b21f2a11.md
- labels: en = coordinate map；zh-CN = 坐标映射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-coordinate-map` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-7c26c2d4483c46368256-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 轨道角动量(orbital angular momentum)

- UUID: `d24133d7-c651-4a09-95d9-bcd4dfecc852` · okf_url: https://limbo137.top/okf/concepts/d24133d7-c651-4a09-95d9-bcd4dfecc852.md
- labels: en = orbital angular momentum；zh-CN = 轨道角动量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.2 · `en/chapter-14/02-quantum-dynamics#first-c302a6cf2fa6e33ec1df-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.2 · `zh/chapter-14/02-quantum-dynamics#first-c302a6cf2fa6e33ec1df-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §34.2 · `books/woit-qtgr/2025-10-20/zh/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-1d0bdb6df591acff7619-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §34.2 · `books/woit-qtgr/2025-10-20/en/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-1d0bdb6df591acff7619-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/02-quantum-dynamics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3.md

## 缩并比安基恒等式(contracted Bianchi identities)

- UUID: `d248e3d5-17b4-4ea4-bd69-c4feffdde743` · okf_url: https://limbo137.top/okf/concepts/d248e3d5-17b4-4ea4-bd69-c4feffdde743.md
- labels: en = contracted Bianchi identities；zh-CN = 缩并比安基恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.6 · `zh/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-c4569c7088780e977aa6-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.6 · `en/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-c4569c7088780e977aa6-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/06-the-riemann-tensor-and-its-symmetries.md

## 自旋向上态(spin-up state)

- UUID: `d2a66ef7-e632-4029-9ec0-07c29ee32e68` · okf_url: https://limbo137.top/okf/concepts/d2a66ef7-e632-4029-9ec0-07c29ee32e68.md
- labels: en = spin-up state；zh-CN = 自旋向上态
- aliases: “spin-up” state
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-098f0fe3af0cd62f4cc9-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 泊松括号(Poisson bracket)

- UUID: `d2b35bcc-bfd9-4e7f-b308-727751f98a08` · okf_url: https://limbo137.top/okf/concepts/d2b35bcc-bfd9-4e7f-b308-727751f98a08.md
- labels: en = Poisson bracket；zh-CN = 泊松括号
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §14.1 · `books/woit-qtgr/2025-10-20/en/chapter-14/01-classical-mechanics-and-the-poisson-bracket#first-dbd69952501b17cbecfe-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §14.1 · `books/woit-qtgr/2025-10-20/zh/chapter-14/01-classical-mechanics-and-the-poisson-bracket#first-0238fe1263ca5c027cd7-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §14.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `相空间 $M=\mathbf R^2$（坐标为 $(q,p)$）上的函数之间有一个称为泊松括号的双线性运算，定义为 $$ (f_1,f_2)\mapsto\{f_1,f_2\}=\frac{\partial f_1}{\partial q}\frac{\partial f_2}{\partial p}-\frac{\partial f_1}{\partial p}\frac{\partial f_2}{\partial q} $$`
  - woit-qtgr/2025-10-20 §14.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The functions on phase space $M=\mathbf R^2$, with coordinates $(q,p)$, have a bilinear operation called the Poisson bracket, defined by $$ (f_1,f_2)\mapsto\{f_1,f_2\}=\frac{\partial f_1}{\partial q}\frac{\partial f_2}{\partial p}-\frac{\partial f_1}{\partial p}\frac{\partial f_2}{\partial q}. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-14/01-classical-mechanics-and-the-poisson-bracket.md

## 不连通空间(disconnected space)

- UUID: `d2b497ed-d240-4e7e-953e-13baf5792584` · okf_url: https://limbo137.top/okf/concepts/d2b497ed-d240-4e7e-953e-13baf5792584.md
- labels: en = disconnected space；zh-CN = 不连通空间
- aliases: disconnected；disconnectedness；不连通；不连通的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.7 · `zh/chapter-10/07-connected-spaces#first-ab48efad4570cfb446cd-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书连通性定义省略了非空条件，来源缺陷保留；不因此改写原文。 · 原文: `Definition 2.8. (a) A topological space $X$ is connected if it cannot be written as $X = X_{1} \cup X_{2}$ , where $X_{1}$ and $X_{2}$ are both open and $X_{1} \cap X_{2} = \emptyset$ . Otherwise $X$ is called disconnected.`
  - szekeres/source-snapshot-bc3404156f6e §10.7 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书逐字条件保留；不同书路径分别记录。 · 原文: `Intuitively, we can think of a topological space $X$ as being ‘disconnected’ if it can be decomposed into two disjoint subsets $X = A \cup B$ without these sets having any boundary points in common. Since the boundary of a set is at the same time the boundary of the complement of that set, the only way such a decomposition can occur is if there exists a s...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/07-connected-spaces.md

## 覆盖映射(covering map)

- UUID: `d3626907-41e1-4ef2-bc68-2d2efdec0b42` · okf_url: https://limbo137.top/okf/concepts/d3626907-41e1-4ef2-bc68-2d2efdec0b42.md
- labels: en = covering map；zh-CN = 覆盖映射
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-cover` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-cover` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md

## 辛形式(Symplectic form)

- UUID: `d48ed7ff-3ca1-404d-9e8e-9c302f532e74` · okf_url: https://limbo137.top/okf/concepts/d48ed7ff-3ca1-404d-9e8e-9c302f532e74.md
- labels: en = Symplectic form；zh-CN = 辛形式
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §14.3 · `books/woit-qtgr/2025-10-20/zh/chapter-14/03-symplectic-geometry#first-1128d1d3771008be8bd2-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §14.3 · `books/woit-qtgr/2025-10-20/en/chapter-14/03-symplectic-geometry#first-0235925bab4334db2e44-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §14.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `A symplectic form $\omega$ on a vector space $V$ is a bilinear map $$ \omega:V\times V\to\mathbf R $$ such that - $\omega$ is antisymmetric: $\omega(v,v^{\prime})=-\omega(v^{\prime},v)$. - $\omega$ is nondegenerate: if $v\ne0$, then $\omega(v,\cdot)\in V^*$ is nonzero.A vector space $V$ with a symplectic form $\omega$ is called a symplectic vector space. ...`
  - woit-qtgr/2025-10-20 §14.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `向量空间 $V$ 上的辛形式 $\omega$ 是一个双线性映射 $$ \omega:V\times V\to\mathbf R $$ 并满足以下条件： - $\omega$ 反对称，即 $\omega(v,v^{\prime})=-\omega(v^{\prime},v)$。 - $\omega$ 非退化，即若 $v\ne0$，则 $\omega(v,\cdot)\in V^*$ 非零。带有辛形式 $\omega$ 的向量空间 $V$ 称为辛向量空间。以辛形式取代内积而得到的几何称为辛几何。这里没有长度的概念，因为反对称性意味着 $\omega(v,v)=0$。与正交群相对应的群称为辛群，由保持 $\omega$ 的线性变换组成；我们将在[第 16 章](../chapter-16/index#sect...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-14/03-symplectic-geometry.md

## 反常(anomaly)

- UUID: `d52824fa-00d0-496c-84fb-ac2c9c371196` · okf_url: https://limbo137.top/okf/concepts/d52824fa-00d0-496c-84fb-ac2c9c371196.md
- labels: en = anomaly；zh-CN = 反常
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §39.3 · `books/woit-qtgr/2025-10-20/zh/chapter-39/03-the-anomaly-and-the-schwinger-term#first-b8882cd26c0201b88129-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §15.3 · `books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map#first-b8882cd26c0201b88129-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/03-the-anomaly-and-the-schwinger-term.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-15/03-group-actions-on-m-and-the-moment-map.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions.md

## 塞曼效应(Zeeman effect)

- UUID: `d550361c-735c-491e-afba-556b6f658c73` · okf_url: https://limbo137.top/okf/concepts/d550361c-735c-491e-afba-556b6f658c73.md
- labels: en = Zeeman effect；zh-CN = 塞曼效应
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §3.3 · `books/woit-qtgr/2025-10-20/en/chapter-03/03-dynamics-of-a-two-state-system#first-feb78058b2228986f547-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §3.3 · `books/woit-qtgr/2025-10-20/zh/chapter-03/03-dynamics-of-a-two-state-system#first-feb78058b2228986f547-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/03-dynamics-of-a-two-state-system.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/02-the-spin-half-particle-in-a-magnetic-field.md

## 朗道能级(Landau levels)

- UUID: `d5c525c9-354b-4589-a5de-06daf0962433` · okf_url: https://limbo137.top/okf/concepts/d5c525c9-354b-4589-a5de-06daf0962433.md
- labels: en = Landau levels；zh-CN = 朗道能级
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §33.1 · `books/woit-qtgr/2025-10-20/zh/chapter-33/01-the-supersymmetric-oscillator#first-a9c45d75e188066931c8-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §33.1 · `books/woit-qtgr/2025-10-20/en/chapter-33/01-the-supersymmetric-oscillator#first-a9c45d75e188066931c8-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-33/01-the-supersymmetric-oscillator.md

## 有限维自伴算符(finite-dimensional self-adjoint operator)

- UUID: `d5f9da17-e94f-479d-90f0-7732d12ee03a` · okf_url: https://limbo137.top/okf/concepts/d5f9da17-e94f-479d-90f0-7732d12ee03a.md
- labels: en = finite-dimensional self-adjoint operator；zh-CN = 有限维自伴算符
- aliases: self-adjoint；self-adjoint operator；自伴算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.4 · `books/woit-qtgr/2025-10-20/zh/chapter-01/04-representations-and-quantum-mechanics#concept-1-self-adjoint-operator` · translation)；woit-qtgr/2025-10-20 (en · §4.5 · `books/woit-qtgr/2025-10-20/en/chapter-04/05-adjoint-operators#first-6c95bef6a9cf492f405a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/05-adjoint-operators.md

## 相容的(compatible)

- UUID: `d61c6e7f-6863-4448-9bb6-4cd1788de624` · okf_url: https://limbo137.top/okf/concepts/d61c6e7f-6863-4448-9bb6-4cd1788de624.md
- labels: en = compatible；zh-CN = 相容的
- aliases: 相容
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-c1ebbf8f7eb6b339461a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 模(module)

- UUID: `d693259b-a955-4659-a88a-5d8e134f2d48` · okf_url: https://limbo137.top/okf/concepts/d693259b-a955-4659-a88a-5d8e134f2d48.md
- labels: en = module；zh-CN = 模
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-module` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 李代数的伴随表示(adjoint Lie algebra representation)

- UUID: `d6add957-677f-4bff-be23-2cb48b7bbaaa` · okf_url: https://limbo137.top/okf/concepts/d6add957-677f-4bff-be23-2cb48b7bbaaa.md
- labels: en = adjoint Lie algebra representation；zh-CN = 李代数的伴随表示
- aliases: Adjoint Lie algebra representation；adjoint representation；伴随李代数表示；伴随表示
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-3ad400dd97590b4ed9ed-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.4 · `zh/chapter-19/04-lie-groups-of-transformations#concept-19-adjoint-representation` · translation)；woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-14b8c474c1d7774b18db-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-79658104cde2957b97a8-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `$(\operatorname{ad},\mathfrak{g})$ 是由下式给出的 伴随李代数表示(adjoint Lie algebra representation)： $$ X\in\mathfrak{g}\mapsto\operatorname{ad}(X) $$ 其中，$\operatorname{ad}(X)$ 是从 $\mathfrak{g}$ 到自身的线性映射，定义为： $$ Y\mapsto[X,Y] $$`
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `$(\operatorname{ad},\mathfrak{g})$ is the adjoint Lie algebra representation given by $$ X\in\mathfrak{g}\mapsto\operatorname{ad}(X) $$ where $\operatorname{ad}(X)$ is the linear map from $\mathfrak{g}$ to itself given by $$ Y\mapsto[X,Y] $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 对偶基(dual basis)

- UUID: `d6eb757d-a21e-4186-8ccf-544d77c8e86d` · okf_url: https://limbo137.top/okf/concepts/d6eb757d-a21e-4186-8ccf-544d77c8e86d.md
- labels: en = dual basis；zh-CN = 对偶基
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.7 · `zh/chapter-03/07-dual-spaces#concept-3-dual-basis` · translation)；woit-qtgr/2025-10-20 (en · §4.2 · `books/woit-qtgr/2025-10-20/en/chapter-04/02-dual-vector-spaces#concept-4-dual-basis` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.2 · `books/woit-qtgr/2025-10-20/zh/chapter-04/02-dual-vector-spaces#first-6323307a7af86f0baa50-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/07-dual-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/02-dual-vector-spaces.md

## 左矢-右矢记号(bra-ket notation)

- UUID: `d6f7741c-ea42-487e-baff-0067243d1990` · okf_url: https://limbo137.top/okf/concepts/d6f7741c-ea42-487e-baff-0067243d1990.md
- labels: en = bra-ket notation；zh-CN = 左矢-右矢记号
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-58f23113063eefbc0b67-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-58f23113063eefbc0b67-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md

## 最大对称性(maximal symmetry)

- UUID: `d74af6bb-82d3-4bbc-a595-a5fc7bbc03ad` · okf_url: https://limbo137.top/okf/concepts/d74af6bb-82d3-4bbc-a595-a5fc7bbc03ad.md
- labels: en = maximal symmetry；zh-CN = 最大对称性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-09882c6c3979faafc464-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-maximal-symmetry` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 最高权定理(Highest weight theorem)

- UUID: `d814193b-cdfb-47f7-bda5-f218747ec2c0` · okf_url: https://limbo137.top/okf/concepts/d814193b-cdfb-47f7-bda5-f218747ec2c0.md
- labels: en = Highest weight theorem；zh-CN = 最高权定理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §8.1 · `books/woit-qtgr/2025-10-20/en/chapter-08/01-representations-of-su2-classification#first-7baa12c03a6dc90dbc68-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §8.1 · `books/woit-qtgr/2025-10-20/zh/chapter-08/01-representations-of-su2-classification#first-6167d3eb963b2666366e-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/01-representations-of-su2-classification.md

## 2-形式(2-form)

- UUID: `d81e5dd1-db46-57ab-b098-41b206c079b3` · okf_url: https://limbo137.top/okf/concepts/d81e5dd1-db46-57ab-b098-41b206c079b3.md
- labels: en = 2-form；zh-CN = 2-形式
- aliases: 2形式；二形式；微分2-形式
- 分书定义状态: david-tong-general-relativity/michaelmas-2019-source-20260923 = not-recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 二元关系(binary relation)

- UUID: `d844c220-b581-45c0-9c00-86dc803161ba` · okf_url: https://limbo137.top/okf/concepts/d844c220-b581-45c0-9c00-86dc803161ba.md
- labels: en = binary relation；zh-CN = 二元关系
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-6163fe0c73c0f1c37023-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §1.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书n元关系与二元关系不同；covector与linear functional明确同义。 · 原文: `Any subset of $S ^ { n }$ is called an $n$-ary relation on a set $S$. For example, $$ \text { unary relation } \equiv 1 \text {-ary relation } = \text { subset of } S $$ $$ \text { binary relation } \equiv 2 \text {-ary relation } = \text { subset of } S ^ {2} = S \times S $$ $$ \text { ternary relation } \equiv 3 \text {-ary relation } = \text { subset o...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 路径积分(path integral)

