{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "labels": { "zh-CN": "双重覆盖", "en": "double cover" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "307cc862ec24f688cb7beffcf22c0e9323ec3bb33083f5081a029a026a7cca1f", "occurrences": [ { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "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-double", "chapter": "6", "section": "6.2", "offset": 1847, "range": [ 10138, 10453 ], "source_revision": "d5776d5bdae66f74cdbd2aaeda0ef4ffb4148f08eeb21a72c16dd61bab582129", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "双重覆盖(double cover)", "context": { "range": [ 9588, 10928 ], "quote": "\n> [!theorem] 拓展\n> *其拓扑原因是：当 $n>2$ 时，$\\mathrm{SO}(n)$ 的[基本群(fundamental group)](../../../../../concepts/topology/fundamental-group)非平凡，且 $\\pi_1(\\mathrm{SO}(n))\\cong\\mathbb Z_2$。特别地，$\\mathrm{SO}(n)$ 中存在一条不可缩的环路，沿它绕行两圈后便可缩。自旋群在拓扑上是 $\\mathrm{SO}(n)$ 的单连通[双重覆盖(double cover)](../../../../../concepts/topology/double-cover)，而[覆盖映射(covering map)](../../../../../concepts/topology/covering-map) $\\Phi\\colon\\operatorname{Spin}(n)\\to\\mathrm{SO}(n)$ 可以选为群同态。*\n", "quote_sha256": "dea483579b808c5e6807bb9c15a43714bb0cfbaf6305b3ac6d8b953b69aed79b", "environment": "theorem", "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-a5989ec2509bfe7d526112f1" }, { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "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-double", "chapter": "6", "section": "6.2", "offset": 2507, "range": [ 9956, 10265 ], "source_revision": "525b7bc4b6973b05449bbbbcbdef1c067e7411218e20b20cb388b95687d1c4d8", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "double cover", "context": { "range": [ 9283, 10795 ], "quote": "\n> [!theorem] Digression\n> *The topological reason is that, for $n>2$, the [fundamental group](../../../../../concepts/topology/fundamental-group) of $\\mathrm{SO}(n)$ is nontrivial: $\\pi_1(\\mathrm{SO}(n))\\cong\\mathbb Z_2$. In particular, $\\mathrm{SO}(n)$ contains a noncontractible loop that becomes contractible when traversed twice. The spin group is the simply connected [double cover](../../../../../concepts/topology/double-cover) of $\\mathrm{SO}(n)$, and the [covering map](../../../../../concepts/topology/covering-map) $\\Phi\\colon\\operatorname{Spin}(n)\\to\\mathrm{SO}(n)$ can be chosen to be a group homomorphism.*\n", "quote_sha256": "ce87774c3310d1aaa59f1b40bf63bbe946c9381253db1be646d265bb3a49a506", "environment": "theorem", "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-8b5270502af8a4d3eb38fed7" }, { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation", "anchor": "concept-7-1-double-cover", "chapter": "7", "section": "7.1", "offset": 4633, "range": [ 12791, 13122 ], "source_revision": "3bf7a61152ec8da38212ac86935b672a5739debafd35daed43c2f6c6fd6e56ef", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "二重覆盖(double cover)", "context": { "range": [ 12309, 13805 ], "quote": "$\\mathrm{SU}(2)$ 的[自旋表示(spin representation)](../../../../../concepts/representation-theory/spinor-representation)并不是 $\\mathrm{SO}(3)$ 的表示。映射 $\\Phi:\\mathrm{SU}(2)\\to\\mathrm{SO}(3)$ 是一个[二重覆盖(double cover)](../../../../../concepts/topology/double-cover)，也是一个[群同态(group homomorphism)](../../../../../concepts/group-theory/group-homomorphism)；因此，将 $\\mathrm{SO}(3)$ 的任一表示 $(\\pi,V)$ 与 $\\Phi$ 复合，就得到 $\\mathrm{SU}(2)$ 的表示 $(\\pi\\circ\\Phi,V)$。