{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
  "labels": {
    "zh-CN": "复Clifford代数",
    "en": "complex Clifford algebra"
  },
  "alternative_labels": [
    {
      "lang": "en",
      "value": "Clifford algebra"
    },
    {
      "lang": "zh-CN",
      "value": "Clifford代数"
    }
  ],
  "scope_note": "§32: Cliff(n,C) in the boson/fermion comparison.",
  "identity_status": "active",
  "identity_revision": "4bc324a7448977be33894fc8d7ae2345da0b52694230780b967a606680d5fc58",
  "occurrences": [
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "first-e703c0ef1873250004e6-zh",
      "chapter": "28",
      "section": "28.1",
      "offset": 3583,
      "range": [
        11192,
        11515
      ],
      "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "复Clifford代数(complex Clifford algebra)",
      "context": {
        "range": [
          11171,
          11759
        ],
        "quote": "\n> [!definition] 定义（[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\" id=\"first-e703c0ef1873250004e6-zh\">复Clifford代数(complex Clifford algebra)</u>](../../../../../concepts/algebra/complex-clifford-algebra)）\n>\n> 单自由度的复Clifford代数 $\\operatorname{Cliff}(2,\\mathbb C)$ 由 $1,a_F,a_F^\\dagger$ 生成，满足正则反对易关系（CAR）：\n>\n> $$\n> [a_F,a_F^\\dagger]_+=1,\\qquad[a_F,a_F]_+=[a_F^\\dagger,a_F^\\dagger]_+=0.\n> $$\n",
        "quote_sha256": "6dd922f0c9ed2a561d5f99f2b5bb196bb8e1d2b130aee6528fd040050a9392a4",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "en",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-a7948789ddd705bf2ee58b77"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "first-e703c0ef1873250004e6-en",
      "chapter": "28",
      "section": "28.1",
      "offset": 4819,
      "range": [
        11529,
        11839
      ],
      "source_revision": "1437e8e29fe07ba5486788969d959b4b958fe09241e74dc8c02204e7032cf358",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "complex Clifford algebra",
      "context": {
        "range": [
          11499,
          12177
        ],
        "quote": "\n> [!definition] Definition ([<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\" id=\"first-e703c0ef1873250004e6-en\">complex Clifford algebra</u>](../../../../../concepts/algebra/complex-clifford-algebra))\n>\n> The complex Clifford algebra for one degree of freedom, $\\operatorname{Cliff}(2,\\mathbb C)$, is generated by $1,a_F,a_F^\\dagger$, subject to the canonical anticommutation relations (CAR):\n>\n> $$\n> [a_F,a_F^\\dagger]_+=1,\\qquad[a_F,a_F]_+=[a_F^\\dagger,a_F^\\dagger]_+=0.\n> $$\n",
        "quote_sha256": "bb2427115dd0e16694f4cafea899ac5814288d8a1639800dcad2eeb5725767b9",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-7a12425dc6377599b0c04cc8"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "source-28-1-complex-clifford-algebra-arbitrary-d",
      "chapter": "28",
      "section": "28.1",
      "offset": 7469,
      "range": [
        15078,
        15366
      ],
      "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "复Clifford代数(complex Clifford algebra)",
      "context": {
        "range": [
          15057,
          15659
        ],
        "quote": "\n> [!definition] 定义（[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\">复Clifford代数(complex Clifford algebra)</u>](../../../../../concepts/algebra/complex-clifford-algebra)）\n>\n> 具有 $d$ 个自由度的复Clifford代数 $\\operatorname{Cliff}(2d,\\mathbb C)$ 由 $1,a_{Fj},a_{Fj}^\\dagger$（$j=1,\\ldots,d$）生成，满足CAR：\n>\n> $$\n> [a_{Fj},a_{Fk}^\\dagger]_+=\\delta_{jk}1,\\qquad[a_{Fj},a_{Fk}]_+=[a_{Fj}^\\dagger,a_{Fk}^\\dagger]_+=0.\n> $$\n",
        "quote_sha256": "a73204cec4bb417de6a6a90133dd6beee262e9515c4983c3934b968be78c2570",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "en",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-552161c3dcf13d4ed492cb1e"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "source-28-1-complex-clifford-algebra-gamma-generators",
