{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:74ae58d7-65e0-4725-947a-20d3cf498508", "labels": { "en": "Groenewold-van Hove" }, "alternative_labels": [ { "lang": "en", "value": "Groenewold–van Hove" } ], "scope_note": "", "identity_status": "active", "identity_revision": "774f64ec21776b6e1403044944a344018548484dc2c03a4124300296755b96e0", "occurrences": [ { "concept_id": "urn:uuid:74ae58d7-65e0-4725-947a-20d3cf498508", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "anchor": "first-ca01412bfa07aab6251e-zh", "chapter": "17", "section": "17.2", "offset": 640, "range": [ 7049, 7691 ], "source_revision": "2c68e922dbf0d44d4637aa030e4d1a740b0ebf594096b2f58480f55342b38632", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "zh-CN", "section": "17.2", "signal": "named-theorem", "label": "Groenewold–van Hove", "english": "", "offset": 570 } }, "label": "> [!theorem] 定理（Groenewold–van Hove）", "context": { "range": [ 7049, 7691 ], "quote": "\n\n> [!theorem] 定理（Groenewold–van Hove）\n> 不存在映射 $f\\mapsto O_f$，将 $\\mathbf R^2$ 上的多项式映到作用在 $L^2(\\mathbf R)$ 上的自伴算符，并满足\n>\n> $$\n> O_{\\{f,g\\}}=-\\frac{i}{\\hbar}[O_f,O_g]\n> $$\n>\n> 以及\n>\n> $$\n> O_p=P,\\quad O_q=Q\n> $$\n>\n> 对于 $\\mathbf R^2$ 上函数所成的任意李子代数，只要次数不超过二的多项式所成子代数是其真子代数，以上条件就不可能同时成立。\n\n> [!proof]- 证明\n> 详细证明见[8]的[第 5.4 节](../chapter-05/04-lie-algebra-representations)、[26]的[第 4.4 节](../chapter-04/04-inner-products)或[37]的[第 16 章](../chapter-16/index#section-16)。概要而言，证明先表明，三次多项式的泊松括号会产生更高次多项式；而且，对于次数三及以上的多项式，不存在次数有统一上界的有限维子代数。定理假设迫使三次项采用某些特定的算符排序；随后利用同一个四次多项式可写成两个不同泊松括号的事实，在四次项中导出矛盾：\n>\n> $$\n> q^2p^2=\\frac13\\{q^2p,p^2q\\}=\\frac19\\{q^3,p^3\\}\n> $$\n", "quote_sha256": "1de06a1d2ae0b64e6d1fdc3c1e7440ddccc919cbdd9dc1b42699775d0f9bfd26", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-716a1ad0ffbefbd47684a292" }, { "concept_id": "urn:uuid:74ae58d7-65e0-4725-947a-20d3cf498508", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "anchor": "first-23a9feb733de92986fe8-en", "chapter": "17", "section": "17.2", "offset": 1588, "range": [ 7150, 8321 ], "source_revision": "0f2a1b7e1d8b680417ec27a91ec344b72ce3099bb0ccb6d8c9f1fdd4b54425a5", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "theorem", "role_basis": { "kind": "registered-source-environment", "registered_position": { "route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "en", "section": "17.2", "signal": "named-theorem", "label": "Groenewold-van Hove", "english": "Groenewold-van Hove", "offset": 1518 } }, "label": "> [!theorem] Theorem (Groenewold-van Hove)", "context": { "range": [ 7150, 8321 ], "quote": "\n\n> [!theorem] Theorem (Groenewold-van Hove)\n> There is no map $f\\mapsto O_f$ from polynomials on $\\mathbf R^2$ to self-adjoint operators on $L^2(\\mathbf R)$ satisfying\n>\n> $$\n> O_{\\{f,g\\}}=-\\frac{i}{\\hbar}[O_f,O_g]\n> $$\n>\n> and\n>\n> $$\n> O_p=P,\\quad O_q=Q\n> $$\n>\n> for any Lie subalgebra of functions on $\\mathbf R^2$ in which the subalgebra of polynomials of degree at most two is a proper subalgebra.\n\n> [!proof]- Proof\n> For a detailed proof, see [section 5.4](../chapter-05/04-lie-algebra-representations) of [8], [section 4.4](../chapter-04/04-inner-products) of [26], or [chapter 16](../chapter-16/index#section-16) of [37]. In outline, the proof begins by showing that Poisson brackets of degree-three polynomials produce higher-degree polynomials; moreover, for degree three and above there are no finite-dimensional subalgebras consisting of polynomials of bounded degree. The theorem’s assumptions force certain specific operator-ordering choices in degree three. These yield a contradiction in degree four, because the same degree-four polynomial has two different