- UUID: `d8842a51-427e-436c-8476-372fb5a37bfd` · okf_url: https://limbo137.top/okf/concepts/d8842a51-427e-436c-8476-372fb5a37bfd.md
- labels: en = path integral；zh-CN = 路径积分
- 分书定义状态: schwartz-qft/local-experiment = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.3 · `books/woit-qtgr/2025-10-20/zh/chapter-35/03-quantization-and-path-integrals#first-ee1f874ac6780dc7b095-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.3 · `books/woit-qtgr/2025-10-20/en/chapter-35/03-quantization-and-path-integrals#first-ee1f874ac6780dc7b095-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): schwartz-qft/local-experiment (2 条, completeness=recorded-only) · 摘录: `Using Eq. (14.12), each term in Eq. (14.10) becomes just a number and the product reduces to`；schwartz-qft/local-experiment (1 条, completeness=recorded-only) · 摘录: `Now we can use the Gaussian integral in Eq. (14.6), $\int dp\exp \left(-\frac{1}{2} ap^2 +Jp\right) = \sqrt{\frac{2\pi}{a}}\exp \left(\frac{J^2}{2a}\right)$ , w...`
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/03-quantization-and-path-integrals.md

## 边界(boundary)

- UUID: `d89858ae-0ae4-44a6-b482-787218d5e179` · okf_url: https://limbo137.top/okf/concepts/d89858ae-0ae4-44a6-b482-787218d5e179.md
- labels: en = boundary；zh-CN = 边界
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-ccd810e4a97cf523defc-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=librarian-source-reviewed; editorial-sense-evidence, dependency_status=not-recorded) · 原文: `The closure of any set $A$, denoted $\overline{A}$ , is the union of the set $A$ and all its accumulation points. The interior of $A$ is the union of all open sets $U \subseteq A ,$ , denoted $A ^ { o }$ . The difference of these two sets, $b ( A ) = { \overline { { A } } } - A ^ { o }$ , is called the boundary of $A$.`
  - szekeres/source-snapshot-bc3404156f6e §10.2 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 点集边界为闭包减内部；不等同于链群的边缘或一般流形的内在边界。 · 原文: `The closure of any set $A$, denoted $\overline{A}$ , is the union of the set $A$ and all its accumulation points. The interior of $A$ is the union of all open sets $U \subseteq A ,$ , denoted $A ^ { o }$ . The difference of these two sets, $b ( A ) = { \overline { { A } } } - A ^ { o }$ , is called the boundary of $A$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## Wick定理(Wick’s theorem)

- UUID: `d898fb1e-b187-459d-b20b-b6ab554b8983` · okf_url: https://limbo137.top/okf/concepts/d898fb1e-b187-459d-b20b-b6ab554b8983.md
- labels: en = Wick’s theorem；zh-CN = Wick定理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §37.3 · `books/woit-qtgr/2025-10-20/zh/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-915860739d788cbd991f-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §37.3 · `books/woit-qtgr/2025-10-20/en/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory#first-915860739d788cbd991f-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-37/03-the-propagator-in-non-relativistic-quantum-field-theory.md

## 置换群(permutation group)

- UUID: `d8b640a2-2675-4f90-bc60-7149b8de3db9` · okf_url: https://limbo137.top/okf/concepts/d8b640a2-2675-4f90-bc60-7149b8de3db9.md
- labels: en = permutation group；zh-CN = 置换群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-permutation-group` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-b147d601c68ff7c731a3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 对称变换(symmetry transformation)

- UUID: `d92b0658-0c85-4297-ade9-05db9fb1b0b3` · okf_url: https://limbo137.top/okf/concepts/d92b0658-0c85-4297-ade9-05db9fb1b0b3.md
- labels: en = symmetry transformation；zh-CN = 对称变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.3 · `zh/chapter-14/03-symmetry-transformations#first-5a807e70c0d085a3125a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.3 · `en/chapter-14/03-symmetry-transformations#first-5a807e70c0d085a3125a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/03-symmetry-transformations.md

## 经验温度(empirical temperature)

- UUID: `d95e9522-e5e3-4131-8d6e-ee0f73b8f273` · okf_url: https://limbo137.top/okf/concepts/d95e9522-e5e3-4131-8d6e-ee0f73b8f273.md
- labels: en = empirical temperature；zh-CN = 经验温度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-a7a2a819846115b20f5c-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-a7a2a819846115b20f5c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 协变相空间(covariant phase space)

- UUID: `d9610d6a-94c7-4d15-9ecf-b32655776b7a` · okf_url: https://limbo137.top/okf/concepts/d9610d6a-94c7-4d15-9ecf-b32655776b7a.md
- labels: en = covariant phase space；zh-CN = 协变相空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-98755a4c1bd6f23001aa-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-98755a4c1bd6f23001aa-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 乘法算符(multiplication operator)

- UUID: `d9cfc0e8-4d07-4144-9b86-a6fa7b076397` · okf_url: https://limbo137.top/okf/concepts/d9cfc0e8-4d07-4144-9b86-a6fa7b076397.md
- labels: en = multiplication operator；zh-CN = 乘法算符
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §13.4 · `zh/chapter-13/04-bounded-linear-operators#concept-13-multiplication-operator` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.4 · `en/chapter-13/04-bounded-linear-operators#first-bf78e829304ec69ff626-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/04-bounded-linear-operators.md

## 相对论光行差(relativistic aberration of light)

- UUID: `d9ea1a64-7c37-476f-aea3-ad513b8349d0` · okf_url: https://limbo137.top/okf/concepts/d9ea1a64-7c37-476f-aea3-ad513b8349d0.md
- labels: en = relativistic aberration of light；zh-CN = 相对论光行差
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.2 · `zh/chapter-09/02-relativistic-kinematics#first-c77836cbbc6e50f73949-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.2 · `en/chapter-09/02-relativistic-kinematics#first-c77836cbbc6e50f73949-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/02-relativistic-kinematics.md

## 矢势(vector potential)

- UUID: `da09116c-bed9-4a1f-b9ff-b0d8d5448cd0` · okf_url: https://limbo137.top/okf/concepts/da09116c-bed9-4a1f-b9ff-b0d8d5448cd0.md
- labels: en = vector potential；zh-CN = 矢势
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-6d8ed988f8999f8d8eae-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-6d8ed988f8999f8d8eae-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 正交矩阵(orthogonal matrix)

- UUID: `da263550-52cd-4a77-a46e-6941bd680d63` · okf_url: https://limbo137.top/okf/concepts/da263550-52cd-4a77-a46e-6941bd680d63.md
- labels: en = orthogonal matrix；zh-CN = 正交矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-orthogonal-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 爱因斯坦引力常数(Einstein’s gravitational constant)

- UUID: `da39fb35-44ff-46b2-8690-4deb1759352d` · okf_url: https://limbo137.top/okf/concepts/da39fb35-44ff-46b2-8690-4deb1759352d.md
- labels: en = Einstein’s gravitational constant；zh-CN = 爱因斯坦引力常数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.8 · `en/chapter-18/08-general-relativity#first-459d51d23d2eb2abbf92-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.8 · `zh/chapter-18/08-general-relativity#first-459d51d23d2eb2abbf92-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/08-general-relativity.md

## 符号(sign)

- UUID: `da4ce753-9ccb-45b5-8261-ebb7a2d228c4` · okf_url: https://limbo137.top/okf/concepts/da4ce753-9ccb-45b5-8261-ebb7a2d228c4.md
- labels: en = sign；zh-CN = 符号
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.2 · `zh/chapter-02/02-transformation-and-permutation-groups#concept-2-sign` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/02-transformation-and-permutation-groups.md

## 基灵方程(Killing’s equations)

- UUID: `da89c755-49c5-48dc-80ea-428c47c11355` · okf_url: https://limbo137.top/okf/concepts/da89c755-49c5-48dc-80ea-428c47c11355.md
- labels: en = Killing’s equations；zh-CN = 基灵方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-fb42322d85fccde270d3-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-killing-s-equations` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 量值(magnitude)

- UUID: `dab77b33-384c-47d5-b9d2-1649b126f195` · okf_url: https://limbo137.top/okf/concepts/dab77b33-384c-47d5-b9d2-1649b126f195.md
- labels: en = magnitude；zh-CN = 量值
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-magnitude` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 有限维标准正交基(finite-dimensional orthonormal basis)

- UUID: `dabe32db-8d84-4c17-9eeb-583f0609a265` · okf_url: https://limbo137.top/okf/concepts/dabe32db-8d84-4c17-9eeb-583f0609a265.md
- labels: en = finite-dimensional orthonormal basis；zh-CN = 有限维标准正交基
- aliases: orthonormal bases；orthonormal basis；标准正交基；正交归一基；正交标准基
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-orthonormal-basis` · translation)；woit-qtgr/2025-10-20 (en · §7.5 · `books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere#first-e134353abe687ae26a43-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-e134353abe687ae26a43-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/07-eigenvalues-and-eigenvectors.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 稳定焦点(stable focus)

- UUID: `dac5e65a-561e-483e-a1a3-0d1cd0217757` · okf_url: https://limbo137.top/okf/concepts/dac5e65a-561e-483e-a1a3-0d1cd0217757.md
- labels: en = stable focus；zh-CN = 稳定焦点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-stable-focus` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 单态射(monomorphism)

- UUID: `db9228a3-e5a9-49b1-b8d6-6fe8691929c2` · okf_url: https://limbo137.top/okf/concepts/db9228a3-e5a9-49b1-b8d6-6fe8691929c2.md
- labels: en = monomorphism；zh-CN = 单态射
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.7 · `zh/chapter-01/07-category-theory#first-d867cbce135267082c1b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/07-category-theory.md

## 第一同构定理(First isomorphism theorem)

- UUID: `dbf8f503-ae98-47ba-af2c-59f26dab3447` · okf_url: https://limbo137.top/okf/concepts/dbf8f503-ae98-47ba-af2c-59f26dab3447.md
- labels: en = First isomorphism theorem；zh-CN = 第一同构定理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §2.5 · `en/chapter-02/05-normal-subgroups-and-factor-groups#first-e1621ec9c46a4701ed44-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#first-e1621ec9c46a4701ed44-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 真空中的麦克斯韦方程组(Maxwell’s equations in vacuo)

- UUID: `dcc39596-780c-4cca-bb89-5cb32df909b8` · okf_url: https://limbo137.top/okf/concepts/dcc39596-780c-4cca-bb89-5cb32df909b8.md
- labels: en = Maxwell’s equations in vacuo；zh-CN = 真空中的麦克斯韦方程组
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §46.1 · `books/woit-qtgr/2025-10-20/en/chapter-46/01-maxwell-s-equations#first-fc3f45008df8c45c0990-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §46.1 · `books/woit-qtgr/2025-10-20/zh/chapter-46/01-maxwell-s-equations#first-ec625250a6dd81771930-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §46.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `真空中电磁场的麦克斯韦方程组为 $$ \nabla \cdot \mathbf {B} = 0\tag{46.1} $$ $$ \nabla \times \mathbf {E} = - \frac {\partial \mathbf {B}}{\partial t}\tag{46.2} $$ $$ \nabla \times \mathbf {B} = \frac {\partial \mathbf {E}}{\partial t}\tag{46.3} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-46/01-maxwell-s-equations.md

## 豪斯多夫空间(Hausdorff space)

- UUID: `dd2edcc6-b7f4-409f-bb17-9ef6212acc44` · okf_url: https://limbo137.top/okf/concepts/dd2edcc6-b7f4-409f-bb17-9ef6212acc44.md
- labels: en = Hausdorff space；zh-CN = 豪斯多夫空间
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.5 · `zh/chapter-10/05-hausdorff-spaces#first-1e9eb98fd11d8e6da435-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.3 (pdf_page=102, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.6：豪斯多夫空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.6. A topological space $(X, \mathcal{T})$ is a Hausdorff space if, for an arbitrary pair of distinct points $x, x' \in X$ , there always exist neighbourhoods $U_x$ of $x$ and $U_{x'}$ of $x'$ such that $U_x \cap U_{x'} = \emptyset$ .`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/05-hausdorff-spaces.md

## 射线(ray)

- UUID: `dd5a80e1-a5d1-416d-a7bc-4bf14e593eb8` · okf_url: https://limbo137.top/okf/concepts/dd5a80e1-a5d1-416d-a7bc-4bf14e593eb8.md
- labels: en = ray；zh-CN = 射线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-909421c7b769a112c204-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-909421c7b769a112c204-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 李代数(Lie algebra)

- UUID: `dd6db1c0-26e6-4777-a6c3-84c28f8fd354` · okf_url: https://limbo137.top/okf/concepts/dd6db1c0-26e6-4777-a6c3-84c28f8fd354.md
- labels: en = Lie algebra；zh-CN = 李代数
- aliases: Lie algebras
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-lie-algebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-d8620fac059d3f8c8aa4-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §2.3 · `books/woit-qtgr/2025-10-20/zh/chapter-02/03-the-charge-operator#first-0f378a6753637eff06e0-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.1 · `books/woit-qtgr/2025-10-20/en/chapter-05/01-lie-algebras#first-0f378a6753637eff06e0-en` · original_transcription)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=230, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.11：李代数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.11. The set of left-invariant vector fields $\mathfrak{g}$ with the Lie bracket $[\quad ,\quad ]:\mathfrak{g}\times \mathfrak{g}\to \mathfrak{g}$ is called the Lie algebra of a Lie group $G$`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=230, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.11：李代数。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `We denote the Lie algebra of a Lie group by the corresponding lower-case German gothic letter. For example $\mathfrak{so}(n)$ is the Lie algebra of $\mathrm{SO}(n)$ .`
  - woit-qtgr/2025-10-20 §5.1 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Lie algebra). For $G$ a Lie group of $n$ by $n$ invertible matrices, the Lie algebra of $G$ (written $Lie(G)$ or $\mathfrak{g}$) is the space of $n$ by $n$ matrices $X$ such that $e^{tX}\in G$ for $t\in\mathbf{R}$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/01-the-rotation-group-in-three-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/07-a-summary.md

## 有界集(bounded set)

- UUID: `de104b91-ca80-4552-9289-c6856691399d` · okf_url: https://limbo137.top/okf/concepts/de104b91-ca80-4552-9289-c6856691399d.md
- labels: en = bounded set；zh-CN = 有界集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.6 · `zh/chapter-10/06-compact-spaces#first-3da3cf9260546d2d08f0-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/06-compact-spaces.md

## 闵可夫斯基型流形(Minkowskian manifold)

- UUID: `de67a981-9d8b-4a60-9553-3d19f3a820b8` · okf_url: https://limbo137.top/okf/concepts/de67a981-9d8b-4a60-9553-3d19f3a820b8.md
- labels: en = Minkowskian manifold；zh-CN = 闵可夫斯基型流形
- aliases: Minkowskian
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-aff4d10862512e91824a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-20d63bae569b760a93d1-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 零锥(null cone)

- UUID: `de88ac36-e07b-4d19-99c2-7a8b00ff4d65` · okf_url: https://limbo137.top/okf/concepts/de88ac36-e07b-4d19-99c2-7a8b00ff4d65.md
- labels: en = null cone；zh-CN = 零锥
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-6e7f24d809c91440033c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-6e7f24d809c91440033c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 共轭作用(conjugation action)