      "chapter": "28",
      "section": "28.1",
      "offset": 8196,
      "range": [
        15805,
        16093
      ],
      "source_revision": "d2a329a1c341b3d909aed292f756d80691f87ff3253e09de676b6b167a5fefb8",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "复Clifford代数(complex Clifford algebra)",
      "context": {
        "range": [
          15784,
          16300
        ],
        "quote": "\n> [!definition] 定义（[<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\">复Clifford代数(complex Clifford algebra)</u>](../../../../../concepts/algebra/complex-clifford-algebra)）\n>\n> $n$ 个变量的复Clifford代数 $\\operatorname{Cliff}(n,\\mathbb C)$ 由 $1,\\gamma_j$（$j=1,\\ldots,n$）生成，满足\n>\n> $$\n> [\\gamma_j,\\gamma_k]_+=2\\delta_{jk}.\n> $$\n",
        "quote_sha256": "53cc4b5bdd0fd5f77a0c9f71d94c8119481d2f23e15bdaef0f4dc4541048a8d9",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "en",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-0e2eae2cf29d9d6e9e841427"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "source-28-1-complex-clifford-algebra-arbitrary-d",
      "chapter": "28",
      "section": "28.1",
      "offset": 10132,
      "range": [
        16842,
        17117
      ],
      "source_revision": "1437e8e29fe07ba5486788969d959b4b958fe09241e74dc8c02204e7032cf358",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "complex Clifford algebra",
      "context": {
        "range": [
          16812,
          17443
        ],
        "quote": "\n> [!definition] Definition ([<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\">complex Clifford algebra</u>](../../../../../concepts/algebra/complex-clifford-algebra))\n>\n> For $d$ degrees of freedom, $\\operatorname{Cliff}(2d,\\mathbb C)$ is generated by $1,a_{Fj},a_{Fj}^\\dagger$ for $j=1,\\ldots,d$, subject to the CAR:\n>\n> $$\n> [a_{Fj},a_{Fk}^\\dagger]_+=\\delta_{jk}1,\\qquad[a_{Fj},a_{Fk}]_+=[a_{Fj}^\\dagger,a_{Fk}^\\dagger]_+=0.\n> $$\n",
        "quote_sha256": "19717cc4fd819ab6559c953e15f56a34c80ccd657cfc837be9e9622557c8687d",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-d30765f38fed17a816e5dd16"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "anchor": "source-28-1-complex-clifford-algebra-gamma-generators",
      "chapter": "28",
      "section": "28.1",
      "offset": 10969,
      "range": [
        17679,
        17954
      ],
      "source_revision": "1437e8e29fe07ba5486788969d959b4b958fe09241e74dc8c02204e7032cf358",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "complex Clifford algebra",
      "context": {
        "range": [
          17649,
          18194
        ],
        "quote": "\n> [!definition] Definition ([<u class=\"concept-term\" data-concept-reference=\"concepts/algebra/complex-clifford-algebra\" data-concept-identity=\"urn:uuid:33849610-9088-454e-a11d-ba3188cc7663\" data-concept-route=\"/concept-identities#concept-33849610-9088-454e-a11d-ba3188cc7663\">complex Clifford algebra</u>](../../../../../concepts/algebra/complex-clifford-algebra))\n>\n> The complex Clifford algebra $\\operatorname{Cliff}(n,\\mathbb C)$ is generated by $1,\\gamma_j$ for $j=1,\\ldots,n$, subject to\n>\n> $$\n> [\\gamma_j,\\gamma_k]_+=2\\delta_{jk}.\n> $$\n",
        "quote_sha256": "479a78a900f7907ac1af871e88cba8f1fcc633c42ea5260b38f5c63675a18ec9",
        "environment": "definition",
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-e096ff603158511034efcc37"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "zh-CN",
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-32/index",
      "anchor": "first-ee7f69a93a291c68572b-zh",
      "chapter": "32",
      "section": "32",
      "offset": 1739,
      "range": [
        8159,
        8473
      ],
      "source_revision": "0d06f91cc3d120333950968ebf19f8c97275db2cacbbb8a4de54cdc403d91ab2",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "Clifford代数(Clifford algebra)",