反过来则不能这样构造：不存在同态 $\\mathrm{SO}(3)\\to\\mathrm{SU}(2)$，能把 $\\mathrm{SU}(2)$ 在 $\\mathbf{C}^2$ 上的自旋表示变成 $\\mathrm{SO}(3)$ 的表示。", "quote_sha256": "e833ec41e05be805a132d6c2736136b75bf6995d6abb86a37c5bebbb9f4b0b12", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d006e8ae955590555c49f40c" }, { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation", "anchor": "concept-7-1-double-cover", "chapter": "7", "section": "7.1", "offset": 5125, "range": [ 12447, 12772 ], "source_revision": "15383506dc3beaec4ce03121843583ac8e897a21ef7acf46a88bffa7c86291e9", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "double cover", "context": { "range": [ 11985, 13631 ], "quote": null, "quote_sha256": "485671457b7b525d0b2ecfda4aca84955780945b3527b190a19c412410999983", "environment": null, "excerpt_omitted": true }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-58c1e5905fc2033b92d673a2" }, { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "anchor": "section-25-3", "chapter": "25", "section": "25.3", "offset": 710, "range": [ 7439, 7697 ], "source_revision": "7b5431388a644a235ec79741120375af2b5cd12628dbcec77ef70f35686bb179", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "二重覆盖(double cover)", "context": { "range": [ 7335, 7871 ], "quote": "这样得到的算符满足正确的 $\\mathfrak{sp}(2d,\\mathbb R)$ 对易关系，尤其是 $a_j^2$ 与 $(a_j^\\dagger)^2$ 的对易关系。但要得到群的真正表示，必须转到一个[二重覆盖(double cover)](../../../../../concepts/topology/double-cover)。因此，Bargmann–Fock 构造给出 $\\mathfrak u(d)$ 在 Fock 空间 $\\mathcal F_d$ 上的酉表示；取指数后，得到的不是 $U(d)$ 本身的表示，而是其二重覆盖 $\\widetilde{U(d)}$ 的表示。", "quote_sha256": "9d4dc5ed9f7aa075e83b1f8c4ce1841a44844063797f7fa2db4e311bee566fec", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-9ff80f32db7b82063379ccd0" }, { "concept_id": "urn:uuid:004c393d-b3c2-4027-957a-d8cea3b92491", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "anchor": "section-25-3", "chapter": "25", "section": "25.3", "offset": 1183, "range": [ 7063, 7315 ], "source_revision": "f5fbd9863ade84e2de966a52a5b07d5eb08cc3100be17057b12b9b9a92e95855", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "double cover", "context": { "range": [ 6852, 7605 ], "quote": "This gives the correct $\\mathfrak{sp}(2d,\\mathbb R)$ commutation relations, in particular for the commutator of $a_j^2$ with $(a_j^\\dagger)^2$. But a true representation of the group then requires passage to a [double cover](../../../../../concepts/topology/double-cover). Thus the Bargmann–Fock construction gives a unitary representation of $\\mathfrak u(d)$ on Fock space $\\mathcal F_d$; after exponentiation, this is a representation of a double cover, denoted $\\widetilde{U(d)}$, rather than of $U(d)$ itself.", "quote_sha256": "8749f01e1bc3c7e5a2bb42beb480e990cd8f015fddf6ee9d129c10eabee07a27", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e01b6fe1d05e55a8dd79f5e4" } ], "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": "double cover", "english": "double cover", "offset": 1883 }, { "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": "双重覆盖", "english": "double cover", "offset": 1379 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/01-the-spinor-representation", "lang": "en", "section": "7.1", "signal": "term-mark", "label": "double cover", "english": "double cover", "offset": 3807 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/01-the-spinor-representation", "lang": "zh-CN", "section": "7.1", "signal": "term-mark", "label": "二重覆盖", "english": "double cover", "offset": 3315 }, { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "lang": "en", "section": "25.3", "signal": "term-mark", "label": "double cover", "english": "double cover", "offset": 1183 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "lang": "zh-CN", "section": "25.3", "signal": "term-mark", "label": "二重覆盖", "english": "double cover", "offset": 710 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-a5989ec2509bfe7d526112f1", "occ-8b5270502af8a4d3eb38fed7", "occ-d006e8ae955590555c49f40c", "occ-58c1e5905fc2033b92d673a2", "occ-9ff80f32db7b82063379ccd0", "occ-e01b6fe1d05e55a8dd79f5e4" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 6, "located_occurrences": 6, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "c6b7afa8cacdc45cb98e605a2dce8c683b77bde3d4e7a0ec5ea773b420fd9b5e", "optional_enrichments": [] }