{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:0c8937be-5b52-431f-a556-31c2c5f610f0", "labels": { "zh-CN": "自旋表示", "en": "spin representation" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "1a07944b5c22f5aab03f1af9e05721b4af8230897d6874c1418947f9d916b5ce", "occurrences": [ { "concept_id": "urn:uuid:0c8937be-5b52-431f-a556-31c2c5f610f0", "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": "first-57f0c39fb5f6b90ccf1a-zh", "chapter": "7", "section": "7.1", "offset": 4170, "range": [ 12328, 12650 ], "source_revision": "3bf7a61152ec8da38212ac86935b672a5739debafd35daed43c2f6c6fd6e56ef", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋表示(spin representation)", "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; 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not-concept-equivalence" } ], "id": "occ-eb611008e5d662a02c3f51b7" }, { "concept_id": "urn:uuid:0c8937be-5b52-431f-a556-31c2c5f610f0", "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-eigenvalue", "chapter": "7", "section": "7.1", "offset": 7380, "range": [ 15538, 15825 ], "source_revision": "3bf7a61152ec8da38212ac86935b672a5739debafd35daed43c2f6c6fd6e56ef", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "自旋表示(spin representation)", "context": { "range": [ 15529, 16681 ], "quote": "从这个意义上说，[自旋表示(spin representation)](../../../../../concepts/representation-theory/spinor-representation)比[向量表示(vector representation)](../../../../../concepts/representation-theory/vector-representation)更为基本：仅凭向量表示无法确定自旋表示，而已知 $\\mathrm{SU}(2)$ 的旋量表示，则可以构造出 $\\mathrm{SO}(3)$ 的向量表示。我们通过把 $\\mathbb{R}^3$ 与 $2\\times2$ 复矩阵空间对应起来，看到了这一点：[方程 6.5](../chapter-06/02-spin-groups-in-three-and-four-dimensions#equation-6-5) 表明，$\\mathbb{R}^3$ 中的旋转对应于旋量表示矩阵的共轭作用。另一种说明方式用到张量积，见[第 9.4.3 节](../chapter-09/04-tensor-products-of-representations#section-9-4-3)。把旋量视为基本对象，意味着不再只用实数来描述三维几何。向量表示是 $\\mathrm{SO}(3)$ 或 $\\mathrm{Spin}(3)$ 的实表示，而旋量表示是复表示。", "quote_sha256": "ab1c8852127b4ef676886302948a82acc841650b34653fbc01c750307dd3c9aa", "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; 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