- UUID: `de96cc1a-7377-4476-ac03-e569f8b5166f` · okf_url: https://limbo137.top/okf/concepts/de96cc1a-7377-4476-ac03-e569f8b5166f.md
- labels: en = conjugation action；zh-CN = 共轭作用
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-a33e38fa15d66d0eb209-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-a33e38fa15d66d0eb209-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 非交换群(non-commutative group)

- UUID: `def3eb05-8eb2-451d-b5fb-2d8076b1e56a` · okf_url: https://limbo137.top/okf/concepts/def3eb05-8eb2-451d-b5fb-2d8076b1e56a.md
- labels: en = non-commutative group；zh-CN = 非交换群
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-48371028521ae40800e1-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 加细(refinement)

- UUID: `defda17a-18a4-441f-bd21-63aeaa375b3f` · okf_url: https://limbo137.top/okf/concepts/defda17a-18a4-441f-bd21-63aeaa375b3f.md
- labels: en = refinement；zh-CN = 加细
- aliases: 精细化
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.1 · `en/chapter-17/01-partitions-of-unity#first-d9fa818df9de9f675c8b-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.1 · `zh/chapter-17/01-partitions-of-unity#first-d9fa818df9de9f675c8b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/01-partitions-of-unity.md

## 多重向量(multivector)

- UUID: `df7c053b-3303-442b-81a3-00ae51a17e3f` · okf_url: https://limbo137.top/okf/concepts/df7c053b-3303-442b-81a3-00ae51a17e3f.md
- labels: en = multivector；zh-CN = 多重向量
- aliases: multivectors
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-multivector` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-cbf08a636acb8a33f607-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md

## 更弱的拓扑(weaker topology)

- UUID: `e1980619-7994-4c07-9945-6a75719f35d0` · okf_url: https://limbo137.top/okf/concepts/e1980619-7994-4c07-9945-6a75719f35d0.md
- labels: en = weaker topology；zh-CN = 更弱的拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.2 · `zh/chapter-10/02-general-topological-spaces#first-ecae0598f69530a51154-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/02-general-topological-spaces.md

## 域上的代数(algebra over a field)

- UUID: `e1f26ebb-28aa-46e5-8109-2a53560d9964` · okf_url: https://limbo137.top/okf/concepts/e1f26ebb-28aa-46e5-8109-2a53560d9964.md
- labels: en = algebra over a field；zh-CN = 域上的代数
- aliases: algebra；代数；域上的代数（不要求结合）
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-4f7dddab87124441fbb0-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-algebra` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 伪黎曼流形(pseudo-Riemannian manifold)

- UUID: `e262a216-77fe-4a42-9099-b3672c04584d` · okf_url: https://limbo137.top/okf/concepts/e262a216-77fe-4a42-9099-b3672c04584d.md
- labels: en = pseudo-Riemannian manifold；zh-CN = 伪黎曼流形
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-ee5625c59c4d344be166-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-ee5625c59c4d344be166-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 联络或矢势(Connection or vector potential)

- UUID: `e295e151-c48e-4e8d-87a8-9c298c638541` · okf_url: https://limbo137.top/okf/concepts/e295e151-c48e-4e8d-87a8-9c298c638541.md
- labels: en = Connection or vector potential；zh-CN = 联络或矢势
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §45.1 · `books/woit-qtgr/2025-10-20/zh/chapter-45/01-u-1-gauge-symmetry#first-d932caa3854c72b9d75d-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §45.1 · `books/woit-qtgr/2025-10-20/en/chapter-45/01-u-1-gauge-symmetry#first-a7b468878b28cce84c9c-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §45.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `$A$ $U ( 1 )$ 联络（数学家的术语）或矢势（物理学家的术语）是时空 $M ^ { 4 }$ 上的一个函数 $A$，取值于 $\mathbf { R } ^ { 4 }$，其分量记为 $$ A _ {\mu} (x) = (A _ {0} (t, \mathbf {x}), \mathbf {A} (t, \mathbf {x})) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-45/01-u-1-gauge-symmetry.md

## 酉矩阵(unitary matrix)

- UUID: `e29afcc7-47f4-48fe-8dfe-d51873f245ba` · okf_url: https://limbo137.top/okf/concepts/e29afcc7-47f4-48fe-8dfe-d51873f245ba.md
- labels: en = unitary matrix；zh-CN = 酉矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-unitary-matrix` · translation)；woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-c96fb0ff3027ce68ac5f-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## “自旋向下”态(spin-down state)

- UUID: `e3156275-3f46-409b-b74c-452208f4fdcb` · okf_url: https://limbo137.top/okf/concepts/e3156275-3f46-409b-b74c-452208f4fdcb.md
- labels: en = spin-down state；zh-CN = “自旋向下”态
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.5 · `books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere#first-d0ebd8ed9f05e242417f-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 幂零矩阵(nilpotent matrix)

- UUID: `e332b565-6ba9-49f9-939f-cb84097991ee` · okf_url: https://limbo137.top/okf/concepts/e332b565-6ba9-49f9-939f-cb84097991ee.md
- labels: en = nilpotent matrix；zh-CN = 幂零矩阵
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.2 · `zh/chapter-04/02-jordan-canonical-form#concept-4-nilpotent-matrix` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/02-jordan-canonical-form.md

## 周期(period)

- UUID: `e372d315-1f44-4cdf-9f85-d01e74317539` · okf_url: https://limbo137.top/okf/concepts/e372d315-1f44-4cdf-9f85-d01e74317539.md
- labels: en = period；zh-CN = 周期
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-2671483f7260b6048199-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-2671483f7260b6048199-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 弧连通空间(arcwise connected space)

- UUID: `e3a3dd64-f1f6-4a49-a701-6c3be5b2c642` · okf_url: https://limbo137.top/okf/concepts/e3a3dd64-f1f6-4a49-a701-6c3be5b2c642.md
- labels: en = arcwise connected space；zh-CN = 弧连通空间
- aliases: arcwise connected；弧连通的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：弧连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(b) A topological space $X$ is called arcwise connected if, for any points $x, y \in X$ , there exists a continuous map $f: [0,1] \to X$ such that $f(0) = x$ and $f(1) = y$ . With a few pathological exceptions, arcwise connectedness is practically equivalent to connectedness. (c) A loop in a topological space $X$ is a continuous map $f: [0,1] \to X$ such ...`
  - nakahara-geometry-topology-physics/second-edition-local §2.3.6 (pdf_page=104, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.8：弧连通空间。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.8. (a) A topological space $X$ is connected if it cannot be written as $X = X_{1} \cup X_{2}$ , where $X_{1}$ and $X_{2}$ are both open and $X_{1} \cap X_{2} = \emptyset$ . Otherwise $X$ is called disconnected.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 第二比安基恒等式(second Bianchi identity)

- UUID: `e3d7ac93-909f-4ff5-9cb6-539fd4533e36` · okf_url: https://limbo137.top/okf/concepts/e3d7ac93-909f-4ff5-9cb6-539fd4533e36.md
- labels: en = second Bianchi identity；zh-CN = 第二比安基恒等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.6 · `zh/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-405e4f80c153b2e1a5dd-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.6 · `en/chapter-18/06-the-riemann-tensor-and-its-symmetries#first-405e4f80c153b2e1a5dd-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/06-the-riemann-tensor-and-its-symmetries.md

## 博雷尔集(Borel set)

- UUID: `e41d2f97-2b81-41a5-a89e-a1df482c0028` · okf_url: https://limbo137.top/okf/concepts/e41d2f97-2b81-41a5-a89e-a1df482c0028.md
- labels: en = Borel set；zh-CN = 博雷尔集
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.1 · `zh/chapter-11/01-measurable-spaces-and-functions#first-cdc12b6ea25268c0b522-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/01-measurable-spaces-and-functions.md

## 共轭四元数(conjugate quaternion)

- UUID: `e43e5270-db6d-4d5d-91f9-3df3742d021b` · okf_url: https://limbo137.top/okf/concepts/e43e5270-db6d-4d5d-91f9-3df3742d021b.md
- labels: en = conjugate quaternion；zh-CN = 共轭四元数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-0aca24809dbc9d12a1bb-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-conjugate-quaternion` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md

## 模子群的等价(equivalence modulo a subgroup)

- UUID: `e4bb126e-1744-448e-9d08-82950ecb4c77` · okf_url: https://limbo137.top/okf/concepts/e4bb126e-1744-448e-9d08-82950ecb4c77.md
- labels: en = equivalence modulo a subgroup；zh-CN = 模子群的等价
- aliases: $x \sim y$；equivalent
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.1 (pdf_page=113, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 本章阿贝尔群G及子群H上x~y当且仅当x-y∈H；原文定义关系，不能仅因旧中文短语而判无概念。与任意等价关系不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `Let $H$ be a subgroup of $G$ . We say $x, y \in G$ are equivalent if`
  - nakahara-geometry-topology-physics/second-edition-local §3.1.1 (pdf_page=113, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 本章阿贝尔群G及子群H上x~y当且仅当x-y∈H；原文定义关系，不能仅因旧中文短语而判无概念。与任意等价关系不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `$$ x - y \in H \tag {3.2} $$`
  - nakahara-geometry-topology-physics/second-edition-local §3.1.1 (pdf_page=113, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 本章阿贝尔群G及子群H上x~y当且仅当x-y∈H；原文定义关系，不能仅因旧中文短语而判无概念。与任意等价关系不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `and write $x \sim y$ . Clearly $\sim$ is an equivalence relation. The equivalence class to which $x$ belongs is denoted by $[x]$ . Let $G / H$ be the quotient space. The group operation $+$ in $G$ naturally induces the group operation $+$ in $G / H$ by`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 等价关系(equivalence relation)

- UUID: `e4dc698e-be80-42e3-aa14-c8df668fb00b` · okf_url: https://limbo137.top/okf/concepts/e4dc698e-be80-42e3-aa14-c8df668fb00b.md
- labels: en = equivalence relation；zh-CN = 等价关系
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-2ebb964b56554a6b0086-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=89, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.2：等价关系。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.2. An equivalence relation $\sim$ is a relation which satisfies the following requirements:`
  - nakahara-geometry-topology-physics/second-edition-local §2.1.2 (pdf_page=89, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.2：等价关系。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(i) $a\sim a$ (reflective). (ii) If $a\sim b$ , then $b\sim a$ (symmetric). (iii) If $a\sim b$ and $b\sim c$ , then $a\sim c$ (transitive).`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 时间平移(time translation)

- UUID: `e4e5b10c-e0be-4e54-aaa6-8ba7699dc29d` · okf_url: https://limbo137.top/okf/concepts/e4e5b10c-e0be-4e54-aaa6-8ba7699dc29d.md
- labels: en = time translation；zh-CN = 时间平移
- aliases: translation in time
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §1.4 · `books/woit-qtgr/2025-10-20/en/chapter-01/04-representations-and-quantum-mechanics#concept-1-time-translation` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/04-representations-and-quantum-mechanics.md

## 正规坐标(normal coordinates)

- UUID: `e5035e1d-d87c-4361-9fe6-66bcfba9ba55` · okf_url: https://limbo137.top/okf/concepts/e5035e1d-d87c-4361-9fe6-66bcfba9ba55.md
- labels: en = normal coordinates；zh-CN = 正规坐标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-b95e4cfd9c00c34c1b17-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-b95e4cfd9c00c34c1b17-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 平行移动(parallel transport)

- UUID: `e57b6201-cfbb-4d85-b4df-5d0ab5b64e07` · okf_url: https://limbo137.top/okf/concepts/e57b6201-cfbb-4d85-b4df-5d0ab5b64e07.md
- labels: en = parallel transport；zh-CN = 平行移动
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.1 · `en/chapter-18/01-linear-connections-and-geodesics#first-ca3f7036c5d35e9c30b4-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.1 · `zh/chapter-18/01-linear-connections-and-geodesics#first-ca3f7036c5d35e9c30b4-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/01-linear-connections-and-geodesics.md

## 坐标邻域(coordinate neighbourhood)

- UUID: `e58c4e16-501f-4eb0-9329-1fcb4c6eaf42` · okf_url: https://limbo137.top/okf/concepts/e58c4e16-501f-4eb0-9329-1fcb4c6eaf42.md
- labels: en = coordinate neighbourhood；zh-CN = 坐标邻域
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-coordinate-neighbourhood` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-0bd6d690f6b5a200d515-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## 有序对(ordered pair)

- UUID: `e5a0d3c6-87bc-459a-b42e-e7d38b8d1601` · okf_url: https://limbo137.top/okf/concepts/e5a0d3c6-87bc-459a-b42e-e7d38b8d1601.md
- labels: en = ordered pair；zh-CN = 有序对
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-5a0640df55285c563005-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §1.3 · `books/woit-qtgr/2025-10-20/zh/chapter-01/03-unitary-group-representations#first-2c46f30ae6e202d6ded9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-01/03-unitary-group-representations.md

## 量词(quantifier)

- UUID: `e5cf34a0-c35c-4078-8194-cbdb83bbc3cf` · okf_url: https://limbo137.top/okf/concepts/e5cf34a0-c35c-4078-8194-cbdb83bbc3cf.md
- labels: en = quantifier；zh-CN = 量词
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.1 · `zh/chapter-01/01-sets-and-logic#first-1003826324061f0695fc-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/01-sets-and-logic.md

## 左理想(left ideal)

- UUID: `e5d1ce04-babc-4a8b-b31d-f65f6ed73aca` · okf_url: https://limbo137.top/okf/concepts/e5d1ce04-babc-4a8b-b31d-f65f6ed73aca.md
- labels: en = left ideal；zh-CN = 左理想
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-e744d74485fad56813dd-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-left-ideal` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 中心(centre)

- UUID: `e5df3ee6-ad9c-4544-bf44-ea5c27385154` · okf_url: https://limbo137.top/okf/concepts/e5df3ee6-ad9c-4544-bf44-ea5c27385154.md
- labels: en = centre；zh-CN = 中心
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.5 · `zh/chapter-02/05-normal-subgroups-and-factor-groups#concept-2-centre` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/05-normal-subgroups-and-factor-groups.md

## 群的共轭作用(group conjugation)

- UUID: `e5f6579c-3285-4510-9961-6c2647fe987c` · okf_url: https://limbo137.top/okf/concepts/e5f6579c-3285-4510-9961-6c2647fe987c.md
- labels: en = group conjugation；zh-CN = 群的共轭作用
- aliases: conjugation；共轭
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-conjugation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 泡利矩阵(Pauli matrices)

- UUID: `e65c3c65-ec76-4453-8b9d-7512c5748f13` · okf_url: https://limbo137.top/okf/concepts/e65c3c65-ec76-4453-8b9d-7512c5748f13.md
- labels: en = Pauli matrices；zh-CN = 泡利矩阵
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-3fd95794b9007286e7b0-zh` · translation)；woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#first-3fd95794b9007286e7b0-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 黎曼球面(Riemann sphere)

