{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "labels": { "zh-CN": "反常", "en": "anomaly" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "a26b6842978fda4979d7f8a7d1cb33853575f3e5f3aa3938422ed5c23624c0dc", "occurrences": [ { "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map", "anchor": "first-b8882cd26c0201b88129-en", "chapter": "15", "section": "15.3", "offset": 3702, "range": [ 9744, 10043 ], "source_revision": "c595b226b342aecf332dd12f82facbf7978a9a402b70e1f0b9b34dc4488d544d", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "anomaly", "context": { "range": [ 9473, 10110 ], "quote": "is a Lie algebra homomorphism from $\\mathfrak g$ to the Lie algebra of functions on $M$. When this is possible, the action is called a Hamiltonian $G$-action. If no such choice of constants is possible, the action of $G$ on the classical phase space is said to have an “[anomaly](../../../../../concepts/symplectic-geometry/moment-map-anomaly)”.", "quote_sha256": "3f498fbda74a1b5392b92f26ead3e429d3de4c12c0a0da9a101eb7b5e59aa769", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-15/03-group-actions-on-m-and-the-moment-map", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e587e400041e128cf6c58a59" }, { "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "anchor": "source-25-5-moment-map-anomaly-title", "chapter": "25", "section": "25.5", "offset": 165, "range": [ 6131, 6395 ], "source_revision": "88f55744ab76ffbe058039b659ef95112b0828406275ef1e219821f846b9ffd3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "anomaly", "context": { "range": [ 6098, 6481 ], "quote": "## 25.5 Normal ordering and the [anomaly](../../../../../concepts/symplectic-geometry/moment-map-anomaly) in finite dimensions", "quote_sha256": "99abc78e7a79e8d80eb8948ca3428d96445354b4d4a0efa21d5e27f9f5658f1d", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-043faee13d24f0c1b7900c5f" }, { "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "anchor": "source-25-5-moment-map-anomaly-definition", "chapter": "25", "section": "25.5", "offset": 3347, "range": [ 9313, 9577 ], "source_revision": "88f55744ab76ffbe058039b659ef95112b0828406275ef1e219821f846b9ffd3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "anomaly", "context": { "range": [ 8960, 9776 ], "quote": "The normal-ordered operators fail to define a Lie algebra homomorphism when extended to $\\mathfrak{sp}(2d,\\mathbb R)$, but the failure is only a constant term. Recall from [Section 15.3](../chapter-15/03-group-actions-on-m-and-the-moment-map#section-15-3) that even classically the moment map is defined up to an additive constant; this can produce an [anomaly](../../../../../concepts/symplectic-geometry/moment-map-anomaly), in which the moment map fails to be a Lie algebra homomorphism by a constant. Here this potential anomaly can be removed by a shift.", "quote_sha256": "06f3fed542ca3fe734dd58cc9e21fd450f96c77620146cc80a6479b9c4304d4c", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-aba74b2e8e14b17be6e38d63" }, { "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-39/03-the-anomaly-and-the-schwinger-term", "anchor": "first-b8882cd26c0201b88129-zh", "chapter": "39", "section": "39.3", "offset": 964, "range": [ 7990, 8293 ], "source_revision": "8f7665700b0b3b987a68af0bce8a88f5c1a3cc9c05a9036419f32e7b7e78bc04", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "反常(anomaly)", "context": { "range": [ 7908, 8515 ], "quote": "Schwinger项给一个标准预期带来了问题：给定某个保持泊松括号作用于相空间的群$G$，人们应当得到$G$在量子态空间$H$上的一个酉表示。这个问题有时被称为“[反常(anomaly)](../../../../../concepts/symplectic-geometry/moment-map-anomaly)”，意思是所预期的李代数酉表示不存在（由于对易关系中的额外标量项）。回顾15.3节，这一潜在问题在经典层面就已经可见，其事实是：给定$L \\in { \\mathfrak { g } } .$，相应的矩映射$\\mu _ { L }$仅在一个常数范围内有定义。