{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:38ecd601-9467-4080-844c-8b76816d3b95", "labels": { "zh-CN": "射影表示", "en": "projective representation" }, "alternative_labels": [ { "lang": "en", "value": "projective representations" } ], "scope_note": "", "identity_status": "active", "identity_revision": "5ca22ddfea951f0773f166c597244c5495c8546972dbcb615ab5b27743fe4a4f", "occurrences": [ { "concept_id": "urn:uuid:38ecd601-9467-4080-844c-8b76816d3b95", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-14/03-symmetry-transformations", "anchor": "first-cc1844bb005888d81b42-zh", "chapter": "14", "section": "14.3", "offset": 7147, "range": [ 14417, 14667 ], "source_revision": "bd46fa60bca66ad8e9e72bb6b593e0bd7bca1ecd3fedf001a84980e27aa738c9", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "射影表示(projective representation)", "context": { "range": [ 14375, 14985 ], "quote": "其中相位$\\varphi$可能依赖于$g$和$h$。这称为群$G$在希尔伯特空间上的射影表示(projective representation)或射线表示。一般而言，不可能选择相位使得所有$\\mathrm { e } ^ { i \\varphi } = 1$，从而给出一个真正的表示。对于连续群（见第10.8节），单位元分支$G _ { 0 }$中的元素必须是酉的，因为它们与恒等元连续相连，而恒等元无疑是酉的。反酉变换只能对应于与恒等元不连续相连的分支中的群元素。", "quote_sha256": "0baef27f66fd6703ab796f8a30f4928d4aa76fa5854cb2be026ea0b2d2ac0515", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-14/03-symmetry-transformations", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-e6737c78c2567468e06a277e" }, { "concept_id": "urn:uuid:38ecd601-9467-4080-844c-8b76816d3b95", "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-projective-representation", "chapter": "7", "section": "7.1", "offset": 6149, "range": [ 14307, 14703 ], "source_revision": "3bf7a61152ec8da38212ac86935b672a5739debafd35daed43c2f6c6fd6e56ef", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "射影表示(projective representation)", "context": { "range": [ 14174, 14862 ], "quote": "双覆盖映射 $\\Phi$ 的非平凡性意味着，无论怎样选择 $\\tilde g$，都无法消除所有负号：沿 $\\mathrm{SO}(3)$ 中一条不可缩闭路，不可能对每个 $g$ 连续地选取 $\\tilde g$。像这样只能在符号差异意义下满足表示性质的例子，称为“[射影表示(projective representation)](../../../../../concepts/representation-theory/projective-representation)”。因此，$\\mathrm{SU}(2)=\\mathrm{Spin}(3)$ 的旋量表示可用来构造 $\\mathrm{SO}(3)$ 的射影表示，但不能构造其真正的表示。", "quote_sha256": "3bc550ca53762488ef2c596de576935592457604fe84fb4265d31d6bd5e41d76", "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-00aa2d93b07b6c95bd518e32" }, { "concept_id": "urn:uuid:38ecd601-9467-4080-844c-8b76816d3b95", "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-projective-representation", "chapter": "7", "section": "7.1", "offset": 7163, "range": [ 14485, 14876 ], "source_revision": "15383506dc3beaec4ce03121843583ac8e897a21ef7acf46a88bffa7c86291e9", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "projective representations", "context": { "range": [ 14173, 15146 ], "quote": "The nontriviality of the double cover $\\Phi$ means that no choice of $\\tilde g$ can eliminate every minus sign: along a noncontractible loop in $\\mathrm{SO}(3)$, one cannot choose $\\tilde g$ continuously for every $g$. Examples that satisfy the representation property only up to a sign ambiguity are known as “[projective representations](../../../../../concepts/representation-theory/projective-representation)”. Thus, the spinor representation of $\\mathrm{SU}(2)=\\mathrm{Spin}(3)$ can be used to construct a projective representation of $\\mathrm{SO}(3)$, but not a true representation of $\\mathrm{SO}(3)$.", "quote_sha256": "72e28735420f966d4761fce0a5d90e7b62f804429a153a5ed648cd5d8c150099", "environment": null, "excerpt_omitted": false }, "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-ffdedc0c8391556c6f9210d2" }, { "concept_id": "urn:uuid:38ecd601-9467-4080-844c-8b76816d3b95", "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": 2197, "range": [ 9223, 9520 ], "source_revision": "8f7665700b0b3b987a68af0bce8a88f5c1a3cc9c05a9036419f32e7b7e78bc04", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "射影表示(projective representation)", "context": { "range": [ 8709, 9608 ], "quote": "注意，对于与复结构对易的 $G$ 来说，这个问题不会出现，因为对于这些对象，正规序李代数表示算符将是 $\\mathfrak { u } ( \\infty ) \\subset \\mathfrak { s p } _ { r e s }$ 的一个真实表示。我们将把与 $J _ { 0 }$ 精确对易、而不仅仅是在模 Hilbert-Schmidt 算符意义下对易的元素构成的子群称为 $U ( \\infty ) \\subset S p _ { r e s }$。结果表明，在我们所关注的大多数情形中都有 $G \\subset U ( \\infty )$，从而允许通过湮灭算符与产生算符的正规序二次组合来构造李代数表示（如 25.6 中那样）。还要注意，由于正规序只是将算符平移一个与常数成比例的量，当这个常数有限时就不会出现反常，因为可以通过对正规序算符作这样的有限平移来得到具有正确对易子的算符。反常本质上是一个无限维问题，因为只有在这种情况下才需要无限平移。当反常确实出现时，它将表现为群表示算符中的相位模糊性（而不仅仅是 $Sp(2d,\\mathbb R)$ 的有限维情形中那样的符号模糊性），并且 $H$ 将是该群的一个[射影表示(projective representation)](../../../../../concepts/representation-theory/projective-representation)（一个在相位意义下的表示）。", "quote_sha256": "8c653fc3f3df9102c512ee4ef3c563590e33f723818a0b773ee95c094c74846d", "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; 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