{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "labels": { "zh-CN": "复结构", "en": "complex structure" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "8415f2b63aac74c8ea08b6a642b788dfe832e4cdab568d2b810666fafd3216f0", "occurrences": [ { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-06/02-complex-numbers-and-complex-structures", "anchor": "concept-6-complex-structure", "chapter": "6", "section": "6.2", "offset": 6369, "range": [ 11055, 11294 ], "source_revision": "f73a45789ab5c5b94bada50b10b7575b97ab50a7f6713e04b91d6862a0f75302", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "复结构(complex structure)", "context": { "range": [ 10968, 11368 ], "quote": "若 $V$ 是实向量空间，则任何满足 $J ^ { 2 } = - \\mathrm { i d } _ { V }$ 的算符 $J : V \\to V$ 称为 $V$ 上的 复结构(complex structure)。复结构 $J$ 可用于将 $V$ 转换为复向量空间 $V _ { J }$，其方法是像在实空间 $V$ 中那样定义向量加法 $u + v$，并通过", "quote_sha256": "10b25518dbf5d955084a08df29512605c094612330154152f3f166074c5f7565", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-06/02-complex-numbers-and-complex-structures", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-20caa989da1b6fcb88be57ff" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-06/02-complex-numbers-and-complex-structures", "anchor": "first-cc6b390cf641456f2750-en", "chapter": "6", "section": "6.2", "offset": 9200, "range": [ 13378, 13614 ], "source_revision": "217ba38664e37b0a5289bc0b639ada0b0059e26f3ae4d9288f83962db4b702ce", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "complex structure", "context": { "range": [ 13258, 13849 ], "quote": "If $V$ is a real vector space, any operator $J : V \\to V$ such that $J ^ { 2 } = - \\mathrm { i d } _ { V }$ is called a complex structure on $V$. A complex structure $J$ can be used to convert $V$ into a complex vector space $V _ { J }$ by defining addition of vectors $u + v$ just as in the real space $V$ , and scalar multiplication of vectors by complex numbers through", "quote_sha256": "895f4a7b62fc5542730f330e84235f6addc446f11fe785976f977931e5edd779", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-06/02-complex-numbers-and-complex-structures", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-fc7e2e52015dac46630df885" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/index", "anchor": "first-da487ccbf2475101f64c-zh", "chapter": "26", "section": "26", "offset": 197, "range": [ 6526, 6834 ], "source_revision": "bae3b5f47bcfe5b25c7b112420d2cba35137fe77eb3c573b5447d04e05efe213", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "复结构(complex structure)", "context": { "range": [ 6329, 7054 ], "quote": "\n$H_{2d+1}$ 的薛定谔表示 $\\Gamma_S$ 使用了经典相空间上的一种额外结构：把坐标分为位置 $q_j$ 和动量 $p_j$。与它酉等价的 Bargmann–Fock 表示需要另一种额外结构，即把相空间坐标分为复坐标 $z_j$ 及其复共轭 $\\bar z_j$。这种分解称为[复结构(complex structure)](../../../../../concepts/linear-algebra/complex-structure) $J$；量子化后，它对应于区分湮灭算符与产生算符的一种选择。此前各章采用了标准选择 $J=J_0$，本章将介绍其他选择。每一种选择都会给出海森堡群表示的 Bargmann–Fock 构造的一个版本 $\\Gamma_J$。后文讨论相对论量子场论时，我们会看到，反粒子现象最好通过该情形下复结构 $J$ 的一种新选择来理解。", "quote_sha256": "6303d7a9ad743c2d15c23388af13fa30d1bf05f8a78d3bee1f67e343773cf273", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/index", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d04916883502c7a950a0e932" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-26/index", "anchor": "first-da487ccbf2475101f64c-en", "chapter": "26", "section": "26", "offset": 478, "range": [ 6001, 6304 ], "source_revision": "569618b8aad3527cbd729faee2dc173edcf9da545cdaf341438fe6a4cb1f6f5f", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "complex structure", "context": { "range": [ 5523, 6849 ], "quote": "\nThe Schrödinger representation $\\Gamma_S$ of $H_{2d+1}$ uses a specific choice of extra structure on classical phase space: a decomposition of its coordinates into positions $q_j$ and momenta $p_j$. For the unitarily equivalent Bargmann–Fock representation a different sort of extra structure is needed, a decomposition into complex coordinates $z_j$ and their complex conjugates $\\bar z_j$. Such a decomposition is called a “[complex structure](../../../../../concepts/linear-algebra/complex-structure)” $J$, and corresponds after quantization to a choice that distinguishes annihilation and creation operators. In previous chapters we used the standard choice $J=J_0$; this chapter describes other choices. Each choice gives a different version $\\Gamma_J$ of the Bargmann–Fock construction of a Heisenberg group representation. In later chapters on relativistic quantum field theory, we will see that antiparticles are best understood in terms of a new possibility for the choice of $J$.", "quote_sha256": "c48f25db2075c06fd1c0f4b24e48962e5e93d4594acc6d41f27527a04013bb56", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/index", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-550e25a18cfba9f6285146a8" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/01-complex-structures-and-phase-space", "anchor": "source-26-1-complex-structure", "chapter": "26", "section": "26.1", "offset": 1596, "range": [ 8799, 9072 ], "source_revision": "fa67bcf044e1ae28d407264338d01ba35e61e76b6dd80a47c6ccb564e466b727", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "复结构(complex structure)", "context": { "range": [ 8778, 9201 ], "quote": "\n> [!definition] 定义（[复结构(complex structure)](../../../../../concepts/linear-algebra/complex-structure)）\n> 实向量空间 $V$ 上的复结构是一个线性算符 $J:V\\to V$，满足\n>\n> $$\n> J^2=-\\mathbf1.