      "context": {
        "range": [
          6541,
          10566
        ],
        "quote": null,
        "quote_sha256": "2f36b287a65d8b0d93ef8b15fd02cb05a4a16ea1220663e9a0ac458f067d4c47",
        "environment": null,
        "excerpt_omitted": true
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-32/index",
          "lang": "en",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-cee367cdca41c1f75282ab1d"
    },
    {
      "concept_id": "urn:uuid:33849610-9088-454e-a11d-ba3188cc7663",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-32/index",
      "anchor": "first-ee7f69a93a291c68572b-en",
      "chapter": "32",
      "section": "32",
      "offset": 2224,
      "range": [
        7853,
        8155
      ],
      "source_revision": "849c0b9aa0f3182348e5286d3530f10f3930f3ef55e3740ca9c16e4e2e1d75f5",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "Clifford algebra",
      "context": {
        "range": [
          5982,
          10441
        ],
        "quote": null,
        "quote_sha256": "04c43fda206ae6198cba9c52c476a32fec321fb5446cf30a0dce21402bbfe26b",
        "environment": null,
        "excerpt_omitted": true
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-32/index",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-83b3a5c7a0c70a17e2a13a14"
    }
  ],
  "registered_positions": [
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "en",
      "section": "28.1",
      "signal": "term-mark",
      "label": "complex Clifford algebra",
      "english": "complex Clifford algebra",
      "offset": 4246
    },
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "en",
      "section": "28.1",
      "signal": "term-mark",
      "label": "complex Clifford algebra",
      "english": "complex Clifford algebra",
      "offset": 9212
    },
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "en",
      "section": "28.1",
      "signal": "term-mark",
      "label": "complex Clifford algebra",
      "english": "complex Clifford algebra",
      "offset": 9893
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "zh-CN",
      "section": "28.1",
      "signal": "term-mark",
      "label": "复Clifford代数",
      "english": "complex Clifford algebra",
      "offset": 2854
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "zh-CN",
      "section": "28.1",
      "signal": "term-mark",
      "label": "复Clifford代数",
      "english": "complex Clifford algebra",
      "offset": 6393
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-28/01-the-complex-weyl-and-cliford-algebras",
      "lang": "zh-CN",
      "section": "28.1",
      "signal": "term-mark",
      "label": "复Clifford代数",
      "english": "complex Clifford algebra",
      "offset": 6964
    },
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-32/index",
      "lang": "en",
      "section": "32",
      "signal": "term-mark",
      "label": "Clifford algebra",
      "english": "Clifford algebra",
      "offset": 1721
    },
    {
      "route": "books/woit-qtgr/2025-10-20/zh/chapter-32/index",
      "lang": "zh-CN",
      "section": "32",
      "signal": "term-mark",
      "label": "Clifford代数",
      "english": "Clifford algebra",
      "offset": 1236
    }
  ],
  "statements": [],
  "resolved_references": [],
  "reading_order": [
    "occ-a7948789ddd705bf2ee58b77",
    "occ-7a12425dc6377599b0c04cc8",
    "occ-552161c3dcf13d4ed492cb1e",
    "occ-0e2eae2cf29d9d6e9e841427",
    "occ-d30765f38fed17a816e5dd16",
    "occ-e096ff603158511034efcc37",
    "occ-cee367cdca41c1f75282ab1d",
    "occ-83b3a5c7a0c70a17e2a13a14"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 8,
    "located_occurrences": 8,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "0ada8e8a628b6eb8d8fe15ffc2bc30cadee120dd2011c6806ce37a62b1f04001",
  "optional_enrichments": []
}