- UUID: `e663b5dd-c1fb-4282-b563-3b00795ebad3` · okf_url: https://limbo137.top/okf/concepts/e663b5dd-c1fb-4282-b563-3b00795ebad3.md
- labels: en = Riemann sphere；zh-CN = 黎曼球面
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.4 · `books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space#first-b695f25c405115c9d415-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.4 · `books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space#first-b695f25c405115c9d415-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/04-complex-projective-space.md

## 子群(subgroup)

- UUID: `e6dd2937-0598-4aa9-bbf2-317c9e35c0e6` · okf_url: https://limbo137.top/okf/concepts/e6dd2937-0598-4aa9-bbf2-317c9e35c0e6.md
- labels: en = subgroup；zh-CN = 子群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-subgroup` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §— (pdf_page=113, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Nakahara 本章限于阿贝尔群，运算记 +，单位元记 0；不把本章约定推广到任意群。 · 原文: `A subset $H \subset G$ is a subgroup if it is a group with respect to the group operation of $G$ . For example,`
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Szekeres 允许非阿贝尔母群。
- 显式依赖组(dependency_groups): nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `From lemma 3.1, it follows that $Z_{r}(K)$ and $B_{r}(K)$ are subgroups of $C_{r}(K)$ . We now prove an important relation between $Z_{r}(K)$ and $B_{r}(K)$ , w...`；nakahara-geometry-topology-physics/second-edition-local (1 条, completeness=recorded-only) · 摘录: `Proof. This is obvious from lemma 3.3. Any element $c$ of $B_r(K)$ is written as $c = \partial_{r+1}d$ for some $d \in C_{r+1}(K)$ . Then we find $\partial_r c ...`
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 完全可积的(completely integrable)

- UUID: `e710238e-7313-48c6-806b-126872e4579a` · okf_url: https://limbo137.top/okf/concepts/e710238e-7313-48c6-806b-126872e4579a.md
- labels: en = completely integrable；zh-CN = 完全可积的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.3 · `zh/chapter-16/03-frobenius-theorem-dual-form#first-43695f3536ee91f7379b-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.3 · `en/chapter-16/03-frobenius-theorem-dual-form#first-43695f3536ee91f7379b-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/03-frobenius-theorem-dual-form.md

## Siegel 上半空间(Siegel upper half-space)

- UUID: `e725ddcb-6d23-450c-be5a-8ec98d7f307b` · okf_url: https://limbo137.top/okf/concepts/e725ddcb-6d23-450c-be5a-8ec98d7f307b.md
- labels: en = Siegel upper half-space；zh-CN = Siegel 上半空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §26.6 · `books/woit-qtgr/2025-10-20/en/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d#first-355d3fda6fbea5309d78-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §26.6 · `books/woit-qtgr/2025-10-20/zh/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d#first-355d3fda6fbea5309d78-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d.md

## 代数同构(algebra isomorphism)

- UUID: `e74a2efd-e969-4049-93b3-d9c7d69f9d11` · okf_url: https://limbo137.top/okf/concepts/e74a2efd-e969-4049-93b3-d9c7d69f9d11.md
- labels: en = algebra isomorphism；zh-CN = 代数同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-algebra-isomorphism` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-72c3b8a32623df7e75cd-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 分次代数(graded algebra)

- UUID: `e77f1eb0-6ba2-47d8-aa76-663ca0eee033` · okf_url: https://limbo137.top/okf/concepts/e77f1eb0-6ba2-47d8-aa76-663ca0eee033.md
- labels: en = graded algebra；zh-CN = 分次代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-eb8b5540f012b88c1c7d-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-graded-algebra` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md

## 不稳定焦点(unstable focus)

- UUID: `e78e5ed4-5e08-4f53-918c-3931da7a7083` · okf_url: https://limbo137.top/okf/concepts/e78e5ed4-5e08-4f53-918c-3931da7a7083.md
- labels: en = unstable focus；zh-CN = 不稳定焦点
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.3 · `zh/chapter-04/03-linear-ordinary-differential-equations#concept-4-unstable-focus` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/03-linear-ordinary-differential-equations.md

## 巴拿赫空间(Banach space)

- UUID: `e7ec5e64-4438-4d30-83bd-f2ea3808a960` · okf_url: https://limbo137.top/okf/concepts/e7ec5e64-4438-4d30-83bd-f2ea3808a960.md
- labels: en = Banach space；zh-CN = 巴拿赫空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.9 · `zh/chapter-10/09-topological-vector-spaces#first-e5d0708204cb2be942a5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/09-topological-vector-spaces.md

## 标量乘法(scalar multiplication)

- UUID: `e802c467-fbf7-473a-9f3a-bc65a40b9b9b` · okf_url: https://limbo137.top/okf/concepts/e802c467-fbf7-473a-9f3a-bc65a40b9b9b.md
- labels: en = scalar multiplication；zh-CN = 标量乘法
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-scalar-multiplication` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 坐标卡(coordinate chart)

- UUID: `e84fb5d6-7851-4bfc-be91-04e92cda6b7e` · okf_url: https://limbo137.top/okf/concepts/e84fb5d6-7851-4bfc-be91-04e92cda6b7e.md
- labels: en = coordinate chart；zh-CN = 坐标卡
- aliases: (coordinate) chart
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.1 · `zh/chapter-15/01-differentiable-manifolds#concept-15-coordinate-chart` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §15.1 · `en/chapter-15/01-differentiable-manifolds#first-e70d67ecb14c4fe2dab8-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/01-differentiable-manifolds.md

## Schwinger项(Schwinger term)

- UUID: `e85447b5-6df6-4a39-8dc5-fc84f01e7e7b` · okf_url: https://limbo137.top/okf/concepts/e85447b5-6df6-4a39-8dc5-fc84f01e7e7b.md
- labels: en = Schwinger term；zh-CN = Schwinger项
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §39.3 · `books/woit-qtgr/2025-10-20/zh/chapter-39/03-the-anomaly-and-the-schwinger-term#first-c62600d6f929dbed3f58-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §39.3 · `books/woit-qtgr/2025-10-20/en/chapter-39/03-the-anomaly-and-the-schwinger-term#first-c62600d6f929dbed3f58-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/03-the-anomaly-and-the-schwinger-term.md

## 自旋二分之一系统(spin-half system)

- UUID: `e88f7c38-759c-4f05-b600-5d36503375fd` · okf_url: https://limbo137.top/okf/concepts/e88f7c38-759c-4f05-b600-5d36503375fd.md
- labels: en = spin-half system；zh-CN = 自旋二分之一系统
- aliases: spin-$1/2$ system
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.3 · `books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture#concept-7-3-spin` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 开球(open ball)

- UUID: `e8a77376-44a4-487c-87d1-0aefde24c786` · okf_url: https://limbo137.top/okf/concepts/e8a77376-44a4-487c-87d1-0aefde24c786.md
- labels: en = open ball；zh-CN = 开球
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.1 · `zh/chapter-10/01-euclidean-topology#first-842836ec103f9db67e61-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## 相对论复结构(Relativistic complex structure)

- UUID: `e8ca6ba1-6ef2-4a3a-a299-bd7c3a6498b5` · okf_url: https://limbo137.top/okf/concepts/e8ca6ba1-6ef2-4a3a-a299-bd7c3a6498b5.md
- labels: en = Relativistic complex structure；zh-CN = 相对论复结构
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §43.2 · `books/woit-qtgr/2025-10-20/en/chapter-43/02-the-symplectic-and-complex-structures-on-m#first-643e72c709a21d83d6b7-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §43.2 · `books/woit-qtgr/2025-10-20/zh/chapter-43/02-the-symplectic-and-complex-structures-on-m#first-74d2487b15a60777c141-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §43.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `实解空间 $\mathcal { M }$ 上的相对论复结构由算符 $J_r$ 给出。将它延拓到 $\mathcal { M }\otimes\mathbf{C}$ 后，它在 $\alpha(\mathbf{p})$ 上的作用为 $+i$，在 $\overline{\alpha}(\mathbf{p})$ 上的作用为 $-i$。 于是 $A(\mathbf{p})$ 构成 $\mathcal{M}^+_{J_r}$ 的连续基，$\overline{A}(\mathbf{p})$ 构成 $\mathcal{M}^-_{J_r}$ 的连续基，并且 $$ \mathcal {M} \otimes \mathbf {C} = \mathcal {M} _ {J _ {r}} ^ {+} \oplus \mathc...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-43/02-the-symplectic-and-complex-structures-on-m.md

## 四元数(quaternion)

- UUID: `e975605a-7577-4615-b024-f78011266eb4` · okf_url: https://limbo137.top/okf/concepts/e975605a-7577-4615-b024-f78011266eb4.md
- labels: en = quaternion；zh-CN = 四元数
- aliases: quaternions
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.3 · `en/chapter-06/03-quaternions-and-clifford-algebras#first-972a5aa65f6fb37f163f-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.3 · `zh/chapter-06/03-quaternions-and-clifford-algebras#concept-6-quaternion` · translation)；woit-qtgr/2025-10-20 (en · §6.2 · `books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-quaternion` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §6.2 · `books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions#concept-6-quaternion` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/03-quaternions-and-clifford-algebras.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-06/02-spin-groups-in-three-and-four-dimensions.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/05-real-cliford-algebras.md

## 长度(length)

- UUID: `e98e09ca-6bd2-49f0-92f7-1860bb397f02` · okf_url: https://limbo137.top/okf/concepts/e98e09ca-6bd2-49f0-92f7-1860bb397f02.md
- labels: en = length；zh-CN = 长度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-length` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 海森堡李代数(Heisenberg Lie algebra)

- UUID: `e992bab1-8a2d-4291-96da-ff7f86a8dcfc` · okf_url: https://limbo137.top/okf/concepts/e992bab1-8a2d-4291-96da-ff7f86a8dcfc.md
- labels: en = Heisenberg Lie algebra；zh-CN = 海森堡李代数
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §5.5 · `books/woit-qtgr/2025-10-20/zh/chapter-05/05-complexification#first-17b999e6f3200fa02e79-zh` · translation)；woit-qtgr/2025-10-20 (en · §5.5 · `books/woit-qtgr/2025-10-20/en/chapter-05/05-complexification#first-17b999e6f3200fa02e79-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §13.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Heisenberg Lie algebra $\mathfrak h_{2d+1}$ is the vector space $\mathbf R^{2d+1}=\mathbf R^{2d}\oplus\mathbf R$, with Lie bracket specified on a basis $X_j,Y_j,Z$ for $j=1,\ldots,d$ by $$ [X_j,Y_k]=\delta_{jk}Z,\qquad [X_j,Z]=[Y_j,Z]=0 $$`
  - woit-qtgr/2025-10-20 §13.1 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `海森堡李代数 $\mathfrak h_{2d+1}$ 是向量空间 $\mathbf R^{2d+1}=\mathbf R^{2d}\oplus\mathbf R$，其李括号在基 $X_j,Y_j,Z$（$j=1,\ldots,d$）上的取值规定为 $$ [X_j,Y_k]=\delta_{jk}Z,\qquad [X_j,Z]=[Y_j,Z]=0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-16/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/01-multiple-degrees-of-freedom.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/05-complexification.md

## 幺半群(monoid)

- UUID: `e9b498c1-1ead-4794-9793-bdac714417f5` · okf_url: https://limbo137.top/okf/concepts/e9b498c1-1ead-4794-9793-bdac714417f5.md
- labels: en = monoid；zh-CN = 幺半群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.6 · `zh/chapter-01/06-structures#first-b0eebe40a9dfea21540b-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/06-structures.md

## 各向同性群(isotropy group)

- UUID: `e9d708ff-0053-47d3-bf1d-21c5b4712799` · okf_url: https://limbo137.top/okf/concepts/e9d708ff-0053-47d3-bf1d-21c5b4712799.md
- labels: en = isotropy group；zh-CN = 各向同性群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.4 · `en/chapter-19/04-lie-groups-of-transformations#first-3ab17ac90df4a1e8293c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-isotropy-group` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.2 (pdf_page=239, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.16：各向同性群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.16. Let $G$ be a Lie group that acts on a manifold $M$ . The isotropy group of $p \in M$ is a subgroup of $G$ defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.2 (pdf_page=239, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.16：各向同性群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$H(p)$ is also called the little group or stabilizer of $p$ .`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.2 (pdf_page=239, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.16：各向同性群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ H (p) = \{g \in G | \sigma (g, p) = p \}. \tag {5.149} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.2 (pdf_page=239, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.16：各向同性群。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `It is easy to see that $H(p)$ is indeed a subgroup. Let $g_1, g_2 \in H(p)$ , then $g_1g_2 \in H(p)$ since $\sigma(g_1g_2, p) = \sigma(g_1, \sigma(g_2, p)) = \sigma(g_1, p) = p$ . Clearly $e \in H(p)$ since $\sigma(e, p) = p$ by definition. If $g \in H(p)$ , then $g^{-1} \in H(p)$ since $p = \sigma(e, p) = \sigma(g^{-1}g, p) = \sigma(g^{-1}, \sigma(g, p))...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/04-lie-groups-of-transformations.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md

## 可数次可加性(countable subadditivity)

- UUID: `ea0d1e41-8e16-4ced-92e0-4c4b1ba5dba9` · okf_url: https://limbo137.top/okf/concepts/ea0d1e41-8e16-4ced-92e0-4c4b1ba5dba9.md
- labels: en = countable subadditivity；zh-CN = 可数次可加性
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-073a5ff25ad2dd1cbc0e-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 零测地线(null geodesic)

- UUID: `ea0e6a8d-6c86-4e6d-b5f8-2a2357a9c182` · okf_url: https://limbo137.top/okf/concepts/ea0e6a8d-6c86-4e6d-b5f8-2a2357a9c182.md
- labels: en = null geodesic；zh-CN = 零测地线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §18.4 · `en/chapter-18/04-pseudo-riemannian-manifolds#first-66eabaf7378d94a01696-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.4 · `zh/chapter-18/04-pseudo-riemannian-manifolds#first-66eabaf7378d94a01696-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md

## 指向未来的(future-pointing)

- UUID: `ea1ef8ed-70b0-4b91-baa3-ae78f479f3a5` · okf_url: https://limbo137.top/okf/concepts/ea1ef8ed-70b0-4b91-baa3-ae78f479f3a5.md
- labels: en = future-pointing；zh-CN = 指向未来的
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-b530498fb686f092dbf9-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-b530498fb686f092dbf9-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 极限(limit)

- UUID: `ea27b1d9-445a-474d-a590-170a34051242` · okf_url: https://limbo137.top/okf/concepts/ea27b1d9-445a-474d-a590-170a34051242.md
- labels: en = limit；zh-CN = 极限
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.1 · `zh/chapter-10/01-euclidean-topology#first-47b0f6a99aa530cc1f8c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## 关系(relation)

- UUID: `ea55b93f-5b50-4f2c-9ba9-7c9507a138e9` · okf_url: https://limbo137.top/okf/concepts/ea55b93f-5b50-4f2c-9ba9-7c9507a138e9.md
- labels: en = relation；zh-CN = 关系
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-c2689ae604a298f7cfe0-zh` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §1.3 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书n元关系与二元关系不同；covector与linear functional明确同义。 · 原文: `Any subset of $S ^ { n }$ is called an $n$-ary relation on a set $S$. For example, $$ \text { unary relation } \equiv 1 \text {-ary relation } = \text { subset of } S $$ $$ \text { binary relation } \equiv 2 \text {-ary relation } = \text { subset of } S ^ {2} = S \times S $$ $$ \text { ternary relation } \equiv 3 \text {-ary relation } = \text { subset o...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 商李代数(factor Lie algebra)

