{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "labels": { "zh-CN": "自旋群", "en": "spin group" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "6feb10cc678ac239594a857ae175fc9057acafca2f6a3313b66cb9ee77d8a78e", "occurrences": [ { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-spin", "chapter": "6", "section": "6.2", "offset": 510, "range": [ 8801, 9111 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋群(spin group)", "context": { "range": [ 8424, 9254 ], "quote": "[正交群(orthogonal group)](../../../../../concepts/linear-algebra/orthogonal-group) $\\mathrm{SO}(n)$ 有一个微妙而引人注目的性质：存在一个与之相关联的群，称为[自旋群(spin group)](../../../../../concepts/group-theory/spin-group) $\\operatorname{Spin}(n)$；$\\mathrm{SO}(n)$ 的每个元素都对应 $\\operatorname{Spin}(n)$ 中恰好两个不同元素。存在一个满射", "quote_sha256": "ee4276f9a1a991877f7b89bc1db27e36b95f46382114b1d21ae1310e3d58d1c9", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-b2745374727ecb9a079f338e" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "anchor": "concept-6-spin", "chapter": "6", "section": "6.2", "offset": 602, "range": [ 8051, 8356 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "spin group", "context": { "range": [ 7612, 8892 ], "quote": "A subtle and remarkable property of the [orthogonal groups](../../../../../concepts/linear-algebra/orthogonal-group) $\\mathrm{SO}(n)$ is that it comes with an associated group, called the [spin group](../../../../../concepts/group-theory/spin-group) $\\operatorname{Spin}(n)$, such that every element of $\\mathrm{SO}(n)$ corresponds to two distinct elements of $\\operatorname{Spin}(n)$. There is a surjective [group homomorphism](../../../../../concepts/group-theory/group-homomorphism)", "quote_sha256": "3ff5b33087b97cb96b866c2d9b35fc40878b83f9135a6e41d83a0cfffa4d6afd", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-5fd62f18440fd405117d14a1" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "anchor": "concept-6-3-spin-group", "chapter": "6", "section": "6.3", "offset": 926, "range": [ 10269, 10595 ], "source_revision": "51224e8d409a684fb67417edf938c1deffb0aa708773316223994618d7a07300", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋群(spin group)", "context": { "range": [ 10266, 12817 ], "quote": null, "quote_sha256": "ccc6c8c7f3b76989ce54f30bbc227a5304b6c0410f33b5205f652eb820bc410b", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-c406a8c062ce93dad6c9db9e" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "anchor": "concept-6-3-spin-group", "chapter": "6", "section": "6.3", "offset": 994, "range": [ 9503, 9824 ], "source_revision": "c6959dd1685931e5d5c3b3ad92a453f6e8e23f74fb6df74bd7839b5bfd7d43ac", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Spin group", "context": { "range": [ 9500, 12157 ], "quote": null, "quote_sha256": "51760142b0b6b47418ad93766ff4c4d84f786e42d2663d2b951cfe7cd1240ff2", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-224aec3d2ef6a299cb78be30" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "anchor": "concept-7-spin-group", "chapter": "7", "section": "7.2", "offset": 6967, "range": [ 17178, 17403 ], "source_revision": "af42ca8c868bc2cdc5dca3ef8a46b1cd266fae227aaba16f410d9185fd6d5068", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋群(spin group)", "context": { "range": [ 17161, 17862 ], "quote": "对于自旋为 $1/2$ 的粒子，[自旋群(spin group) Spin(3)](../../../../../concepts/group-theory/spin-group)与 $SU(2)$ 同构，并通过[旋量表示(spinor representation)](../../../../../concepts/representation-theory/spinor-representation)作用于量子态，其中元素 $\\Omega ( \\theta , \\mathbf { w } ) \\in S U ( 2 )$ 的作用为", "quote_sha256": "dd1a12971a4672bcfe84033769777516e0b91c49bb4292f42556838bd0c264cc", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-c14fb6e610208851c63078f5" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "anchor": "source-7-2-spinor-representation", "chapter": "7", "section": "7.2", "offset": 9068, "range": [ 18389, 18583 ], "source_revision": "2f4e030ded93d226da1b050014630cb67a384fcab12bb0f30e051144b3d12eec", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Spin group", "context": { "range": [ 18357, 19063 ], "quote": "For a spin-$1/2$ particle, the [Spin group (Spin(3))](../../../../../concepts/group-theory/spin-group) group, which is isomorphic to $SU(2)$, acts on states through the [spinor representation](../../../../../concepts/representation-theory/spinor-representation) with the element $\\Omega ( \\theta , \\mathbf { w } ) \\in S U ( 2 )$ acting as", "quote_sha256": "8912f517a254304db371d7f228eafa4a433322e99b4641120bad0b4a4b8ca1d4", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d846555048c6f1432724ae62" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry", "anchor": "source-29-2-spin-group", "chapter": "29", "section": "29.2", "offset": 6471, "range": [ 14135, 14392 ], "source_revision": "ed4796086da35969c2b515beee623bc0e11b40ba7b7008e2a2a0a7060912d0f9", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋群(spin group)", "context": { "range": [ 14114, 14745 ], "quote": "\n> [!definition] 定义（[自旋群(spin group)](../../../../../concepts/group-theory/spin-group) $\\operatorname{Spin}(n)$）\n>\n> 群 $\\operatorname{Spin}(n,\\mathbb R)$ 由 $\\operatorname{Cliff}(n)$ 中形如下式的可逆元素组成：\n>\n> $$\n> \\not{\\mathbf w}_1\\not{\\mathbf w}_2\\cdots\\not{\\mathbf w}_k\n> $$\n>\n> 其中 $\\mathbf w_j\\in\\mathbb R^n$（$j=1,\\ldots,k$），$k\\le n$，$\\lVert\\mathbf w_j\\rVert^2=1$，且 $k$ 为偶数。群乘法就是Clifford代数中的乘法。\n", "quote_sha256": "d51c4e4b921226a9d50aebf6b58bb453909cbf6a99c5e5e40e8ae4fc8f683af2", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-fd3064acb60f329be5e50a92" }, { "concept_id": "urn:uuid:8adadd91-0a76-4463-a472-62ff66c5bfb3", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry", "anchor": "source-29-2-spin-group", "chapter": "29", "section": "29.2", "offset": 8918, "range": [ 15742, 15994 ], "source_revision": "f227ff63312143df03766b8ec5fd76e099e4b68faf0df92b2f24255b52929a29", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "spin group", "context": { "range": [ 15712, 16426 ], "quote": "\n> [!definition] Definition ([spin group](../../../../../concepts/group-theory/spin-group) $\\operatorname{Spin}(n)$)\n>\n> $\\operatorname{Spin}(n,\\mathbb R)$ is the group of invertible elements of $\\operatorname{Cliff}(n)$ of the form\n>\n> $$\n> \\not{\\mathbf w}_1\\not{\\mathbf w}_2\\cdots\\not{\\mathbf w}_k\n> $$\n>\n> where $\\mathbf w_j\\in\\mathbb R^n$ for $j=1,\\ldots,k$, $k\\le n$, $\\lVert\\mathbf w_j\\rVert^2=1$, and $k$ is even. Multiplication is Clifford-algebra multiplication.\n", "quote_sha256": "a16e7c94b19c47753f580dd068d1b4ecc66982bd483e2f49d7effe601c629e21", "environment": "definition", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-297ebc93abc2a2c8d68ddf95" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "en", "section": "6.2", "signal": "term-mark", "label": "spin group", "english": "spin group", "offset": 446 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/02-spin-groups-in-three-and-four-dimensions", "lang": "zh-CN", "section": "6.2", "signal": "term-mark", "label": "自旋群", "english": "spin group", "offset": 354 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-06/07-a-summary", "lang": "en", "section": "6.3", "signal": "term-mark", "label": "Spin group", "english": "Spin group", "offset": 838 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-06/07-a-summary", "lang": "zh-CN", "section": "6.3", "signal": "term-mark", "label": "自旋群", "english": "spin group", "offset": 770 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "lang": "en", "section": "7.2", "signal": "term-mark", "label": "Spin group", "english": "Spin group", "offset": 7489 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/02-the-spin-half-particle-in-a-magnetic-field", "lang": "zh-CN", "section": "7.2", "signal": "term-mark", "label": "自旋群", "english": "spin group", "offset": 5563 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-29/02-cliford-algebras-and-geometry", "lang": "en", "section": "29.2", "signal": "term-mark", "label": "spin group", "english": "spin group", "offset": 8154 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-29/02-cliford-algebras-and-geometry", "lang": "zh-CN", "section": "29.2", "signal": "term-mark", "label": "自旋群", "english": "spin group", "offset": 5707 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-b2745374727ecb9a079f338e", "occ-5fd62f18440fd405117d14a1", "occ-c406a8c062ce93dad6c9db9e", "occ-224aec3d2ef6a299cb78be30", "occ-c14fb6e610208851c63078f5", "occ-d846555048c6f1432724ae62", "occ-fd3064acb60f329be5e50a92", "occ-297ebc93abc2a2c8d68ddf95" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 8, "located_occurrences": 8, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "7ff09058d64b47ad9f22caff3dd9418ff67cbe2b91a365097ea793d1c3fdca6d", "optional_enrichments": [] }