{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:c985d06a-9bb8-4419-8598-30d94e1b91f8",
  "labels": {
    "zh-CN": "正交复结构",
    "en": "orthogonal complex structure"
  },
  "alternative_labels": [],
  "scope_note": "Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。",
  "identity_status": "active",
  "identity_revision": "6217816558683e08030ccf563f8c0536f0dc685f686c2b2b7c8197873ac70258",
  "occurrences": [],
  "registered_positions": [],
  "statements": [],
  "resolved_references": [],
  "reading_order": [],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 0,
    "located_occurrences": 0,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "reviewed_definitions": [
    {
      "concept_id": "urn:uuid:c985d06a-9bb8-4419-8598-30d94e1b91f8",
      "definition_id": "urn:uuid:6e3aa928-40f9-5a16-afec-58e029ae6915",
      "definition_group": "reviewed-unit-6e3aa928-40f9-5a16-afec-58e029ae6915",
      "path": "library/concepts/source-evidence/mcp/14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38.txt",
      "range": [
        0,
        588
      ],
      "quote": "The construction of the spinor representation given here has used a specific choice of the $\\theta _ { j } , \\overline { { \\theta } } _ { j }$ (see equations 31.2) and the fermionic annihilation and creation operators. This corresponds to a standard choice of complex structure $J _ { 0 }$ , which appears in a manner closely parallel to that of the Bargmann-Fock case of section 26.1. The diference here is that, for the analogous construction of spinors, the complex structure J must be chosen so as to preserve not an antisymmetric bilinear form Ω, but the inner product, and one has\n\n",
      "document_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "quote_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "scope": {
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20"
      },
      "section": "31.4",
      "pdf_page_index": 360,
      "node": "legacy36-orthogonal-complex-structure-0",
      "source_node_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "upstream": {
        "path": "library/concepts/source-evidence/mcp/d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc.json",
        "document_sha256": "d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc",
        "node_id": "legacy36-orthogonal-complex-structure-0",
        "node_source_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38"
      },
      "conditions": "Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。",
      "review_status": "librarian-original-source-reviewed",
      "source_locator": {
        "path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json",
        "sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20",
        "nodes": [
          "legacy36-orthogonal-complex-structure-0",
          "legacy36-orthogonal-complex-structure-1",
          "legacy36-orthogonal-complex-structure-2"
        ],
        "node_hashes": {
          "legacy36-orthogonal-complex-structure-0": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
          "legacy36-orthogonal-complex-structure-1": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
          "legacy36-orthogonal-complex-structure-2": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad"
        },
        "unit_kind": "reviewed-prose-equations"
      },
      "definition_part": "reviewed-definition-unit",
      "environment_basis": {
        "path": "library/concepts/source-evidence/environment-basis/28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f.json.txt",
        "document_sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "source_path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json"
      },
      "role": "definition",
      "first_in_book": "not-established",
      "evidence_origin": "explicit-reviewed-source-manifest",
      "definition_part_basis": "reviewed-source-unit-membership",
      "reading_links": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "en",
          "basis": "registered-section-context"
        },
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "zh-CN",
          "basis": "registered-section-context"
        }
      ]
    },
    {
      "concept_id": "urn:uuid:c985d06a-9bb8-4419-8598-30d94e1b91f8",
      "definition_id": "urn:uuid:6e3aa928-40f9-5a16-afec-58e029ae6915",
      "definition_group": "reviewed-unit-6e3aa928-40f9-5a16-afec-58e029ae6915",
      "path": "library/concepts/source-evidence/mcp/30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6.txt",
      "range": [
        0,
        46
      ],
      "quote": "$$\n(J (\\cdot), J (\\cdot)) = (\\cdot , \\cdot)\n$$",
      "document_sha256": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
      "quote_sha256": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "scope": {
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20"
      },
      "section": "31.4",
      "pdf_page_index": 360,
      "node": "legacy36-orthogonal-complex-structure-1",
      "source_node_sha256": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
      "upstream": {
        "path": "library/concepts/source-evidence/mcp/d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc.json",
        "document_sha256": "d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc",
        "node_id": "legacy36-orthogonal-complex-structure-1",
        "node_source_sha256": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6"
      },
      "conditions": "Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。",
      "review_status": "librarian-original-source-reviewed",
      "source_locator": {
        "path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json",
        "sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20",
        "nodes": [
          "legacy36-orthogonal-complex-structure-0",
          "legacy36-orthogonal-complex-structure-1",
          "legacy36-orthogonal-complex-structure-2"
        ],
        "node_hashes": {
          "legacy36-orthogonal-complex-structure-0": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
          "legacy36-orthogonal-complex-structure-1": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
          "legacy36-orthogonal-complex-structure-2": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad"
        },
        "unit_kind": "reviewed-prose-equations"
      },
      "definition_part": "reviewed-definition-unit",
      "environment_basis": {
        "path": "library/concepts/source-evidence/environment-basis/28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f.json.txt",
        "document_sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "source_path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json"
      },
      "role": "definition",
      "first_in_book": "not-established",