\n> $$\n", "quote_sha256": "42db3c69fd812b78db92279ed5a02ece5ece94a878f2070a782dd8d2a3536d49", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/01-complex-structures-and-phase-space", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-567643a7b689eb64a26c5c0c" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-26/01-complex-structures-and-phase-space", "anchor": "source-26-1-complex-structure", "chapter": "26", "section": "26.1", "offset": 2624, "range": [ 8988, 9256 ], "source_revision": "3bb7a27e4592fbdcf5e39bd60a7e02773abc88ac53672e499d1d531982fae242", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "complex structure", "context": { "range": [ 8958, 9437 ], "quote": "\n> [!definition] Definition ([complex structure](../../../../../concepts/linear-algebra/complex-structure))\n> A complex structure on a real vector space $V$ is a linear operator $J:V\\to V$ such that\n>\n> $$\n> J^2=-\\mathbf1.\n> $$\n", "quote_sha256": "2556864119bf4d859ef27bb8a9b5795088ae18aa8b6e3cd66c5f8693b3d9b8c9", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/01-complex-structures-and-phase-space", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b382d78c25e90cc89980db0a" }, { "concept_id": "urn:uuid:9e862121-659d-4ec4-8d9f-204bdd709d9f", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation", "anchor": "xref-05-complex-structures-u-d-so-2-d-and-the-spinor-representation-2", "chapter": "31", "section": "31.4", "offset": 1090, "range": [ 7965, 8238 ], "source_revision": "daf0b01d6904e9be0a685a9af91464a200fd942d9951f2dad5f90816b7a4109d", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "complex structure", "context": { "range": [ 7091, 8419 ], "quote": "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 Bargmann-Fock case of section 26.1. The diference here is that, for the analogous construction of spinors, the [complex structure](../../../../../concepts/geometry/orthogonal-complex-structure) $J$ must be chosen so as to preserve not an antisymmetric bilinear form $\\Omega$, but the inner product, and one has", "quote_sha256": "613bba33fde2d39392cdf4c632d0edad0d87d11e586de352e750ed31ff0f7b4d", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-abe4d11e4cba193f0b3d68a4" } ], "registered_positions": [ { "route": "en/chapter-06/02-complex-numbers-and-complex-structures", "lang": "en", "section": "6.2", "signal": "term-mark", "label": "complex structure", "english": "complex structure", "offset": 8818 }, { "route": "zh/chapter-06/02-complex-numbers-and-complex-structures", "lang": "zh-CN", "section": "6.2", "signal": "term-mark", "label": "复结构", "english": "complex structure", "offset": 6057 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/index", "lang": "en", "section": "26", "signal": "term-mark", "label": "complex structure", "english": "complex structure", "offset": 478 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/index", "lang": "zh-CN", "section": "26", "signal": "term-mark", "label": "复结构", "english": "complex structure", "offset": 197 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/01-complex-structures-and-phase-space", "lang": "en", "section": "26.1", "signal": "term-mark", "label": "complex structure", "english": "complex structure", "offset": 2624 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/01-complex-structures-and-phase-space", "lang": "zh-CN", "section": "26.1", "signal": "term-mark", "label": "复结构", "english": "complex structure", "offset": 1440 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation", "lang": "en", "section": "31.4", "signal": "term-mark", "label": "complex structure", "english": "complex structure", "offset": 1090 } ], "statements": [], "resolved_references": [ { "raw": "href=\"04-spinors-and-the-bargmann-fock-construction#equation-31-2\"", "href": "04-spinors-and-the-bargmann-fock-construction#equation-31-2", "range": [ 7242, 7308 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-31/04-spinors-and-the-bargmann-fock-construction", "target_anchor": "equation-31-2", "resolved": true, "reason": null, "source_revision": "daf0b01d6904e9be0a685a9af91464a200fd942d9951f2dad5f90816b7a4109d", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation", "occurrence_id": "occ-abe4d11e4cba193f0b3d68a4" }, { "raw": "href=\"../chapter-26/01-complex-structures-and-phase-space#section-26-1\"", "href": "../chapter-26/01-complex-structures-and-phase-space#section-26-1", "range": [ 7670, 7741 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-26/01-complex-structures-and-phase-space", "target_anchor": "section-26-1", "resolved": true, "reason": null, "source_revision": "daf0b01d6904e9be0a685a9af91464a200fd942d9951f2dad5f90816b7a4109d", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation", "occurrence_id": "occ-abe4d11e4cba193f0b3d68a4" } ], "reading_order": [ "occ-20caa989da1b6fcb88be57ff", "occ-fc7e2e52015dac46630df885", "occ-d04916883502c7a950a0e932", "occ-550e25a18cfba9f6285146a8", "occ-567643a7b689eb64a26c5c0c", "occ-b382d78c25e90cc89980db0a", "occ-abe4d11e4cba193f0b3d68a4" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 7, "located_occurrences": 7, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "be0c4ebceaed545361566dfef4b2c837ae6c9dcd797d2eefa8f48bd56e0e63f0", "optional_enrichments": [] }