而对于我们所研究的有限维情形，可以选择这些常数，使得该映射", "quote_sha256": "327f1fe78f0bdd9f704cec579e327183f371777b6c0943efdea1a777a1a7aa33", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-39/03-the-anomaly-and-the-schwinger-term", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-6ab18831ed30c64a3ff48d10" }, { "concept_id": "urn:uuid:d52824fa-00d0-496c-84fb-ac2c9c371196", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-39/03-the-anomaly-and-the-schwinger-term", "anchor": "source-39-3-moment-map-anomaly", "chapter": "39", "section": "39.3", "offset": 1549, "range": [ 7730, 7994 ], "source_revision": "40fde8ead2a344961c7bd9ab86bad94a100680204c2caf4d78fc89290c14fb59", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "anomaly", "context": { "range": [ 7462, 8716 ], "quote": "The Schwinger term causes a problem with the standard expectation that given some group $G$ acting on the phase space preserving the Poisson bracket, one should get a unitary representation of $G$ on the quantum state space $H$. This problem is sometimes called the “[anomaly](../../../../../concepts/symplectic-geometry/moment-map-anomaly)”, meaning that the expected unitary Lie algebra representation does not exist (due to extra scalar terms in the commutation relations). Recall from section 15.3 that this potential problem was already visible at the classical level, in the fact that given $L \\in { \\mathfrak { g } } .$ the corresponding moment map $\\mu _ { L }$ is only well-defined up to a constant. While for the finite dimensional cases we studied, the constants could be chosen so as to make the map", "quote_sha256": "22c4b082f96dbdc9aaea4d6b4b5c815554bd6e31104c18b5cf242907014807c7", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-39/03-the-anomaly-and-the-schwinger-term", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-07cec6d3b2c5c72d467f2e25" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map", "lang": "en", "section": "15.3", "signal": "term-mark", "label": "anomaly", "english": "anomaly", "offset": 3631 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "lang": "en", "section": "25.5", "signal": "term-mark", "label": "anomaly", "english": "anomaly", "offset": 165 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "lang": "en", "section": "25.5", "signal": "term-mark", "label": "anomaly", "english": "anomaly", "offset": 3191 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-39/03-the-anomaly-and-the-schwinger-term", "lang": "en", "section": "39.3", "signal": "term-mark", "label": "anomaly", "english": "anomaly", "offset": 1358 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-39/03-the-anomaly-and-the-schwinger-term", "lang": "zh-CN", "section": "39.3", "signal": "term-mark", "label": "反常", "english": "anomaly", "offset": 773 } ], "statements": [], "resolved_references": [ { "raw": "[Section 15.3](../chapter-15/03-group-actions-on-m-and-the-moment-map#section-15-3)", "href": "../chapter-15/03-group-actions-on-m-and-the-moment-map#section-15-3", "range": [ 9132, 9215 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map", "target_anchor": "section-15-3", "resolved": true, "reason": null, "source_revision": "88f55744ab76ffbe058039b659ef95112b0828406275ef1e219821f846b9ffd3", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-25/07-normal-ordering-and-the-anomaly-in-finite-dimensions", "occurrence_id": "occ-aba74b2e8e14b17be6e38d63" }, { "raw": "href=\"../chapter-15/03-group-actions-on-m-and-the-moment-map#section-15-3\"", "href": "../chapter-15/03-group-actions-on-m-and-the-moment-map#section-15-3", "range": [ 8211, 8285 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-15/03-group-actions-on-m-and-the-moment-map", "target_anchor": "section-15-3", "resolved": true, "reason": null, "source_revision": "40fde8ead2a344961c7bd9ab86bad94a100680204c2caf4d78fc89290c14fb59", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-39/03-the-anomaly-and-the-schwinger-term", "occurrence_id": "occ-07cec6d3b2c5c72d467f2e25" } ], "reading_order": [ "occ-e587e400041e128cf6c58a59", "occ-043faee13d24f0c1b7900c5f", "occ-aba74b2e8e14b17be6e38d63", "occ-6ab18831ed30c64a3ff48d10", "occ-07cec6d3b2c5c72d467f2e25" ], "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": "feb199dd4a32bffe08a41e0030b43f4c250bc1fd9d9990d05ecad32365e41d40", "optional_enrichments": [] }