- UUID: `ea81721c-fa04-44d4-82ca-3e47570700d4` · okf_url: https://limbo137.top/okf/concepts/ea81721c-fa04-44d4-82ca-3e47570700d4.md
- labels: en = factor Lie algebra；zh-CN = 商李代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.5 · `en/chapter-06/05-lie-algebras-and-lie-groups#first-5c61ee7977d5eba8597c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.5 · `zh/chapter-06/05-lie-algebras-and-lie-groups#concept-6-factor-lie-algebra` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/05-lie-algebras-and-lie-groups.md

## 复化(complexification)

- UUID: `eaa8baed-19fe-4db6-a610-2a84829b114d` · okf_url: https://limbo137.top/okf/concepts/eaa8baed-19fe-4db6-a610-2a84829b114d.md
- labels: en = complexification；zh-CN = 复化
- aliases: Complexification
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §6.2 · `en/chapter-06/02-complex-numbers-and-complex-structures#first-a931acf7f73fdaa13883-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.2 · `zh/chapter-06/02-complex-numbers-and-complex-structures#concept-6-complexification` · translation)；woit-qtgr/2025-10-20 (en · §5.5 · `books/woit-qtgr/2025-10-20/en/chapter-05/05-complexification#first-b40c8c00e19d679adde9-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.5 · `books/woit-qtgr/2025-10-20/zh/chapter-05/05-complexification#first-b40c8c00e19d679adde9-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §5.5 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition. The complexification $V_{\mathbf{C}}$ of a real vector space $V$ is the space of pairs $(v_1,v_2)$ of elements of $V$ with multiplication by $a+bi\in\mathbf{C}$ given by $$ (a+ib)(v_1,v_2)=\left(av_1-bv_2,\,av_2+bv_1\right) $$`
  - woit-qtgr/2025-10-20 §5.5 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义。实向量空间 $V$ 的 复化(complexification) $V_{\mathbf{C}}$ 是 $V$ 中元素的配对 $(v_1,v_2)$ 构成的空间，其与 $a+bi\in\mathbf{C}$ 的乘法由下式给出 $$ (a+ib)(v_1,v_2)=\left(av_1-bv_2,\,av_2+bv_1\right) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/02-complex-numbers-and-complex-structures.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-26/01-complex-structures-and-phase-space.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/05-complexification.md

## 标准单纯形(standard simplex)

- UUID: `eb02922d-db9b-40f3-82d7-5efdff8f6a0a` · okf_url: https://limbo137.top/okf/concepts/eb02922d-db9b-40f3-82d7-5efdff8f6a0a.md
- labels: en = standard simplex；zh-CN = 标准单纯形
- aliases: standard
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.4 · `zh/chapter-17/04-homology-and-cohomology#first-5c41f4360a795b0315ca-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §17.4 · `en/chapter-17/04-homology-and-cohomology#first-25242d5ab28856d4128d-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/04-homology-and-cohomology.md

## 不可约实表示(irreducible real representation)

- UUID: `eb527126-d5d0-417c-a57e-1131e5b8f893` · okf_url: https://limbo137.top/okf/concepts/eb527126-d5d0-417c-a57e-1131e5b8f893.md
- labels: en = irreducible real representation；zh-CN = 不可约实表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §7.1 · `books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation#concept-7-1-irreducible-representation` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §7.1 · `books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation#concept-7-1-irreducible-representation` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/01-the-spinor-representation.md

## 元辛表示(metaplectic representation)

- UUID: `eb959135-fcc2-4572-a09a-1267ad1e544c` · okf_url: https://limbo137.top/okf/concepts/eb959135-fcc2-4572-a09a-1267ad1e544c.md
- labels: en = metaplectic representation；zh-CN = 元辛表示
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §25 · `books/woit-qtgr/2025-10-20/zh/chapter-25/index#first-5933e2dbb4e3939e827f-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §25 · `books/woit-qtgr/2025-10-20/en/chapter-25/index#first-5933e2dbb4e3939e827f-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-32/index.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-25/index.md

## 向量空间同态(vector space homomorphism)

- UUID: `eba919b4-fdf9-44e7-85e3-f560aa4c7943` · okf_url: https://limbo137.top/okf/concepts/eba919b4-fdf9-44e7-85e3-f560aa4c7943.md
- labels: en = vector space homomorphism；zh-CN = 向量空间同态
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.3 · `zh/chapter-03/03-vector-space-homomorphisms#concept-3-vector-space-homomorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md

## 对偶麦克斯韦2-形式(dual Maxwell 2-form)

- UUID: `ec27abc0-a495-455c-887a-04ad0f3a5e76` · okf_url: https://limbo137.top/okf/concepts/ec27abc0-a495-455c-887a-04ad0f3a5e76.md
- labels: en = dual Maxwell 2-form；zh-CN = 对偶麦克斯韦2-形式
- aliases: 对偶麦克斯韦二形式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §17.5 · `en/chapter-17/05-the-poincar-e-lemma#first-d00e7cacdc59f8be9b04-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §17.5 · `zh/chapter-17/05-the-poincar-e-lemma#first-d00e7cacdc59f8be9b04-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-17/05-the-poincar-e-lemma.md

## 场作用量原理(field action principle)

- UUID: `ec41f559-b8c4-48a6-9df1-6274d5965185` · okf_url: https://limbo137.top/okf/concepts/ec41f559-b8c4-48a6-9df1-6274d5965185.md
- labels: en = field action principle；zh-CN = 场作用量原理
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.10 · `zh/chapter-18/10-variation-principles-in-space-time#first-65b80d6b9151fedb30c4-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.10 · `en/chapter-18/10-variation-principles-in-space-time#first-65b80d6b9151fedb30c4-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/10-variation-principles-in-space-time.md

## 右矢(ket)

- UUID: `ec53c000-7589-4e23-86a7-e670b510905f` · okf_url: https://limbo137.top/okf/concepts/ec53c000-7589-4e23-86a7-e670b510905f.md
- labels: en = ket；zh-CN = 右矢
- aliases: ket vector；kets
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-c70c407bcffc733f3dab-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-b856a4d7dff72fc84951-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §4.4 · `books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products#first-78b64e180d2a96799ae6-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.4 · `books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products#first-78b64e180d2a96799ae6-en` · original_transcription)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §14.1 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `In this chapter we will adopt Dirac’s bra-ket notation which, though slightly quirky, has largely become the convention of choice among physicists. Vectors $\psi \in { \mathcal { H } }$ are written as kets $|\psi\rangle$ and one makes the identification $| \lambda \psi \rangle = \lambda | \psi \rangle$ . By the Riesz representation theorem 13.10, to each ...`
  - woit-qtgr/2025-10-20 §4.4 (pdf_page=not-recorded, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 具体书中条件保留在原文；不将本次核对声称为首次出现。 · 原文: `Physicists have a useful notation due to Dirac for elements of a vector space and its dual, for the case when $V$ is a complex vector space with a Hermitian inner product (such as the state space $H$ for a quantum theory). An element of such a vector space $V$ is written as a “ket vector” $$ |\alpha\rangle $$ where $\alpha$ is a label for a vector in $V$....`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/04-inner-products.md

## Cartan–Dieudonné定理(Cartan–Dieudonné theorem)

- UUID: `ec5695e9-5589-4bda-bdd1-786f7d96125c` · okf_url: https://limbo137.top/okf/concepts/ec5695e9-5589-4bda-bdd1-786f7d96125c.md
- labels: en = Cartan–Dieudonné theorem；zh-CN = Cartan–Dieudonné定理
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §29.2 · `books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry#first-d0e98c2775cd151b9da4-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §29.2 · `books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry#first-d0e98c2775cd151b9da4-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/02-cliford-algebras-and-geometry.md

## 右平移(right translation)

- UUID: `ec9113bb-b878-409e-a9a3-73ae2ac1eaab` · okf_url: https://limbo137.top/okf/concepts/ec9113bb-b878-409e-a9a3-73ae2ac1eaab.md
- labels: en = right translation；zh-CN = 右平移
- aliases: right trans lation
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-9de39a355921a9426500-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 左平移(left translation)

- UUID: `ec95f882-e96f-4231-99b7-8e9903ed60a8` · okf_url: https://limbo137.top/okf/concepts/ec95f882-e96f-4231-99b7-8e9903ed60a8.md
- labels: en = left translation；zh-CN = 左平移
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.1 · `en/chapter-19/01-lie-groups#first-0ebadeea28f766cc7e69-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.1 · `zh/chapter-19/01-lie-groups#concept-19-left-translation` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：左平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ L _ {a} g = a g. \tag {5.109b} $$`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：左平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.9. Let $a$ and $g$ be elements of a Lie group $G$ . The right-translation $R_{a}: G \to G$ and the left-translation $L_{a}: G \to G$ of $g$ by $a$ are defined by`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：左平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `By definition, $R_{a}$ and $L_{a}$ are diffeomorphisms from $G$ to $G$ . Hence, the maps $L_{a}: G \to G$ and $R_{a}: G \to G$ induce $L_{a*}: T_{g}G \to T_{ag}G$ and $R_{a*}: T_{g}G \to T_{ga}G$ ; see section 5.2. Since these translations give equivalent theories, we are concerned mainly with the left-translation in the following. The analysis based on t...`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：左平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Given a Lie group $G$ , there exists a special class of vector fields characterized by an invariance under group action. [On the usual manifold there is no canonical way of discriminating some vector fields from the others.]`
  - nakahara-geometry-topology-physics/second-edition-local §5.6.2 (pdf_page=228, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.9：左平移。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `$$ R _ {a} g = g a \tag {5.109a} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/01-lie-groups.md

## 邻域(neighbourhood)

- UUID: `ed082212-4450-43ff-be65-96e35fa3d058` · okf_url: https://limbo137.top/okf/concepts/ed082212-4450-43ff-be65-96e35fa3d058.md
- labels: en = neighbourhood；zh-CN = 邻域
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.1 · `zh/chapter-10/01-euclidean-topology#first-45266bf66f5e877e61c9-zh` · translation)
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §2.3.3 (pdf_page=101, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 2.5：邻域。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 2.5. Suppose $\mathcal{T}$ gives a topology to $X$ . $N$ is a neighbourhood of a point $x \in X$ if $N$ is a subset of $X$ and $N$ contains some (at least one) open set $U_i$ to which $x$ belongs. (The subset $N$ need not be an open set. If $N$ happens to be an open set in $\mathcal{T}$ , it is called an open neighbourhood.)`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/01-euclidean-topology.md

## Trotter乘积公式(Trotter product formula)

- UUID: `ed1b2064-8ab4-4ffb-a2c6-a818e2b81c2d` · okf_url: https://limbo137.top/okf/concepts/ed1b2064-8ab4-4ffb-a2c6-a818e2b81c2d.md
- labels: en = Trotter product formula；zh-CN = Trotter乘积公式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §35.3 · `books/woit-qtgr/2025-10-20/en/chapter-35/03-quantization-and-path-integrals#first-d4727b385671a203377a-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §35.3 · `books/woit-qtgr/2025-10-20/zh/chapter-35/03-quantization-and-path-integrals#first-d4727b385671a203377a-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/03-quantization-and-path-integrals.md

## 路径(path)

- UUID: `ee5d2cd4-b49e-5536-9ed7-e7c8c959f0cb` · okf_url: https://limbo137.top/okf/concepts/ee5d2cd4-b49e-5536-9ed7-e7c8c959f0cb.md
- labels: en = path；zh-CN = 路径
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.1：路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 4.1. Let $X$ be a topological space and let $I = [0,1]$ . A continuous map $\alpha : I \to X$ is called a path with an initial point $x_0$ and an end point $x_1$ if $\alpha(0) = x_0$ and $\alpha(1) = x_1$ . If $\alpha(0) = \alpha(1) = x_0$ , the path is called a loop with base point $x_0$ (or a loop at $x_0$ ).`
  - nakahara-geometry-topology-physics/second-edition-local §4.1.2 (pdf_page=141, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 4.1：路径。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `For $x \in X$ , a constant path $c_{x}: I \to X$ is defined by $c_{x}(s) = x, s \in I$ . A constant path is also a constant loop since $c_{x}(0) = c_{x}(1) = x$ . The set of paths or loops in a topological space $X$ may be endowed with an algebraic structure as follows.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 等温线(isotherm)

- UUID: `ee640353-55cc-42c2-9143-5dc6ddb0d74e` · okf_url: https://limbo137.top/okf/concepts/ee640353-55cc-42c2-9143-5dc6ddb0d74e.md
- labels: en = isotherm；zh-CN = 等温线
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-40d94badb74cb92c2740-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-40d94badb74cb92c2740-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 自由指标(free index)

- UUID: `ee67ea84-09a6-429a-9539-9a7bdf49ef37` · okf_url: https://limbo137.top/okf/concepts/ee67ea84-09a6-429a-9539-9a7bdf49ef37.md
- labels: en = free index；zh-CN = 自由指标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-free-index` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 单位元分支(component of the identity)

- UUID: `eecc1ebf-eaa6-4cf7-8236-131513de65d2` · okf_url: https://limbo137.top/okf/concepts/eecc1ebf-eaa6-4cf7-8236-131513de65d2.md
- labels: en = component of the identity；zh-CN = 单位元分支
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.8 · `zh/chapter-10/08-topological-groups#first-dcffa27cdcd67fd87f55-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/08-topological-groups.md

## 约束哈密顿动力学(constrained Hamiltonian dynamics)

- UUID: `ef1131c2-08bf-4d12-92e8-b36397a1407f` · okf_url: https://limbo137.top/okf/concepts/ef1131c2-08bf-4d12-92e8-b36397a1407f.md
- labels: en = constrained Hamiltonian dynamics；zh-CN = 约束哈密顿动力学
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-18d489bc92714fd6a5b2-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-18d489bc92714fd6a5b2-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 正则分布(regular distribution)

- UUID: `ef60bda0-3f62-4112-9dfa-81af9938599d` · okf_url: https://limbo137.top/okf/concepts/ef60bda0-3f62-4112-9dfa-81af9938599d.md
- labels: en = regular distribution；zh-CN = 正则分布
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.1 · `zh/chapter-12/01-test-functions-and-distributions#first-e85f7b05eafc32f7475c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/01-test-functions-and-distributions.md

## 重整化(renormalization)

- UUID: `efea2076-78f7-4fb4-8c9b-42f8bc2b9d1a` · okf_url: https://limbo137.top/okf/concepts/efea2076-78f7-4fb4-8c9b-42f8bc2b9d1a.md
- labels: en = renormalization；zh-CN = 重整化
- aliases: 重正化
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §39.5 · `books/woit-qtgr/2025-10-20/en/chapter-39/05-higher-order-operators-and-renormalization#first-aa5b1d5b7ad2ca94ce0c-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §39.5 · `books/woit-qtgr/2025-10-20/zh/chapter-39/05-higher-order-operators-and-renormalization#first-aa5b1d5b7ad2ca94ce0c-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-39/05-higher-order-operators-and-renormalization.md