      "evidence_origin": "explicit-reviewed-source-manifest",
      "definition_part_basis": "reviewed-source-unit-membership",
      "reading_links": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "en",
          "basis": "registered-section-context"
        },
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "zh-CN",
          "basis": "registered-section-context"
        }
      ]
    },
    {
      "concept_id": "urn:uuid:c985d06a-9bb8-4419-8598-30d94e1b91f8",
      "definition_id": "urn:uuid:6e3aa928-40f9-5a16-afec-58e029ae6915",
      "definition_group": "reviewed-unit-6e3aa928-40f9-5a16-afec-58e029ae6915",
      "path": "library/concepts/source-evidence/mcp/76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad.txt",
      "range": [
        0,
        174
      ],
      "quote": "\n\nWe will here restrict to the case of $( \\cdot , \\cdot )$ positive definite, and unlike in the bosonic case, no additional positivity condition on $J$ will then be required.",
      "document_sha256": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad",
      "quote_sha256": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "scope": {
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20"
      },
      "section": "31.4",
      "pdf_page_index": 360,
      "node": "legacy36-orthogonal-complex-structure-2",
      "source_node_sha256": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad",
      "upstream": {
        "path": "library/concepts/source-evidence/mcp/d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc.json",
        "document_sha256": "d53b0a75862d6bb2951fd2716c64d491b9404d616145b725a25a987149ccc0dc",
        "node_id": "legacy36-orthogonal-complex-structure-2",
        "node_source_sha256": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad"
      },
      "conditions": "Woit §31.4，实内积空间上的复结构J保持度量g(Jv,Jw)=g(v,w)；此处度量正定。与保持辛形式Omega的相容复结构分开，不声称首次出现。",
      "review_status": "librarian-original-source-reviewed",
      "source_locator": {
        "path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json",
        "sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20",
        "nodes": [
          "legacy36-orthogonal-complex-structure-0",
          "legacy36-orthogonal-complex-structure-1",
          "legacy36-orthogonal-complex-structure-2"
        ],
        "node_hashes": {
          "legacy36-orthogonal-complex-structure-0": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
          "legacy36-orthogonal-complex-structure-1": "30eb81e2bac62a79c0c6df9192fafb8200a5545b4bda84337f7c20ac60b63aa6",
          "legacy36-orthogonal-complex-structure-2": "76dbf4785c90ddaa02b7e7ab3c98f6be84587f184720eb8fe580a3ed9fada2ad"
        },
        "unit_kind": "reviewed-prose-equations"
      },
      "definition_part": "reviewed-definition-unit",
      "environment_basis": {
        "path": "library/concepts/source-evidence/environment-basis/28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f.json.txt",
        "document_sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "source_path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json"
      },
      "role": "definition",
      "first_in_book": "not-established",
      "evidence_origin": "explicit-reviewed-source-manifest",
      "definition_part_basis": "reviewed-source-unit-membership",
      "reading_links": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "en",
          "basis": "registered-section-context"
        },
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "zh-CN",
          "basis": "registered-section-context"
        }
      ]
    }
  ],
  "reviewed_mentions": [
    {
      "concept_id": "urn:uuid:c985d06a-9bb8-4419-8598-30d94e1b91f8",
      "path": "library/concepts/source-evidence/mcp/14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38.txt",
      "range": [
        0,
        588
      ],
      "quote": "The construction of the spinor representation given here has used a specific choice of the $\\theta _ { j } , \\overline { { \\theta } } _ { j }$ (see equations 31.2) and the fermionic annihilation and creation operators. This corresponds to a standard choice of complex structure $J _ { 0 }$ , which appears in a manner closely parallel to that of the Bargmann-Fock case of section 26.1. The diference here is that, for the analogous construction of spinors, the complex structure J must be chosen so as to preserve not an antisymmetric bilinear form Ω, but the inner product, and one has\n\n",
      "document_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "quote_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "scope": {
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20"
      },
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "section": "31.4",
      "pdf_page_index": 360,
      "node": "legacy36-orthogonal-complex-structure-0",
      "source_node_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38",
      "source_locator": {
        "path": "reports/coordination-20261025/legacy-keep-resolution/source-packets/orthogonal-complex-structure.json",
        "sha256": "28ee219c565dd3ec2b765c5594cd8684266f29a2e4b6e1d747a7304c3233bb4f",
        "book_id": "woit-qtgr",
        "edition_id": "2025-10-20",
        "nodes": [
          "legacy36-orthogonal-complex-structure-0"
        ],
        "node_hashes": {
          "legacy36-orthogonal-complex-structure-0": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38"
        }
      },
      "conditions": "librarian codex-3e702767: 用户授权按概念维护#25-B；逐页核原书定义，区分度量/辛相容、基/数、非退化/正定及表示/元素；不按标签合并，不声称首次出现。",
      "review_status": "librarian-original-source-mention-reviewed",
      "upstream": {
        "path": "library/concepts/source-evidence/mcp/20bcd3655ca5e61b93c5064f7f4fd259634faaf1420aa56ac15b0cd46db9b456.json",
        "document_sha256": "20bcd3655ca5e61b93c5064f7f4fd259634faaf1420aa56ac15b0cd46db9b456",
        "node_id": "legacy36-orthogonal-complex-structure-0",
        "node_source_sha256": "14798a9d66cfe2131e5bcef31ead88cbbceb12625491b2d2ddd8049a67e69a38"
      },
      "role": "mention",
      "first_in_book": "not-established",
      "evidence_origin": "explicit-reviewed-source-manifest",
      "reading_links": [
        {
          "route": "books/woit-qtgr/2025-10-20/en/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "en",
          "basis": "registered-section-context"
        },
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-31/05-complex-structures-u-d-so-2-d-and-the-spinor-representation",
          "anchor": "section-31-4",
          "lang": "zh-CN",
          "basis": "registered-section-context"
        }
      ]
    }
  ],
  "part_of_speech": "noun",
  "base_revision": "41343657dbc6de8152994564579ae157d83a4fbdd09f8fc3edaa82331d5c0e5f",
  "optional_enrichments": []
}