{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:676173e7-6104-41f9-b2f2-95d235a3a95c", "labels": { "zh-CN": "Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示", "en": "Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "4ba8c91f65655f68b3939085c18c18c6046a77557609e7d715fd49236cd718a4", "occurrences": [ { "concept_id": "urn:uuid:676173e7-6104-41f9-b2f2-95d235a3a95c", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-22/03-the-bargmann-fock-representation", "anchor": "first-bb4b3a2dd83c090bc814-zh", "chapter": "22", "section": "22.3", "offset": 326, "range": [ 6816, 7358 ], "source_revision": "926d95c2d6151adee1dd616c64a59a054392c02d0923410b8acc0825e4d8fcd1", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-22/03-the-bargmann-fock-representation", "lang": "zh-CN", "section": "22.3", "signal": "named-definition", "label": "Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示", "english": "", "offset": 256 } }, "label": "> [!definition] 定义（Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示）", "context": { "range": [ 6816, 7358 ], "quote": "\n\n> [!definition] 定义（Bargmann–Fock 表示，也称振子表示、全纯表示或 Segal–Shale–Weil 表示）\n> 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\n>\n> \n>\n> $$\n> \\langle\\psi_1|\\psi_2\\rangle=\\frac{1}{\\pi}\\int_{\\mathbb{C}}\\overline{\\psi_1(z)}\\psi_2(z)e^{-|z|^2}\\,d^2z\\tag{22.4}\n> $$\n>\n> 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：\n>\n> $$\n> a=\\frac{d}{dz},\\qquad a^\\dagger=z\n> $$\n", "quote_sha256": "33cb567e1662e58804df2f7fe007f3c8b0488c7408049e0096c8908ad5e15763", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-22/03-the-bargmann-fock-representation", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-35e11f71b23828ea592e6026" }, { "concept_id": "urn:uuid:676173e7-6104-41f9-b2f2-95d235a3a95c", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-22/03-the-bargmann-fock-representation", "anchor": "first-bfc0d9585e87a8beeba5-en", "chapter": "22", "section": "22.3", "offset": 642, "range": [ 6284, 7028 ], "source_revision": "0c0f0c6be497d20d42817864c0ddbc6e601be69aeea39b4e2d25110a6e58b6c8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "definition", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/en/chapter-22/03-the-bargmann-fock-representation", "lang": "en", "section": "22.3", "signal": "named-definition", "label": "Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation", "english": "Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation", "offset": 572 } }, "label": "> [!definition] Definition (Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation)", "context": { "range": [ 6284, 7028 ], "quote": "\n\n> [!definition] Definition (Bargmann–Fock, oscillator, holomorphic, or Segal–Shale–Weil representation)\n> 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\n>\n> \n>\n> $$\n> \\langle\\psi_1|\\psi_2\\rangle=\\frac{1}{\\pi}\\int_{\\mathbb{C}}\\overline{\\psi_1(z)}\\psi_2(z)e^{-|z|^2}\\,d^2z\\tag{22.4}\n> $$\n>\n> where $d^2z=d\\operatorname{Re}(z)\\,d\\operatorname{Im}(z)$. The space $\\mathcal{F}$ is sometimes called “Fock space”. 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