{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "labels": { "zh-CN": "正规序", "en": "normal ordering" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "f047d38c3eb1fdaebbb82184cae7006495ad9b8889d971094f2b1566a1dd220b", "occurrences": [ { "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/index", "anchor": "first-b89fd35829a573405729-zh", "chapter": "25", "section": "25", "offset": 775, "range": [ 7965, 8272 ], "source_revision": "91f39772e5e3c858c5ace0e20ab58d838e6a1cb7c455bdd074ffea132432c699", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "正规序(normal ordering)", "context": { "range": [ 7414, 8407 ], "quote": "本章把第24章的 $d=1$ 情形推广到任意 $d$。选取 $d$ 个湮灭算符和 $d$ 个产生算符，会突出子群 $U(d)\\subset Sp(2d,\\mathbb R)$：其中的变换不混合湮灭算符与产生算符。[元辛表示(metaplectic representation)](../../../../../concepts/representation-theory/metaplectic-representation)给出该子群的一个双覆盖表示。我们将看到，对湮灭算符与产生算符的乘积作[正规序(normal ordering)](../../../../../concepts/quantum-mechanics/normal-ordering)后，得到的是 $U(d)$ 本身的表示，而非其双覆盖群的表示。由此，相空间上 $U(d)$ 的作用在 $d$ 维谐振子的态空间上诱导出一个无限维表示。", "quote_sha256": "82971a53b558f130838741d9b1823fa519efb027665c8443b229c6c2a26c924a", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-8986234780f8fb6b513e68a6" }, { "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index", "anchor": "first-b89fd35829a573405729-en", "chapter": "25", "section": "25", "offset": 1051, "range": [ 7472, 7774 ], "source_revision": "a4fabc3ade7e18eac393871eaed9571dfcf75d61baf7220e6eb15075ae13412d", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "normal ordering", "context": { "range": [ 6709, 8127 ], "quote": "This chapter generalizes the $d=1$ case of Chapter 24 to arbitrary $d$. Choosing $d$ annihilation operators and $d$ creation operators singles out a distinguished subgroup $U(d)\\subset Sp(2d,\\mathbb R)$: the transformations that do not mix annihilation and creation operators. The [metaplectic representation](../../../../../concepts/representation-theory/metaplectic-representation) gives a representation of a double cover of this subgroup. We will see that [normal ordering](../../../../../concepts/quantum-mechanics/normal-ordering) products of annihilation and creation operators yields a representation of $U(d)$ itself, rather than of the double cover. Thus the action of $U(d)$ on finite-dimensional phase space induces an infinite-dimensional representation on the state space of the $d$-dimensional harmonic oscillator.", "quote_sha256": "992e865c17638eca517475a6cd0d7850b409044499039055aa51975115b47f97", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/index", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-989ee41fe6c19a607c495051" }, { "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "anchor": "section-26-6", "chapter": "26", "section": "26.6", "offset": 371, "range": [ 7363, 7635 ], "source_revision": "c3a9cc96f7c6dbef555811a2ad5bd45eee8efdc63abedfe8145b6bf5225a5614", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "正规序(normal ordering)", "context": { "range": [ 7236, 7724 ], "quote": "- 对 $\\mathcal M\\otimes\\mathbb C$ 中元素的量子化，分为来自 $\\mathcal M_J^+$ 的产生算符和来自 $\\mathcal M_J^-$ 的湮灭算符。相应地，由这类算符相乘得到的算符 $\\mathcal O$ 的[正规序(normal ordering)](../../../../../concepts/quantum-mechanics/normal-ordering)也依赖于 $J$，记为 $:\\mathcal O:_J$。", "quote_sha256": "cb79680e9b454745ab37131bac7c33024a40e59edc7caaf037bc9194a49bac6d", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-c5f12bfa2a87626b2e0ad2fd" }, { "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "anchor": "section-26-6", "chapter": "26", "section": "26.6", "offset": 672, "range": [ 6806, 7073 ], "source_revision": "1dabffb388c16dfb4234ff6292dfa54cae2c028fa1903605632c56764979b3da", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "normal ordering", "context": { "range": [ 6608, 7206 ], "quote": "- As a division of the quantizations of $\\mathcal M\\otimes\\mathbb C$ into creation operators from $\\mathcal M_J^+$ and annihilation operators from $\\mathcal M_J^-$. The corresponding $J$-dependent [normal ordering](../../../../../concepts/quantum-mechanics/normal-ordering) of a product $\\mathcal O$ of such operators is denoted $:\\mathcal O:_J$.", "quote_sha256": "97cca05bd6f7c7eca17406f2dfdc1df87be1276d57b86ba286199eff53038a4e", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-ec9380efb3f9546f84222d88" }, { "concept_id": "urn:uuid:43457138-2721-4162-870f-85006ba3e6be", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras", "anchor": "first-f5ce9dbff97d7055f7b0-zh", "chapter": "28", "section": "28.1", "offset": 1888, "range": [ 9497, 9769 ], "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "正规序(normal ordering)", "context": { "range": [ 9483, 9837 ], "quote": "这个代数的任意元素都可写为[正规序(normal ordering)](../../../../../concepts/quantum-mechanics/normal-ordering)单项式之和，形如", "quote_sha256": "71668ada2f480f6aba09047d4a107127a44bd7c4d14ebe37125bc82b84c3dffd", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-636c88f455f23c30867d86c0" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index", "lang": "en", "section": "25", "signal": "term-mark", "label": "normal ordering", "english": "normal ordering", "offset": 860 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/index", "lang": "zh-CN", "section": "25", "signal": "term-mark", "label": "正规序", "english": "normal ordering", "offset": 584 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "lang": "en", "section": "26.6", "signal": "term-mark", "label": "normal ordering", "english": "normal ordering", "offset": 672 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-26/06-complex-structures-and-bargmann-fock-quantization-for-arbitrary-d", "lang": "zh-CN", "section": "26.6", "signal": "term-mark", "label": "正规序", "english": "normal ordering", "offset": 371 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras", "lang": "zh-CN", "section": "28.1", "signal": "term-mark", "label": "正规序", "english": "normal ordering", "offset": 1315 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-8986234780f8fb6b513e68a6", "occ-989ee41fe6c19a607c495051", "occ-c5f12bfa2a87626b2e0ad2fd", "occ-ec9380efb3f9546f84222d88", "occ-636c88f455f23c30867d86c0" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 5, "located_occurrences": 5, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "1f751a204299664a83c53ddb8996f2fad080949af4d3851f8ca8334b5b0600c5", "optional_enrichments": [] }