{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:eb959135-fcc2-4572-a09a-1267ad1e544c", "labels": { "zh-CN": "元辛表示", "en": "metaplectic representation" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "d928c3cb09fa5725d34610c629b63daf05a41f8fc539311c091b66d5822fd07c", "occurrences": [ { "concept_id": "urn:uuid:eb959135-fcc2-4572-a09a-1267ad1e544c", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/index", "anchor": "first-5933e2dbb4e3939e827f-zh", "chapter": "25", "section": "25", "offset": 331, "range": [ 7521, 7855 ], "source_revision": "91f39772e5e3c858c5ace0e20ab58d838e6a1cb7c455bdd074ffea132432c699", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "元辛表示(metaplectic representation)", "context": { "range": [ 7414, 8407 ], "quote": "本章把第24章的 $d=1$ 情形推广到任意 $d$。选取 $d$ 个湮灭算符和 $d$ 个产生算符，会突出子群 $U(d)\\subset Sp(2d,\\mathbb R)$：其中的变换不混合湮灭算符与产生算符。[元辛表示(metaplectic representation)](../../../../../concepts/representation-theory/metaplectic-representation)给出该子群的一个双覆盖表示。我们将看到，对湮灭算符与产生算符的乘积作[正规序(normal ordering)](../../../../../concepts/quantum-mechanics/normal-ordering)后，得到的是 $U(d)$ 本身的表示，而非其双覆盖群的表示。由此，相空间上 $U(d)$ 的作用在 $d$ 维谐振子的态空间上诱导出一个无限维表示。", "quote_sha256": "82971a53b558f130838741d9b1823fa519efb027665c8443b229c6c2a26c924a", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-caa4c5ca814e95aa90dd024c" }, { "concept_id": "urn:uuid:eb959135-fcc2-4572-a09a-1267ad1e544c", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index", "anchor": "first-5933e2dbb4e3939e827f-en", "chapter": "25", "section": "25", "offset": 570, "range": [ 6991, 7319 ], "source_revision": "a4fabc3ade7e18eac393871eaed9571dfcf75d61baf7220e6eb15075ae13412d", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "metaplectic representation", "context": { "range": [ 6709, 8127 ], "quote": "This chapter generalizes the $d=1$ case of Chapter 24 to arbitrary $d$. Choosing $d$ annihilation operators and $d$ creation operators singles out a distinguished subgroup $U(d)\\subset Sp(2d,\\mathbb R)$: the transformations that do not mix annihilation and creation operators. The [metaplectic representation](../../../../../concepts/representation-theory/metaplectic-representation) gives a representation of a double cover of this subgroup. We will see that [normal ordering](../../../../../concepts/quantum-mechanics/normal-ordering) products of annihilation and creation operators yields a representation of $U(d)$ itself, rather than of the double cover. 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