## 李子群(Lie subgroup)

- UUID: `f0104c54-899b-429c-808b-f923ff1a7f81` · okf_url: https://limbo137.top/okf/concepts/f0104c54-899b-429c-808b-f923ff1a7f81.md
- labels: en = Lie subgroup；zh-CN = 李子群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §19.3 · `en/chapter-19/03-lie-subgroups#first-15e2e825d739e6fbadb8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.3 · `zh/chapter-19/03-lie-subgroups#concept-19-lie-subgroup` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/03-lie-subgroups.md

## 抽象李代数(Abstract Lie algebra)

- UUID: `f0166e21-db33-4458-b2e7-ed19c7f71ecb` · okf_url: https://limbo137.top/okf/concepts/f0166e21-db33-4458-b2e7-ed19c7f71ecb.md
- labels: en = Abstract Lie algebra；zh-CN = 抽象李代数
- aliases: abstract Lie algebra
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §5.4 · `books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations#first-b57a994f16ccbbae9609-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §5.4 · `books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations#first-b55b5444a0dd77c981a6-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Abstract Lie algebra). An abstract Lie algebra over a field $k$ is a vector space $A$ over $k$, with a bilinear operation $$ [\cdot,\cdot]:(X,Y)\in A\times A\mapsto[X,Y]\in A $$ satisfying 1. Antisymmetry: $$ [X,Y]=-[Y,X] $$ 2. Jacobi identity: $$ [[X,Y],Z]+[[Z,X],Y]+[[Y,Z],X]=0 $$`
  - woit-qtgr/2025-10-20 §5.4 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（抽象李代数）。域 $k$ 上的一个 抽象李代数(abstract Lie algebra) 是 $k$ 上的一个向量空间 $A$，带有一个双线性运算 $$ [\cdot,\cdot]:(X,Y)\in A\times A\mapsto[X,Y]\in A $$ 满足 1. 反对称性： $$ [X,Y]=-[Y,X] $$ 2. [雅可比恒等式(Jacobi identity)](/concepts/lie-theory/jacobi-identity)： $$ [[X,Y],Z]+[[Z,X],Y]+[[Y,Z],X]=0 $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-05/04-lie-algebra-representations.md

## 雅可比群(Jacobi group)

- UUID: `f033ef18-bc06-41cd-99de-7d70e80ee200` · okf_url: https://limbo137.top/okf/concepts/f033ef18-bc06-41cd-99de-7d70e80ee200.md
- labels: en = Jacobi group；zh-CN = 雅可比群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §18.2 · `books/woit-qtgr/2025-10-20/zh/chapter-18/02-semi-direct-product-groups#first-7f70b92ed826589a6d11-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §18.2 · `books/woit-qtgr/2025-10-20/en/chapter-18/02-semi-direct-product-groups#first-dedfb71bf65f1a9333d3-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §18.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The Jacobi group in $d$ dimensions is the semi-direct product group $$ G ^ {J} (d) = H _ {2 d + 1} \rtimes S p (2 d, \mathbf {R}) $$ If we write elements of the group as $$ \left(\left(\binom{\mathbf {c} _ {q}}{\mathbf {c} _ {p}}, c\right), k\right) $$ where $k \in S p ( 2 d , { \bf R } )$ , then the automorphism $\Phi _ { k }$ that defines the Jacobi gro...`
  - woit-qtgr/2025-10-20 §18.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `$d$ 维的雅可比群是半直积群 $$ G ^ {J} (d) = H _ {2 d + 1} \rtimes S p (2 d, \mathbf {R}) $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-18/02-semi-direct-product-groups.md

## 光子(photon)

- UUID: `f04b510e-b8bd-425c-8eb1-73dd541662f9` · okf_url: https://limbo137.top/okf/concepts/f04b510e-b8bd-425c-8eb1-73dd541662f9.md
- labels: en = photon；zh-CN = 光子
- aliases: photons
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-b55a62f84298838c3801-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §14.1 · `en/chapter-14/01-basic-concepts#first-2de513922e64cc7fbe33-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 环(ring)

- UUID: `f05f6f83-0600-4f4e-951c-348c86414c39` · okf_url: https://limbo137.top/okf/concepts/f05f6f83-0600-4f4e-951c-348c86414c39.md
- labels: en = ring；zh-CN = 环
- aliases: 环（不要求交换）
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §28.1 · `books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-bb6b914065450a4d0c14-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §28.1 · `books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras#first-bb6b914065450a4d0c14-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-28/01-the-complex-weyl-and-cliford-algebras.md

## 交换代数(commutative algebra)

- UUID: `f0e269d8-bae5-4032-9611-4e7c7b5f8518` · okf_url: https://limbo137.top/okf/concepts/f0e269d8-bae5-4032-9611-4e7c7b5f8518.md
- labels: en = commutative algebra；zh-CN = 交换代数
- aliases: commutative
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.1 · `zh/chapter-06/01-algebras-and-ideals#concept-6-commutative-algebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.1 · `en/chapter-06/01-algebras-and-ideals#first-5ae721da6d748fa5a23f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/01-algebras-and-ideals.md

## 反对称化算子(antisymmetrization operator)

- UUID: `f146a3dd-50b7-455b-a3a5-c0f3aff52fe9` · okf_url: https://limbo137.top/okf/concepts/f146a3dd-50b7-455b-a3a5-c0f3aff52fe9.md
- labels: en = antisymmetrization operator；zh-CN = 反对称化算子
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §8.1 · `zh/chapter-08/01-r-vectors-and-r-forms#first-9f869e4580ce50f2547a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §8.1 · `en/chapter-08/01-r-vectors-and-r-forms#first-9f869e4580ce50f2547a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/01-r-vectors-and-r-forms.md

## 格拉姆–施密特正交归一化(Gram–Schmidt orthonormalization)

- UUID: `f1871003-5561-4817-861b-c7259f0f4e9e` · okf_url: https://limbo137.top/okf/concepts/f1871003-5561-4817-861b-c7259f0f4e9e.md
- labels: en = Gram–Schmidt orthonormalization；zh-CN = 格拉姆–施密特正交归一化
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#concept-5-gram-schmidt-orthonormalization` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 推迟传播子(Retarded propagator)

- UUID: `f1ac8fda-6f61-4308-a401-d899c91ace3f` · okf_url: https://limbo137.top/okf/concepts/f1ac8fda-6f61-4308-a401-d899c91ace3f.md
- labels: en = Retarded propagator；zh-CN = 推迟传播子
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §12.6 · `books/woit-qtgr/2025-10-20/zh/chapter-12/06-propagators-in-frequency-momentum-space#first-0984b2747e6672fb293c-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §12.6 · `books/woit-qtgr/2025-10-20/en/chapter-12/06-propagators-in-frequency-momentum-space#first-356c5c06b722b245fb2e-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §12.6 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `推迟传播子由下式给出 $$ U _ {+} (t, q _ {t} - q _ {0}) = \left\{ \begin{array}{l l} 0 & t 0 \end{array} \right. $$`
  - woit-qtgr/2025-10-20 §12.6 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The retarded propagator is given by $$ U _ {+} (t, q _ {t} - q _ {0}) = \left\{ \begin{array}{l l} 0 & t 0 \end{array} \right. $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-12/06-propagators-in-frequency-momentum-space.md

## 柯西–施瓦茨不等式(Cauchy–Schwarz inequality)

- UUID: `f1bc887f-aba2-4c03-a6e1-1bd0892e42e5` · okf_url: https://limbo137.top/okf/concepts/f1bc887f-aba2-4c03-a6e1-1bd0892e42e5.md
- labels: en = Cauchy–Schwarz inequality；zh-CN = 柯西–施瓦茨不等式
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §5.2 · `en/chapter-05/02-complex-inner-product-spaces#first-7c1c64b0f6b394e116a8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.2 · `zh/chapter-05/02-complex-inner-product-spaces#first-7c1c64b0f6b394e116a8-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/02-complex-inner-product-spaces.md

## 海森堡不确定性(Heisenberg uncertainty)

- UUID: `f1d8e0c0-3496-4449-ac6f-acf05f78feee` · okf_url: https://limbo137.top/okf/concepts/f1d8e0c0-3496-4449-ac6f-acf05f78feee.md
- labels: en = Heisenberg uncertainty；zh-CN = 海森堡不确定性
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §12.4 · `books/woit-qtgr/2025-10-20/zh/chapter-12/04-heisenberg-uncertainty#first-f92de24c6423d076daf3-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §12.4 · `books/woit-qtgr/2025-10-20/en/chapter-12/04-heisenberg-uncertainty#first-8010396109cc918803d2-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-12/04-heisenberg-uncertainty.md

## 有限维(finite dimensional)

- UUID: `f214d7a0-7106-43ff-bc0a-3801debd64c2` · okf_url: https://limbo137.top/okf/concepts/f214d7a0-7106-43ff-bc0a-3801debd64c2.md
- labels: en = finite dimensional；zh-CN = 有限维
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-finite-dimensional` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## “自旋向上”态(spin-up state)

- UUID: `f279d758-ab16-4805-a63d-086d2d235785` · okf_url: https://limbo137.top/okf/concepts/f279d758-ab16-4805-a63d-086d2d235785.md
- labels: en = spin-up state；zh-CN = “自旋向上”态
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.5 · `books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere#first-04fc72557d53faf6f9f4-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/05-the-bloch-sphere.md

## 自由群作用(free group action)

- UUID: `f2cf8a92-cabe-5899-bc9c-422857c86780` · okf_url: https://limbo137.top/okf/concepts/f2cf8a92-cabe-5899-bc9c-422857c86780.md
- labels: en = free group action；zh-CN = 自由群作用
- aliases: free；自由的
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：自由群作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `Definition 5.15. Let $G$ be a Lie group that acts on a manifold $M$ by $\sigma : G \times M \to M$ . The action $\sigma$ is said to be`
  - nakahara-geometry-topology-physics/second-edition-local §5.7.1 (pdf_page=238, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: Nakahara Definition 5.15：自由群作用。完整条件以所列原文节点为准；不将同框中作为前提提及的概念视为被定义。 · 原文: `(a) transitive if, for any $p_1, p_2 \in M$ , there exists an element $g \in G$ such that $\sigma(g, p_1) = p_2$ ; (b) free if every non-trivial element $g \neq e$ of $G$ has no fixed points in $M$ , that is, if there exists an element $p \in M$ such that $\sigma(g, p) = p$ , then $g$ must be the unit element $e$ ; and (c) effective if the unit element $e...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 直和表示(Direct sum representation)

- UUID: `f2fbd84b-a1e7-4739-a004-babb445170aa` · okf_url: https://limbo137.top/okf/concepts/f2fbd84b-a1e7-4739-a004-babb445170aa.md
- labels: en = Direct sum representation；zh-CN = 直和表示
- aliases: direct sum representation
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §2.1 · `books/woit-qtgr/2025-10-20/en/chapter-02/01-some-representation-theory#first-eedaf6dfd40fd18b9e7c-en` · original_transcription)；woit-qtgr/2025-10-20 (zh-CN · §2.1 · `books/woit-qtgr/2025-10-20/zh/chapter-02/01-some-representation-theory#first-eedaf6dfd40fd18b9e7c-zh` · translation)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §2.1 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Direct sum representation). Given representations $\pi _ { 1 }$ and $\pi _ { 2 }$ of dimensions $n _ { 1 }$ and $n _ { 2 }$ , there is a representation of dimension $n _ { 1 } + n _ { 2 }$ called the direct sum of the two representations, denoted by $\pi _ { 1 } \oplus \pi _ { 2 }$ . This representation is given by the homomorphism $$ (\pi_ {1...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-02/01-some-representation-theory.md

## 等距变换(isometry)

- UUID: `f2fd5f2e-a289-4b81-842e-25df99d026cf` · okf_url: https://limbo137.top/okf/concepts/f2fd5f2e-a289-4b81-842e-25df99d026cf.md
- labels: en = isometry；zh-CN = 等距变换
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §19.5 · `zh/chapter-19/05-groups-of-isometries#concept-19-isometry` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.5 · `en/chapter-19/05-groups-of-isometries#first-3a409aca97312bf1a0ee-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/05-groups-of-isometries.md

## 玻色子Fock空间(bosonic Fock space)

- UUID: `f307ef9a-a1cb-4747-9a3a-93f549e9f325` · okf_url: https://limbo137.top/okf/concepts/f307ef9a-a1cb-4747-9a3a-93f549e9f325.md
- labels: en = bosonic Fock space；zh-CN = 玻色子Fock空间
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §36.1 · `books/woit-qtgr/2025-10-20/zh/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator#first-ab53fe8616fa2f0759f5-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-36/01-multi-particle-quantum-systems-as-quanta-of-a-harmonic-oscillator.md

## 勒贝格可积(Lebesgue integrable)

- UUID: `f311149d-8641-4993-b10c-1279d63e9f69` · okf_url: https://limbo137.top/okf/concepts/f311149d-8641-4993-b10c-1279d63e9f69.md
- labels: en = Lebesgue integrable；zh-CN = 勒贝格可积
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-6b9402f8e7bface19154-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 主动(active transformation)

- UUID: `f37c195f-7103-4aa4-968c-6fbb8cd23223` · okf_url: https://limbo137.top/okf/concepts/f37c195f-7103-4aa4-968c-6fbb8cd23223.md
- labels: en = active transformation；zh-CN = 主动
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.6 · `zh/chapter-03/06-summation-convention-and-transformation-of-bases#concept-3-active-transformation` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/06-summation-convention-and-transformation-of-bases.md

## 酉变换(unitary transformation)

- UUID: `f3d0516f-16c8-42bf-9b2d-e348d15f5539` · okf_url: https://limbo137.top/okf/concepts/f3d0516f-16c8-42bf-9b2d-e348d15f5539.md
- labels: en = unitary transformation；zh-CN = 酉变换
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §3.1 · `books/woit-qtgr/2025-10-20/zh/chapter-03/01-the-two-state-quantum-system#first-19dbc282fc310e616b43-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-03/01-the-two-state-quantum-system.md

## 四维势(4-potential)

- UUID: `f3f90259-d32d-41f1-9a3b-28ee65093e23` · okf_url: https://limbo137.top/okf/concepts/f3f90259-d32d-41f1-9a3b-28ee65093e23.md
- labels: en = 4-potential；zh-CN = 四维势
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.4 · `zh/chapter-09/04-electrodynamics#first-295659ff0e40e723f497-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.4 · `en/chapter-09/04-electrodynamics#first-295659ff0e40e723f497-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/04-electrodynamics.md

