{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:22d066f1-dd17-450c-8457-1f856104d052", "labels": { "zh-CN": "Schwinger 玻色子方法", "en": "Schwinger boson method" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "67cef2014739182390e57862ad8385ee904a8834235018157c326509e31fc5a9", "occurrences": [ { "concept_id": "urn:uuid:22d066f1-dd17-450c-8457-1f856104d052", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3", "anchor": "first-20e213766c5b80def17e-zh", "chapter": "25", "section": "25.4", "offset": 583, "range": [ 7369, 7706 ], "source_revision": "fde0eda8fcbb75fe7ab83d9b5b198d9b490d8a5c6cff10e56002be8f84b78829", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Schwinger 玻色子方法(Schwinger boson method)", "context": { "range": [ 7020, 7779 ], "quote": "在 $d=2$ 时，群 $U(2)\\subset Sp(4,\\mathbb R)$ 在[第25.3节](03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3)中讨论。它在谐振子态空间上的作用与标准谐振子哈密顿算符对易，因此是一组保持各能量本征子空间不变的对称变换。将作用限制到子群 $SU(2)\\subset U(2)$，便可在新的语境中重现此前用齐次多项式构造 $SU(2)$ 表示的方法（见[第8.2节](../chapter-08/04-representations-of-su2-construction#section-8-2)）。物理学文献将利用二维谐振子的能量本征态研究 $SU(2)$ 表示的这一做法称为“[Schwinger 玻色子方法(Schwinger boson method)](../../../../../concepts/representation-theory/schwinger-boson-method)”。", "quote_sha256": "226ae86dce4e200510967c1877eef008d618c6f44559ac5c9b602b704565eb71", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-40d0626e9836ddb50036f1d1" }, { "concept_id": "urn:uuid:22d066f1-dd17-450c-8457-1f856104d052", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3", "anchor": "first-20e213766c5b80def17e-en", "chapter": "25", "section": "25.4", "offset": 1013, "range": [ 6863, 7183 ], "source_revision": "87bed67ee3760b5fe20376ae7bbabe76a75ad57dbc24add3fe3fdf7ad4fc41f4", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Schwinger boson method", "context": { "range": [ 6111, 7299 ], "quote": "In the case $d = 2$ , the action of the group $U ( 2 ) \\subset S p ( 4 , \\mathbf { R } )$ discussed in [section 25.3](03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3) commutes with the standard harmonic oscillator Hamiltonian and thus acts as symmetries on the quantum harmonic oscillator state space, preserving energy eigenspaces. Restricting to the subgroup $S U ( 2 ) \\subset U ( 2 )$ , we’ll see that we can recover our earlier (see [section 8.2](../chapter-08/04-representations-of-su2-construction#section-8-2)) construction of $S U ( 2 )$ representations in terms of homogeneous polynomials, in a new context. This use of the energy eigenstates of a two dimensional harmonic oscillator appears in the physics literature as the “[Schwinger boson method](../../../../../concepts/representation-theory/schwinger-boson-method)” for studying representations of $S U ( 2 )$", "quote_sha256": "a1b18188913fd5d06f2ab2a2cdad115c9c01e7587ff95f50bca82b28b63c2d48", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-191c0600bfeee579f654ccc5" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3", "lang": "en", "section": "25.4", "signal": "term-mark", "label": "Schwinger boson method", "english": "Schwinger boson method", "offset": 1013 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3", "lang": "zh-CN", "section": "25.4", "signal": "term-mark", "label": "Schwinger 玻色子方法", "english": "Schwinger boson method", "offset": 583 } ], "statements": [], "resolved_references": [ { "raw": "[第25.3节](03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3)", "href": "03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3", "range": [ 7063, 7135 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "target_anchor": "section-25-3", "resolved": true, "reason": null, "source_revision": "fde0eda8fcbb75fe7ab83d9b5b198d9b490d8a5c6cff10e56002be8f84b78829", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3", "occurrence_id": "occ-40d0626e9836ddb50036f1d1" }, { "raw": "[第8.2节](../chapter-08/04-representations-of-su2-construction#section-8-2)", "href": "../chapter-08/04-representations-of-su2-construction#section-8-2", "range": [ 7253, 7326 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-08/04-representations-of-su2-construction", "target_anchor": "section-8-2", "resolved": true, "reason": null, "source_revision": "fde0eda8fcbb75fe7ab83d9b5b198d9b490d8a5c6cff10e56002be8f84b78829", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-25/04-examples-in-d-2-and-3", "occurrence_id": "occ-40d0626e9836ddb50036f1d1" }, { "raw": "[section 25.3](03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3)", "href": "03-the-metaplectic-representation-and-u-d-sp-2d-r#section-25-3", "range": [ 6214, 6292 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-25/03-the-metaplectic-representation-and-u-d-sp-2d-r", "target_anchor": "section-25-3", "resolved": true, "reason": null, "source_revision": "87bed67ee3760b5fe20376ae7bbabe76a75ad57dbc24add3fe3fdf7ad4fc41f4", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3", "occurrence_id": "occ-191c0600bfeee579f654ccc5" }, { "raw": "[section 8.2](../chapter-08/04-representations-of-su2-construction#section-8-2)", "href": "../chapter-08/04-representations-of-su2-construction#section-8-2", "range": [ 6564, 6643 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-08/04-representations-of-su2-construction", "target_anchor": "section-8-2", "resolved": true, "reason": null, "source_revision": "87bed67ee3760b5fe20376ae7bbabe76a75ad57dbc24add3fe3fdf7ad4fc41f4", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-25/04-examples-in-d-2-and-3", "occurrence_id": "occ-191c0600bfeee579f654ccc5" } ], "reading_order": [ "occ-40d0626e9836ddb50036f1d1", "occ-191c0600bfeee579f654ccc5" ], "recorded_dependencies": [], "reading_paths": [], "coverage": { "registered_positions": 2, "located_occurrences": 2, "whole_book_search_complete": false, "registered_positions_truncated": false, "independent_human_review_implied": false }, "unresolved": [], "base_revision": "34fb1dc7c81dc3a3f51685aed7866da909f54f8ca8e7879d05acc07730b4773c", "optional_enrichments": [] }