{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e", "labels": { "zh-CN": "对易子", "en": "commutator" }, "alternative_labels": [ { "lang": "en", "value": "commutators" } ], "scope_note": "", "identity_status": "active", "identity_revision": "64f05e04bf90c80315d39189b7182668aaa57f290c651f9116acd2de0b8aa5e0", "occurrences": [ { "concept_id": "urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-15/05-commutators-flows-and-lie-derivatives", "anchor": "concept-15-commutator", "chapter": "15", "section": "15.5", "offset": 383, "range": [ 7960, 8186 ], "source_revision": "2f0608b8eac69ae4caf530afbfcbc72c0bbf08ce94d9db8746758c0032aafdce", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "对易子(commutator)", "context": { "range": [ 7916, 8447 ], "quote": "设 $X$ 与 $Y$ 是可微流形 $M$ 的开子集 $U$ 上的光滑向量场。我们将其 对易子(commutator) 或 李括号(Lie bracket) [$X$, $Y$] 定义为 $U$ 上由下式定义的向量场", "quote_sha256": "7d1a21530f8a03e6eaf1c37a84774b47fae82a715502b1fc2399bbbfd1820e64", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-15/05-commutators-flows-and-lie-derivatives", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d58d358367e533bb598f1a44" }, { "concept_id": "urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-15/05-commutators-flows-and-lie-derivatives", "anchor": "first-3cb8369bd67f180c0676-en", "chapter": "15", "section": "15.5", "offset": 459, "range": [ 7525, 7754 ], "source_revision": "b8c69f025d8d682aca940e9c3c2c1374cac6eaec1671872b5a7ec7b3301a1baa", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "commutator", "context": { "range": [ 7413, 8037 ], "quote": "Let $X$ and $Y$ be smooth vector fields on an open subset $U$ of a differentiable manifold $M$. We define their commutator or Lie bracket [$X$, $Y$] as the vector field on $U$ defined by", "quote_sha256": "312f5397f187a003661122788a220d42d511bc6907c3d0c758ad94af66ccd62c", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-15/05-commutators-flows-and-lie-derivatives", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-172aeb73e4a4f4599d1b4404" }, { "concept_id": "urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices", "anchor": "first-c01d6e6a0ebd504fbd78-zh", "chapter": "3", "section": "3.2", "offset": 2649, "range": [ 11307, 11572 ], "source_revision": "59260a454f4e5877261b20806d461df669f06f442a99b7596955f4ec200babfb", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "对易子(commutator)", "context": { "range": [ 11305, 11616 ], "quote": "这些对易子(commutator)并非全为零，这反映了该群的非交换性。接近单位元的群元素 $U \\in SU(2)$ 满足", "quote_sha256": "371c44bcc89e0d79200be6e40948747edbdacf0ecb33dd6afd4cc9f8149c61f1", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-f56d281059563f15b0c62937" }, { "concept_id": "urn:uuid:1a15b2f0-5129-43ae-a79b-291a51fb196e", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices", "anchor": "first-522bb45847f0c2c64d7a-en", "chapter": "3", "section": "3.2", "offset": 3384, "range": [ 10623, 10923 ], "source_revision": "d94860156429173a676b4ffcdc432f70d9a5648fad559409281d5b9cb75de2ec", "source_category": "original_transcription", "review_status": "ai-transcription-compared", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "commutators", "context": { "range": [ 10545, 11041 ], "quote": "The fact that the commutators do not all vanish reflects the non-commutativity of the group. Group elements $U \\in SU(2)$ near the identity satisfy", "quote_sha256": "61f8b107872f2eed8fb17cebfb56f2fd3b4d1be9f34b706f66793b1828d17924", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-dfa11d545a977255bf53c745" } ], "registered_positions": [ { "route": "en/chapter-15/05-commutators-flows-and-lie-derivatives", "lang": "en", "section": "15.5", "signal": "term-mark", "label": "commutator", "english": "commutator", "offset": 459 }, { "route": "zh/chapter-15/05-commutators-flows-and-lie-derivatives", "lang": "zh-CN", "section": "15.5", "signal": "term-mark", "label": "对易子", "english": "commutator", "offset": 383 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-03/02-commutation-relations-for-pauli-matrices", "lang": "en", "section": "3.2", "signal": "term-mark", "label": "commutators", "english": "commutators", "offset": 2802 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-03/02-commutation-relations-for-pauli-matrices", "lang": "zh-CN", "section": "3.2", "signal": "term-mark", "label": "对易子", "english": "commutator", "offset": 2267 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-d58d358367e533bb598f1a44", "occ-172aeb73e4a4f4599d1b4404", "occ-f56d281059563f15b0c62937", "occ-dfa11d545a977255bf53c745" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 4, "located_occurrences": 4, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "30eb285533e2e1cf62982554e1942aa589e959e582cc37053f357feb0f75831d", "optional_enrichments": [] }