## 伴随算符(Adjoint operator)

- UUID: `f410ba39-f137-4631-8236-e8799213561e` · okf_url: https://limbo137.top/okf/concepts/f410ba39-f137-4631-8236-e8799213561e.md
- labels: en = Adjoint operator；zh-CN = 伴随算符
- aliases: adjoint operator
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §4.5 · `books/woit-qtgr/2025-10-20/zh/chapter-04/05-adjoint-operators#first-b89d65840ee145776b31-zh` · translation)；woit-qtgr/2025-10-20 (en · §4.5 · `books/woit-qtgr/2025-10-20/en/chapter-04/05-adjoint-operators#first-b89d65840ee145776b31-en` · original_transcription)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §4.5 (pdf_page=not-recorded, status=unreviewed, dependency_status=not-recorded) · 原文: `定义（伴随算符(adjoint operator)）。线性算符$L : V \to V$的伴随是满足以下条件的算符$L ^ { \dagger }$ $$ \langle L v, w \rangle = \langle v, L ^ {\dagger} w \rangle $$ 对于所有 $v , w \in V$。`
  - woit-qtgr/2025-10-20 §4.5 (pdf_page=not-recorded, status=ai-transcription-compared, dependency_status=not-recorded) · 原文: `Definition (Adjoint operator). The adjoint of a linear operator $L : V \to V$ is the operator $L ^ { \dagger }$ satisfying $$ \langle L v, w \rangle = \langle v, L ^ {\dagger} w \rangle $$ for all $v , w \in V$.`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-04/05-adjoint-operators.md

## 弗里德曼方程(Friedmann equation)

- UUID: `f43defde-548e-4e2b-8aa8-076312a3487d` · okf_url: https://limbo137.top/okf/concepts/f43defde-548e-4e2b-8aa8-076312a3487d.md
- labels: en = Friedmann equation；zh-CN = 弗里德曼方程
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §18.9 · `zh/chapter-18/09-cosmology#first-b81d2b1d693a8012dec2-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §18.9 · `en/chapter-18/09-cosmology#first-b81d2b1d693a8012dec2-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/09-cosmology.md

## 逆变次数(contravariant degree)

- UUID: `f497b918-3708-409f-b659-4a96fef55088` · okf_url: https://limbo137.top/okf/concepts/f497b918-3708-409f-b659-4a96fef55088.md
- labels: en = contravariant degree；zh-CN = 逆变次数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-6cad317a9e3ca1defea5-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-6cad317a9e3ca1defea5-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 螺旋度(helicity)

- UUID: `f4ddac0c-c765-44bd-b2a9-e5be5de5ed13` · okf_url: https://limbo137.top/okf/concepts/f4ddac0c-c765-44bd-b2a9-e5be5de5ed13.md
- labels: en = helicity；zh-CN = 螺旋度
- 分书定义状态: schwartz-qft/local-experiment = not-recorded；woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §34.2 · `books/woit-qtgr/2025-10-20/zh/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-e405bb29e9ba22b21fa6-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §34.2 · `books/woit-qtgr/2025-10-20/en/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3#first-e405bb29e9ba22b21fa6-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
  - woit-qtgr/2025-10-20 §47.3 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `算符 $$ \frac {1}{2} \frac {\boldsymbol {\sigma} \cdot \mathbf {p}}{| \mathbf {p} |}\tag{47.18} $$`
  - woit-qtgr/2025-10-20 §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: 原有范围、语言、来源类别及审核状态全部保留；本次只确认阅读身份对应，不提升证据资格。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-38/05-spatial-symmetries.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-34/02-solutions-of-the-pauli-equation-and-representations-of-e-3.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-47/05-weyl-spinors.md

## 正洛伦兹变换(proper Lorentz transformation)

- UUID: `f54dfdc2-da4c-4991-a744-ac989d49e9ea` · okf_url: https://limbo137.top/okf/concepts/f54dfdc2-da4c-4991-a744-ac989d49e9ea.md
- labels: en = proper Lorentz transformation；zh-CN = 正洛伦兹变换
- aliases: proper Lorentz transformations
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §9.1 · `zh/chapter-09/01-minkowski-space-time#first-d0b60aabb42cb69cee1d-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §9.1 · `en/chapter-09/01-minkowski-space-time#first-0a4b07842dc5fbeacc2c-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-09/01-minkowski-space-time.md

## 粘合拓扑(identification topology)

- UUID: `f5be2df2-a716-470d-93b8-2043adf53100` · okf_url: https://limbo137.top/okf/concepts/f5be2df2-a716-470d-93b8-2043adf53100.md
- labels: en = identification topology；zh-CN = 粘合拓扑
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-b36dbe6e996de25a22e9-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 逆度规张量(inverse metric tensor)

- UUID: `f61ac22e-d472-4b48-a199-14591405deeb` · okf_url: https://limbo137.top/okf/concepts/f61ac22e-d472-4b48-a199-14591405deeb.md
- labels: en = inverse metric tensor；zh-CN = 逆度规张量
- aliases: 逆度规；逆度量张量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.2 · `zh/chapter-07/02-multilinear-maps-and-tensors#first-2dde5af67305dc9832f3-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-2dde5af67305dc9832f3-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-18/04-pseudo-riemannian-manifolds.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 幺模群(unimodular group)

- UUID: `f64cefe3-d339-4f0e-ae61-35c7b848b740` · okf_url: https://limbo137.top/okf/concepts/f64cefe3-d339-4f0e-ae61-35c7b848b740.md
- labels: en = unimodular group；zh-CN = 幺模群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.3 · `zh/chapter-02/03-matrix-groups#concept-2-unimodular-group` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §19.3 · `en/chapter-19/03-lie-subgroups#first-09ccb6356583aab53039-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-19/03-lie-subgroups.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/03-matrix-groups.md

## 逆变张量(contravariant tensor)

- UUID: `f6bfef51-6914-4e59-85f0-6ef326bbef9d` · okf_url: https://limbo137.top/okf/concepts/f6bfef51-6914-4e59-85f0-6ef326bbef9d.md
- labels: en = contravariant tensor；zh-CN = 逆变张量
- aliases: contravariant tensors
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.1 · `zh/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-8b5a95404e6afec200fa-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.1 · `en/chapter-07/01-free-vector-spaces-and-tensor-spaces#first-cfa6026f59f7138bd267-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/01-free-vector-spaces-and-tensor-spaces.md

## 阿贝尔群中的整系数线性无关(linear independence over the integers)

- UUID: `f70f06da-ee42-4dea-8e56-c3bd034a31c8` · okf_url: https://limbo137.top/okf/concepts/f70f06da-ee42-4dea-8e56-c3bd034a31c8.md
- labels: en = linear independence over the integers；zh-CN = 阿贝尔群中的整系数线性无关
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.2 (pdf_page=115, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书3.1.2在阿贝尔群中以整系数ni判Σnixi=0仅有零解；不是域上向量空间线性无关，不按同名合并。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `form a subgroup of $G$ , which we denote $H$ . $H$ is called a subgroup of $G$ generated by the generators $x_{1},\ldots ,x_{r}$ . If $G$ itself is generated by finite elements $x_{1},\ldots ,x_{r}$ , $G$ is said to be finitely generated. If $n_1x_1 + \dots +n_rx_r = 0$ is satisfied only when $n_1 = \dots = n_r = 0$ , $x_{1},\ldots ,x_{r}$ are said to be ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 向量空间同构(vector space isomorphism)

- UUID: `f72b5959-0f5c-4871-ad33-b1809f9920cb` · okf_url: https://limbo137.top/okf/concepts/f72b5959-0f5c-4871-ad33-b1809f9920cb.md
- labels: en = vector space isomorphism；zh-CN = 向量空间同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.3 · `zh/chapter-03/03-vector-space-homomorphisms#concept-3-vector-space-isomorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/03-vector-space-homomorphisms.md

## 阿贝尔群(abelian group)

- UUID: `f7a3ed41-d4ca-42c5-91ea-dfc1a0dc4b7b` · okf_url: https://limbo137.top/okf/concepts/f7a3ed41-d4ca-42c5-91ea-dfc1a0dc4b7b.md
- labels: en = abelian group；zh-CN = 阿贝尔群
- aliases: commutative；commutative group；交换的；交换群
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-abelian-group` · translation)
- 已审定义(definitions):
  - szekeres/source-snapshot-bc3404156f6e §— (pdf_page=not-recorded, status=codex-source-reviewed; pending-independent-human-review, dependency_status=not-recorded) · 条件: Szekeres 先讨论一般群，再明确加法记号的交换约定。
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 内能(internal energy)

- UUID: `f807579a-57c3-403d-aada-e8bec3d29d6e` · okf_url: https://limbo137.top/okf/concepts/f807579a-57c3-403d-aada-e8bec3d29d6e.md
- labels: en = internal energy；zh-CN = 内能
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-4d1ca1566ddc76193750-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-4d1ca1566ddc76193750-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 升指标(raising the index)

- UUID: `f867d1e9-f2d3-4090-965b-3f76f6179b36` · okf_url: https://limbo137.top/okf/concepts/f867d1e9-f2d3-4090-965b-3f76f6179b36.md
- labels: en = raising the index；zh-CN = 升指标
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §7.4 · `zh/chapter-07/04-operations-on-tensors#first-887edb11b8b1f60cbd2a-zh` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §7.4 · `en/chapter-07/04-operations-on-tensors#first-887edb11b8b1f60cbd2a-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/04-operations-on-tensors.md

## 余切空间(cotangent space)

- UUID: `f871eb88-c0a5-4b52-8d60-f5daef5ddef1` · okf_url: https://limbo137.top/okf/concepts/f871eb88-c0a5-4b52-8d60-f5daef5ddef1.md
- labels: en = cotangent space；zh-CN = 余切空间
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.3 · `en/chapter-15/03-tangent-cotangent-and-tensor-spaces#first-469b2fbb724b75494d6c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.3 · `zh/chapter-15/03-tangent-cotangent-and-tensor-spaces#concept-15-cotangent-space` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/03-tangent-cotangent-and-tensor-spaces.md

## 塞格雷特征(Segré characteristics)

- UUID: `f91d641b-1b8e-4482-9b47-7180cea2ef53` · okf_url: https://limbo137.top/okf/concepts/f91d641b-1b8e-4482-9b47-7180cea2ef53.md
- labels: en = Segré characteristics；zh-CN = 塞格雷特征
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §4.2 · `zh/chapter-04/02-jordan-canonical-form#concept-4-segre-characteristics` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-04/02-jordan-canonical-form.md

## 非退化双线性形式(non-degenerate bilinear form)

- UUID: `f99d2218-e64e-42aa-ac15-b1adf3fb020d` · okf_url: https://limbo137.top/okf/concepts/f99d2218-e64e-42aa-ac15-b1adf3fb020d.md
- labels: en = non-degenerate bilinear form；zh-CN = 非退化双线性形式
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §29.1 · `books/woit-qtgr/2025-10-20/zh/chapter-29/01-non-degenerate-bilinear-forms#first-db368e14c04f50e9d4aa-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §29.1 · `books/woit-qtgr/2025-10-20/en/chapter-29/01-non-degenerate-bilinear-forms#first-db368e14c04f50e9d4aa-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-29/01-non-degenerate-bilinear-forms.md

## 自旋向量可观测量(spin-vector observable)

- UUID: `f9e4f49c-fba8-4cc3-8560-b5dadb3c85b2` · okf_url: https://limbo137.top/okf/concepts/f9e4f49c-fba8-4cc3-8560-b5dadb3c85b2.md
- labels: en = spin-vector observable；zh-CN = 自旋向量可观测量
- aliases: 自旋(spin)向量可观测量
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §7.3 · `books/woit-qtgr/2025-10-20/zh/chapter-07/03-the-heisenberg-picture#first-3cf3a62fadfcf8c4f252-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §7.3 · `books/woit-qtgr/2025-10-20/en/chapter-07/03-the-heisenberg-picture#first-e692888af7a65e614a5e-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-07/03-the-heisenberg-picture.md

## 广义动量(generalized momentum)

- UUID: `fa2ca729-0b3e-4cc8-8d07-b3c46fd2db3f` · okf_url: https://limbo137.top/okf/concepts/fa2ca729-0b3e-4cc8-8d07-b3c46fd2db3f.md
- labels: en = generalized momentum；zh-CN = 广义动量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-6c194196dbcd9b6544b3-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md

## 控制函数(dominating function)

- UUID: `fa326fe3-78cc-44bb-94ae-3aac6356d8e4` · okf_url: https://limbo137.top/okf/concepts/fa326fe3-78cc-44bb-94ae-3aac6356d8e4.md
- labels: en = dominating function；zh-CN = 控制函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.3 · `zh/chapter-11/03-lebesgue-integration#first-4e8c49427d5d544d4f8c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/03-lebesgue-integration.md

## 余维(codimension)

- UUID: `fa3d2ab8-a4c9-4465-af9e-abe1be4c093d` · okf_url: https://limbo137.top/okf/concepts/fa3d2ab8-a4c9-4465-af9e-abe1be4c093d.md
- labels: en = codimension；zh-CN = 余维
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.5 · `zh/chapter-03/05-bases-of-a-vector-space#concept-3-codimension` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/05-bases-of-a-vector-space.md

## 映射的像(image of a mapping)

- UUID: `fab6468b-8179-4027-92d5-5cb6648106f9` · okf_url: https://limbo137.top/okf/concepts/fab6468b-8179-4027-92d5-5cb6648106f9.md
- labels: en = image of a mapping；zh-CN = 映射的像
- aliases: image；image of a map；mapping image；像
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.4 · `zh/chapter-01/04-mappings#first-3bc26d51aea2bffb8857-mapping-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/04-mappings.md

## $\mathfrak{sl}(2,\mathbb C)$ 有限维表示的分类(Classification of finite-dimensional $\mathfrak{sl}(2,\mathbb C)$ representations)

- UUID: `fb37442e-a89f-4d2d-be7e-93cacc21a726` · okf_url: https://limbo137.top/okf/concepts/fb37442e-a89f-4d2d-be7e-93cacc21a726.md
- labels: en = Classification of finite-dimensional $\mathfrak{sl}(2,\mathbb C)$ representations；zh-CN = $\mathfrak{sl}(2,\mathbb C)$ 有限维表示的分类
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §41.1 · `books/woit-qtgr/2025-10-20/zh/chapter-41/01-representations-of-the-lorentz-group#first-a1bb9c82b24af2e5fefb-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §41.1 · `books/woit-qtgr/2025-10-20/en/chapter-41/01-representations-of-the-lorentz-group#first-d95e1eb1ef58beadaad1-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-41/01-representations-of-the-lorentz-group.md

## 绝对温度(absolute temperature)

- UUID: `fbc344e8-0790-492c-b854-78bb211050ff` · okf_url: https://limbo137.top/okf/concepts/fbc344e8-0790-492c-b854-78bb211050ff.md
- labels: en = absolute temperature；zh-CN = 绝对温度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.4 · `en/chapter-16/04-thermodynamics#first-0a79094078355dd0e41c-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.4 · `zh/chapter-16/04-thermodynamics#first-0a79094078355dd0e41c-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/04-thermodynamics.md