expressions as a Poisson bracket:\n>\n> $$\n> q^2p^2=\\frac13\\{q^2p,p^2q\\}=\\frac19\\{q^3,p^3\\}\n> $$\n", "quote_sha256": "62c6e03a7e9906d369384453dc8853bf99cf69137f1c29f965c14a48184a182b", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-87e6ae74b2a1190bea801b54" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "en", "section": "17.2", "signal": "named-theorem", "label": "Groenewold-van Hove", "english": "Groenewold-van Hove", "offset": 1518 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "lang": "zh-CN", "section": "17.2", "signal": "named-theorem", "label": "Groenewold–van Hove", "english": "", "offset": 570 } ], "statements": [], "resolved_references": [ { "raw": "[第 5.4 节](../chapter-05/04-lie-algebra-representations)", "href": "../chapter-05/04-lie-algebra-representations", "range": [ 7356, 7411 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-05/04-lie-algebra-representations", "target_anchor": "", "resolved": true, "reason": null, "source_revision": "2c68e922dbf0d44d4637aa030e4d1a740b0ebf594096b2f58480f55342b38632", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-716a1ad0ffbefbd47684a292" }, { "raw": "[第 4.4 节](../chapter-04/04-inner-products)", "href": "../chapter-04/04-inner-products", "range": [ 7417, 7459 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-04/04-inner-products", "target_anchor": "", "resolved": true, "reason": null, "source_revision": "2c68e922dbf0d44d4637aa030e4d1a740b0ebf594096b2f58480f55342b38632", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-716a1ad0ffbefbd47684a292" }, { "raw": "[第 16 章](../chapter-16/index#section-16)", "href": "../chapter-16/index#section-16", "range": [ 7465, 7505 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/zh/chapter-16/index", "target_anchor": "section-16", "resolved": true, "reason": null, "source_revision": "2c68e922dbf0d44d4637aa030e4d1a740b0ebf594096b2f58480f55342b38632", "source_route": "books/woit-qtgr/2025-10-20/zh/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-716a1ad0ffbefbd47684a292" }, { "raw": "[section 5.4](../chapter-05/04-lie-algebra-representations)", "href": "../chapter-05/04-lie-algebra-representations", "range": [ 7600, 7659 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-05/04-lie-algebra-representations", "target_anchor": "", "resolved": true, "reason": null, "source_revision": "0f2a1b7e1d8b680417ec27a91ec344b72ce3099bb0ccb6d8c9f1fdd4b54425a5", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-87e6ae74b2a1190bea801b54" }, { "raw": "[section 4.4](../chapter-04/04-inner-products)", "href": "../chapter-04/04-inner-products", "range": [ 7668, 7714 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-04/04-inner-products", "target_anchor": "", "resolved": true, "reason": null, "source_revision": "0f2a1b7e1d8b680417ec27a91ec344b72ce3099bb0ccb6d8c9f1fdd4b54425a5", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-87e6ae74b2a1190bea801b54" }, { "raw": "[chapter 16](../chapter-16/index#section-16)", "href": "../chapter-16/index#section-16", "range": [ 7727, 7771 ], "relation": "references", "target_route": "books/woit-qtgr/2025-10-20/en/chapter-16/index", "target_anchor": "section-16", "resolved": true, "reason": null, "source_revision": "0f2a1b7e1d8b680417ec27a91ec344b72ce3099bb0ccb6d8c9f1fdd4b54425a5", "source_route": "books/woit-qtgr/2025-10-20/en/chapter-17/02-the-groenewold-van-hove-no-go-theorem", "occurrence_id": "occ-87e6ae74b2a1190bea801b54" } ], "reading_order": [ "occ-716a1ad0ffbefbd47684a292", "occ-87e6ae74b2a1190bea801b54" ], "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": "670090f8a2d51172685d74a8df830673de922b07040f07fea2c31718a3f97124", "optional_enrichments": [] }