{
"schema_version": 1,
"kind": "ConceptView",
"producer": "concept-view-1",
"concept_id": "urn:uuid:35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9",
"labels": {
"zh-CN": "圆群",
"en": "circle group"
},
"alternative_labels": [],
"scope_note": "",
"identity_status": "active",
"identity_revision": "0e2fca77a46bef086da02fb2a102d2d3954361e27ea3bcdba8f2db57ec1583a7",
"occurrences": [
{
"concept_id": "urn:uuid:35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9",
"book_id": "woit-qtgr",
"edition_id": "2025-10-20",
"lang": "zh-CN",
"route": "books/woit-qtgr/2025-10-20/zh/chapter-02/index",
"anchor": "first-c205b531a9556fbe5a3d-zh",
"chapter": "2",
"section": "2",
"offset": 741,
"range": [
7999,
8294
],
"source_revision": "419ac63f327dd8ec1940b506658884008ceb07824f738873312b5a720d3bf3b1",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "圆群(circle group)",
"context": {
"range": [
7614,
8915
],
"quote": "最简单的[李群(Lie group)](../../../../../concepts/group-theory/lie-group)例子是平面的旋转群，其元素由单个参数即旋转角$\\theta$来参数化。将这类群元素识别为复平面中的单位向量 $e^{i\\theta}$，会更方便。该群即[圆群(circle group)](../../../../../concepts/group-theory/circle-group)，记作 $U(1)$；这是因为这样的复数可视为 $1\\times1$ 的酉矩阵。我们将在本章中看到[第1](../chapter-01/index#section-1)章所描述的一般图景如何在这个简单情形中实现。状态空间将是 $U(1)$ 群的[酉表示(unitary representation)](../../../../../concepts/representation-theory/unitary-representation)，并且我们将看到任何这样的表示都分解为一系列一维表示之和。这些一维表示由整数 $q$ 标记；这些整数是自伴算符 $Q$ 的特征值，而 $Q$ 是量子理论中的一个可观测量。",
"quote_sha256": "8f0b15d615ffaf0d34d636754f40ddf925a418068c41c623801d07a86b151d80",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "books/woit-qtgr/2025-10-20/en/chapter-02/index",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-1de412b01e123ee19ed59a52"
},
{
"concept_id": "urn:uuid:35b13cd0-fcbd-4d42-aa3a-04bf19d2fbb9",
"book_id": "woit-qtgr",
"edition_id": "2025-10-20",
"lang": "zh-CN",
"route": "books/woit-qtgr/2025-10-20/zh/chapter-02/02-the-group-u-1-and-its-representations",
"anchor": "source-block-2050",
"chapter": "2",
"section": "2.2",
"offset": 522,
"range": [
9735,
9995
],
"source_revision": "a3ac9e2453498f3e54871a0eea32d79b92c27d625bee0884bdc82db6f567d38c",
"source_category": "translation",
"review_status": "unreviewed",
"role": "occurrence",
"role_basis": {
"kind": "explicit-source-uuid-marker"
},
"label": "圆群(circle group)",
"context": {
"range": [
9575,
10073
],
"quote": "人们可能会认为最简单的李群是一维加法群 $\\mathbf{R}$，我们将在[第 10](../chapter-10/index#section-10) 章开始连同其表示一起研究这个群。事实上，加入周期性条件后，便可避开参数趋于 $\\pm\\infty$ 时的行为问题，并得到一个更易于分析的李群——由单位圆上的点组成的“[圆群(circle group)](../../../../../concepts/group-theory/circle-group)”。每个这样的点由一个角刻画，而群法则就是角的加法。",
"quote_sha256": "abf6221cd0e4e049ff752446bc76ccf005ac9e0704056df766ca678362ba3f6b",
"environment": null,
"excerpt_omitted": false
},
"parallel_readings": [
{
"route": "books/woit-qtgr/2025-10-20/en/chapter-02/02-the-group-u-1-and-its-representations",
"lang": "en",
"source_category": "original_transcription",
"basis": "registered-book-edition-section-language-context; not-concept-equivalence"
}
],
"id": "occ-3f32d5fbf4beca3e8ad75d25"
}
],
"registered_positions": [
{
"route": "books/woit-qtgr/2025-10-20/zh/chapter-02/index",
"lang": "zh-CN",
"section": "2",
"signal": "term-mark",
"label": "圆群",
"english": "circle group",
"offset": 585
},
{
"route": "books/woit-qtgr/2025-10-20/zh/chapter-02/02-the-group-u-1-and-its-representations",
"lang": "zh-CN",
"section": "2.2",
"signal": "term-mark",
"label": "圆群",
"english": "circle group",
"offset": 522
}
],
"statements": [],
"resolved_references": [
{
"raw": "[第1](../chapter-01/index#section-1)",
"href": "../chapter-01/index#section-1",
"range": [
8395,
8430
],
"relation": "references",
"target_route": "books/woit-qtgr/2025-10-20/zh/chapter-01/index",
"target_anchor": "section-1",
"resolved": true,
"reason": null,
"source_revision": "419ac63f327dd8ec1940b506658884008ceb07824f738873312b5a720d3bf3b1",
"source_route": "books/woit-qtgr/2025-10-20/zh/chapter-02/index",
"occurrence_id": "occ-1de412b01e123ee19ed59a52"
},
{
"raw": "[第 10](../chapter-10/index#section-10)",
"href": "../chapter-10/index#section-10",
"range": [
9612,
9650
],
"relation": "references",
"target_route": "books/woit-qtgr/2025-10-20/zh/chapter-10/index",
"target_anchor": "section-10",
"resolved": true,
"reason": null,
"source_revision": "a3ac9e2453498f3e54871a0eea32d79b92c27d625bee0884bdc82db6f567d38c",
"source_route": "books/woit-qtgr/2025-10-20/zh/chapter-02/02-the-group-u-1-and-its-representations",
"occurrence_id": "occ-3f32d5fbf4beca3e8ad75d25"
}
],
"reading_order": [
"occ-1de412b01e123ee19ed59a52",
"occ-3f32d5fbf4beca3e8ad75d25"
],
"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": "4a7055d777881cecb2bb945b9d5b316c73fb97bb6d526bd4c5a3fde1c27573d9",
"optional_enrichments": []
}