## 齐次多项式(Homogeneous polynomials)

- UUID: `fbc5dab1-019e-483b-ba02-749641eeca97` · okf_url: https://limbo137.top/okf/concepts/fbc5dab1-019e-483b-ba02-749641eeca97.md
- labels: en = Homogeneous polynomials；zh-CN = 齐次多项式
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §8.2 · `books/woit-qtgr/2025-10-20/zh/chapter-08/04-representations-of-su2-construction#first-34a5a36c6ba2d4c60d80-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §8.2 · `books/woit-qtgr/2025-10-20/en/chapter-08/04-representations-of-su2-construction#first-babda79b00c70739df3c-en` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §8.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `次数为 $n$ 的齐次多项式在两个复变量 $z _ { 1 } , z _ { 2 }$ 中的复向量空间是 $\mathbf { C } ^ { 2 }$ 上具有以下形式的函数空间 $$ f (z _ {1}, z _ {2}) = a _ {0} z _ {1} ^ {n} + a _ {1} z _ {1} ^ {n - 1} z _ {2} + \dots + a _ {n - 1} z _ {1} z _ {2} ^ {n - 1} + a _ {n} z _ {2} ^ {n} $$`
  - woit-qtgr/2025-10-20 §8.2 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The complex vector space of homogeneous polynomials of degree $n$ in two complex variables $z _ { 1 } , z _ { 2 }$ is the space of functions on $\mathbf { C } ^ { 2 }$ of the form $$ f (z _ {1}, z _ {2}) = a _ {0} z _ {1} ^ {n} + a _ {1} z _ {1} ^ {n - 1} z _ {2} + \dots + a _ {n - 1} z _ {1} z _ {2} ^ {n - 1} + a _ {n} z _ {2} ^ {n} $$`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-08/04-representations-of-su2-construction.md

## 稳定子群或小群(Stabilizer group or little group)

- UUID: `fbccb3cc-baae-4284-a87f-3eb0e06d6070` · okf_url: https://limbo137.top/okf/concepts/fbccb3cc-baae-4284-a87f-3eb0e06d6070.md
- labels: en = Stabilizer group or little group；zh-CN = 稳定子群或小群
- 分书定义状态: woit-qtgr/2025-10-20 = recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (en · §20.4 · `books/woit-qtgr/2025-10-20/en/chapter-20/08-representations-of-nk-n-commutative#first-2112aa74fb08bb842825-en` · machine_draft)；woit-qtgr/2025-10-20 (zh-CN · §20.4 · `books/woit-qtgr/2025-10-20/zh/chapter-20/08-representations-of-nk-n-commutative#first-e1af6d61cef320fe0c00-zh` · machine_draft)
- 已审定义(definitions):
  - woit-qtgr/2025-10-20 §20.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `对给定的 $\alpha\in\widehat{N}$，由满足下式的元素 $k\in K$ 构成的子群 $K_\alpha\subset K$，称为稳定子群（数学上的叫法）或小群（物理学上的叫法）： $$ \widehat{\Phi}_k(\alpha)=\alpha. $$`
  - woit-qtgr/2025-10-20 §20.4 (pdf_page=not-recorded, status=machine-draft-not-reviewed, dependency_status=not-recorded) · 原文: `The subgroup $K _ { \alpha } \subset K$ of elements $k \in K$ such that $$ \widehat {\Phi} _ {k} (\alpha) = \alpha $$ for a given $\alpha \in \widehat { N }$ is called the stabilizer subgroup (by mathematicians) or little subgroup (by physicists).`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-20/08-representations-of-nk-n-commutative.md

## 西尔维斯特惯性定律(Sylvester’s law of inertia)

- UUID: `fc0ae0df-4043-4beb-be49-1664bb2bfb3c` · okf_url: https://limbo137.top/okf/concepts/fc0ae0df-4043-4beb-be49-1664bb2bfb3c.md
- labels: en = Sylvester’s law of inertia；zh-CN = 西尔维斯特惯性定律
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §5.1 · `en/chapter-05/01-real-inner-product-spaces#first-777e78796e7eacd431b7-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.1 · `zh/chapter-05/01-real-inner-product-spaces#first-777e78796e7eacd431b7-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/01-real-inner-product-spaces.md

## 有限生成阿贝尔群(finitely generated Abelian group)

- UUID: `fc4d4ed6-1e21-45ac-81f8-30621814489a` · okf_url: https://limbo137.top/okf/concepts/fc4d4ed6-1e21-45ac-81f8-30621814489a.md
- labels: en = finitely generated Abelian group；zh-CN = 有限生成阿贝尔群
- aliases: finitely generated
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = recorded
- 首次出现(first_mentions): not-recorded
- 已审定义(definitions):
  - nakahara-geometry-topology-physics/second-edition-local §3.1.2 (pdf_page=115, status=librarian-original-source-reviewed, dependency_status=not-recorded) · 条件: 原书3.1.2中由有限多个元素整系数生成的阿贝尔群；与自由阿贝尔群及其结构定理不同。管理员核对的原书定义/陈述；不声明首次出现，不把Theorem环境伪作Definition编号。 · 原文: `form a subgroup of $G$ , which we denote $H$ . $H$ is called a subgroup of $G$ generated by the generators $x_{1},\ldots ,x_{r}$ . If $G$ itself is generated by finite elements $x_{1},\ldots ,x_{r}$ , $G$ is said to be finitely generated. If $n_1x_1 + \dots +n_rx_r = 0$ is satisfied only when $n_1 = \dots = n_r = 0$ , $x_{1},\ldots ,x_{r}$ are said to be ...`
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）

## 柯西序列(Cauchy sequence)

- UUID: `fc7ee2f4-4b46-4d4c-9f66-cdc82f1fb7ec` · okf_url: https://limbo137.top/okf/concepts/fc7ee2f4-4b46-4d4c-9f66-cdc82f1fb7ec.md
- labels: en = Cauchy sequence；zh-CN = 柯西序列
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.3 · `zh/chapter-10/03-metric-spaces#first-428a9830fa9dfd65cd14-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/03-metric-spaces.md

## 除代数(division algebra)

- UUID: `fcfb759a-7e9c-4fdf-aafb-dc2c25046515` · okf_url: https://limbo137.top/okf/concepts/fcfb759a-7e9c-4fdf-aafb-dc2c25046515.md
- labels: en = division algebra；zh-CN = 除代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.2 · `zh/chapter-06/02-complex-numbers-and-complex-structures#concept-6-division-algebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.2 · `en/chapter-06/02-complex-numbers-and-complex-structures#first-8ae0cb960e45f4773fdc-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/02-complex-numbers-and-complex-structures.md

## 费米子数算符(fermion number operator)

- UUID: `fd3ec660-2fd6-44c1-9079-3606dad2ac73` · okf_url: https://limbo137.top/okf/concepts/fd3ec660-2fd6-44c1-9079-3606dad2ac73.md
- labels: en = fermion number operator；zh-CN = 费米子数算符
- 分书定义状态: woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): woit-qtgr/2025-10-20 (zh-CN · §27.1 · `books/woit-qtgr/2025-10-20/zh/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator#first-f8f6aaa7138d2b4e5f5a-zh` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-27/01-canonical-anticommutation-relations-and-the-fermionic-oscillator.md

## 标量(scalar)

- UUID: `fd5c405e-c47e-4504-8fe9-fd9eaa0ea089` · okf_url: https://limbo137.top/okf/concepts/fd5c405e-c47e-4504-8fe9-fd9eaa0ea089.md
- labels: en = scalar；zh-CN = 标量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §3.2 · `zh/chapter-03/02-vector-spaces#concept-3-scalar` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-03/02-vector-spaces.md

## 正交补(orthogonal complement)

- UUID: `fdb013fe-7377-4518-9301-f74c3a834eca` · okf_url: https://limbo137.top/okf/concepts/fdb013fe-7377-4518-9301-f74c3a834eca.md
- labels: en = orthogonal complement；zh-CN = 正交补
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §5.3 · `zh/chapter-05/03-representations-of-finite-groups#concept-5-orthogonal-complement` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §13.3 · `en/chapter-13/03-linear-functionals#first-2cc5aa0ef8a38a55799f-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-13/03-linear-functionals.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-05/03-representations-of-finite-groups.md

## 互补可观测量(complementary observable)

- UUID: `fdc31736-bf4c-4021-8259-dc9238e311cc` · okf_url: https://limbo137.top/okf/concepts/fdc31736-bf4c-4021-8259-dc9238e311cc.md
- labels: en = complementary observable；zh-CN = 互补可观测量
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §14.1 · `zh/chapter-14/01-basic-concepts#first-f00a3afbf7a562fb6077-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-14/01-basic-concepts.md

## 内自同构(inner automorphism)

- UUID: `fe4e122c-475a-40c8-9152-313db9f50745` · okf_url: https://limbo137.top/okf/concepts/fe4e122c-475a-40c8-9152-313db9f50745.md
- labels: en = inner automorphism；zh-CN = 内自同构
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.4 · `zh/chapter-02/04-homomorphisms-and-isomorphisms#concept-2-inner-automorphism` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/04-homomorphisms-and-isomorphisms.md

## 等价类(equivalence class)

- UUID: `fe80fb2d-f836-47e5-9297-3c634d6df6af` · okf_url: https://limbo137.top/okf/concepts/fe80fb2d-f836-47e5-9297-3c634d6df6af.md
- labels: en = equivalence class；zh-CN = 等价类
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = candidate-only；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §1.3 · `zh/chapter-01/03-products-and-relations#first-59f17f4ed9db05598cbf-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-01/03-products-and-relations.md

## 多线性的(multilinear)

- UUID: `fe88d3de-93f7-4d30-842e-252bfa61c2ec` · okf_url: https://limbo137.top/okf/concepts/fe88d3de-93f7-4d30-842e-252bfa61c2ec.md
- labels: en = multilinear；zh-CN = 多线性的
- aliases: 多线性
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §7.2 · `en/chapter-07/02-multilinear-maps-and-tensors#first-ab2593ca5b40dd94cf25-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-07/02-multilinear-maps-and-tensors.md

## 概率测度(probability measure)

- UUID: `fe8f24c1-1db7-4f49-8858-a645d47c4c7f` · okf_url: https://limbo137.top/okf/concepts/fe8f24c1-1db7-4f49-8858-a645d47c4c7f.md
- labels: en = probability measure；zh-CN = 概率测度
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §11.2 · `zh/chapter-11/02-measure-spaces#first-8023c8f2d22ec9489962-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-11/02-measure-spaces.md

## 推迟格林函数(retarded Green’s function)

- UUID: `fea39ea6-6745-42be-a5cf-5ad1ad507793` · okf_url: https://limbo137.top/okf/concepts/fea39ea6-6745-42be-a5cf-5ad1ad507793.md
- labels: en = retarded Green’s function；zh-CN = 推迟格林函数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.4 · `zh/chapter-12/04-green-s-functions#first-0c6a5a0c9b978798f3ad-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/04-green-s-functions.md

## 外代数(exterior algebra)

- UUID: `fec421d2-e46d-4258-acc3-ef2dfb231874` · okf_url: https://limbo137.top/okf/concepts/fec421d2-e46d-4258-acc3-ef2dfb231874.md
- labels: en = exterior algebra；zh-CN = 外代数
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §6.4 · `zh/chapter-06/04-grassmann-algebras#concept-6-exterior-algebra` · translation)；szekeres/source-snapshot-bc3404156f6e (en · §6.4 · `en/chapter-06/04-grassmann-algebras#first-6cac602420dc2ecf58ec-en` · original_transcription)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-08/03-exterior-product.md；https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-06/04-grassmann-algebras.md

## 环面(torus)

- UUID: `fee87ca7-4cd9-4a1f-ab6b-9f29f4e871d4` · okf_url: https://limbo137.top/okf/concepts/fee87ca7-4cd9-4a1f-ab6b-9f29f4e871d4.md
- labels: en = torus；zh-CN = 环面
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §10.4 · `zh/chapter-10/04-induced-topologies#first-e32baf3c671c03264106-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-10/04-induced-topologies.md

## 作用量(action)

- UUID: `ff25d777-490f-4ae6-b7f9-2071832df195` · okf_url: https://limbo137.top/okf/concepts/ff25d777-490f-4ae6-b7f9-2071832df195.md
- labels: en = action；zh-CN = 作用量
- 分书定义状态: nakahara-geometry-topology-physics/second-edition-local = not-recorded；szekeres/source-snapshot-bc3404156f6e = not-recorded；woit-qtgr/2025-10-20 = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §16.5 · `en/chapter-16/05-classical-mechanics#first-bee57b2516883280ed58-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §16.5 · `zh/chapter-16/05-classical-mechanics#first-bee57b2516883280ed58-zh` · translation)；woit-qtgr/2025-10-20 (zh-CN · §35.1 · `books/woit-qtgr/2025-10-20/zh/chapter-35/01-lagrangian-mechanics#first-57bd7d538c682f8ac524-zh` · machine_draft)；woit-qtgr/2025-10-20 (en · §35.1 · `books/woit-qtgr/2025-10-20/en/chapter-35/01-lagrangian-mechanics#first-57bd7d538c682f8ac524-en` · machine_draft)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-16/05-classical-mechanics.md；https://limbo137.top/okf/books/woit-qtgr/2025-10-20/chapter-35/01-lagrangian-mechanics.md

## 循环群(cyclic group)

- UUID: `ff7730a6-70b3-41c7-82df-49ecbed5b2ad` · okf_url: https://limbo137.top/okf/concepts/ff7730a6-70b3-41c7-82df-49ecbed5b2ad.md
- labels: en = cyclic group；zh-CN = 循环群
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.1 · `zh/chapter-02/01-elements-of-group-theory#concept-2-cyclic-group` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/01-elements-of-group-theory.md

## 正则嵌入(regular embedding)

- UUID: `ff9a93aa-d509-45d6-9110-f7207970069a` · okf_url: https://limbo137.top/okf/concepts/ff9a93aa-d509-45d6-9110-f7207970069a.md
- labels: en = regular embedding；zh-CN = 正则嵌入
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (en · §15.4 · `en/chapter-15/04-tangent-map-and-submanifolds#first-7bd5b812501557d1ecd8-en` · original_transcription)；szekeres/source-snapshot-bc3404156f6e (zh-CN · §15.4 · `zh/chapter-15/04-tangent-map-and-submanifolds#concept-15-regular-embedding` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-15/04-tangent-map-and-submanifolds.md

## 出射波条件(outgoing wave condition)

- UUID: `ff9f20ff-7034-4ebd-b37a-fa0a5de1c451` · okf_url: https://limbo137.top/okf/concepts/ff9f20ff-7034-4ebd-b37a-fa0a5de1c451.md
- labels: en = outgoing wave condition；zh-CN = 出射波条件
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §12.4 · `zh/chapter-12/04-green-s-functions#first-86476f24d7196bf84708-zh` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-12/04-green-s-functions.md

## 左作用(left action)

- UUID: `ffc34fd9-952c-491c-a016-5b629c2b2135` · okf_url: https://limbo137.top/okf/concepts/ffc34fd9-952c-491c-a016-5b629c2b2135.md
- labels: en = left action；zh-CN = 左作用
- 分书定义状态: szekeres/source-snapshot-bc3404156f6e = not-recorded
- 首次出现(first_mentions): szekeres/source-snapshot-bc3404156f6e (zh-CN · §2.6 · `zh/chapter-02/06-group-actions#concept-2-left-action` · translation)
- 已审定义(definitions): not-recorded（当前已登记来源未提供正式定义记录；未断言原书没有定义）
- 显式依赖组(dependency_groups): not-recorded（未登记显式作者陈述依赖组；回答时必须声明未登记，不得将正文推导脉络混充已登记 dependency_groups）
- 原文小节包: https://limbo137.top/okf/books/szekeres/source-snapshot-bc3404156f6e/chapter-02/06-group-actions.md