{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:b6e6358a-67f2-4f17-abe3-6047c1590084", "labels": { "zh-CN": "Hopf 纤维化", "en": "Hopf fibration" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "b2d196595ad285b34b99f5a52b9f08c8dc95274b5f0fc3ed297a4d0bb1ab5f51", "occurrences": [ { "concept_id": "urn:uuid:b6e6358a-67f2-4f17-abe3-6047c1590084", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "anchor": "first-7dcdf66d410d25744005-zh", "chapter": "7", "section": "7.4", "offset": 8362, "range": [ 16442, 16743 ], "source_revision": "dece3546a425ea2c52b709472898bb8128e5d5d751614a636ab79e1e1bbfe61c", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Hopf 纤维化(Hopf fibration)", "context": { "range": [ 15955, 16853 ], "quote": "\n> [!theorem] 拓展\n>\n> *Digression. 还可以从另一角度看 $\\mathbf{C}P^1$：分两步构造 $\\mathbf{C}^2$ 对复数乘法的商。乘以实数会改变态的归一化；我们常利用这一自由度，把态取为满足下式的归一化态：*\n>\n> $$\n> \\langle\\psi|\\psi\\rangle=|z_1|^2+|z_2|^2=1.\n> $$\n>\n> *这些归一化态是 $\\mathbf{C}^2$ 中的单位向量，可由单位球面 $S^3\\subset\\mathbf{R}^4=\\mathbf{C}^2$ 上的点表示。*\n>\n> *接下来还要商去相位乘法的作用，也就是把只相差一个 $e^{i\\theta}$ 因子的 $S^3$ 中的点视为等同。所得空间常记为 $S^3/U(1)$。相位作用的轨道是彼此不相交的圆；这种以圆为纤维的结构称为[Hopf 纤维化(Hopf fibration)](../../../../../concepts/topology/hopf-fibration.md)。可以在网上搜索相关几何图像：将无穷远点加入三维空间得到 $S^3$，会得到一种由互不相交曲线构成的奇妙分解。*\n", "quote_sha256": "fc60e7b2b730de5f528609a5355d7de7915ccbfb7fad0701477e6e2b1dfebac7", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-ce096f4b198a7c772a9b5296" }, { "concept_id": "urn:uuid:b6e6358a-67f2-4f17-abe3-6047c1590084", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "en", "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "anchor": "first-7dcdf66d410d25744005-en", "chapter": "7", "section": "7.4", "offset": 10185, "range": [ 17598, 17889 ], "source_revision": "56ee7f1727deaa19e151a716a8ad06b4eb7d62437a0e4e97e582b9d242bc09a3", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Hopf fibration", "context": { "range": [ 16772, 18125 ], "quote": "\n> [!theorem] Digression\n>\n> *Digression. One can construct the quotient defining $\\mathbf{C}P^1$ in two steps. Multiplication by a real scalar changes the normalization of a state, so we can use this freedom to work with normalized states satisfying:*\n>\n> $$\n> \\langle\\psi|\\psi\\rangle=|z_1|^2+|z_2|^2=1.\n> $$\n>\n> *These normalized states are unit vectors in $\\mathbf{C}^2$, represented by points on the unit sphere $S^3\\subset\\mathbf{R}^4=\\mathbf{C}^2$.*\n>\n> *We must still quotient by multiplication by a phase, identifying points of $S^3$ that differ by a factor $e^{i\\theta}$. The resulting space is often written $S^3/U(1)$. Its circle fibers are disjoint orbits of the phase action, and this structure is called the [Hopf fibration](../../../../../concepts/topology/hopf-fibration.md). Try searching online for visualizations of this geometry: adding a point at infinity to three-dimensional space gives $S^3$, which is surprisingly decomposed into disjoint curves.*\n", "quote_sha256": "45bcacf3089c084500902de0dd481e9633354e6eaaaf7cd45066ebeb8977ae43", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-ac68b4b8eda851f5fc7758b3" } ], "registered_positions": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/04-complex-projective-space", "lang": "en", "section": "7.4", "signal": "term-mark", "label": "Hopf fibration", "english": "Hopf fibration", "offset": 7565 }, { "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/04-complex-projective-space", "lang": "zh-CN", "section": "7.4", "signal": "term-mark", "label": "Hopf 纤维化", "english": "Hopf fibration", "offset": 5777 } ], "statements": [], "resolved_references": [], "reading_order": [ "occ-ce096f4b198a7c772a9b5296", "occ-ac68b4b8eda851f5fc7758b3" ], "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": "6f9808b60c008d02db71e69fb9f37f115063cce9716e9fc7f74f819393563d9e", "optional_enrichments": [] }