{
  "schema_version": 1,
  "kind": "ConceptView",
  "producer": "concept-view-1",
  "concept_id": "urn:uuid:74059dab-b498-4887-9978-fad6081fc63d",
  "labels": {
    "en": "unitary representations"
  },
  "alternative_labels": [],
  "scope_note": "",
  "identity_status": "active",
  "identity_revision": "0f9171d43815aa03a3fa46dee9182952fb19c827692eef060329b780cb53802e",
  "occurrences": [
    {
      "concept_id": "urn:uuid:74059dab-b498-4887-9978-fad6081fc63d",
      "book_id": "woit-qtgr",
      "edition_id": "2025-10-20",
      "lang": "en",
      "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index",
      "anchor": "first-79b3af86386f7fc2ae7b-en",
      "chapter": "25",
      "section": "25",
      "offset": 1928,
      "range": [
        8349,
        8670
      ],
      "source_revision": "a4fabc3ade7e18eac393871eaed9571dfcf75d61baf7220e6eb15075ae13412d",
      "source_category": "machine_draft",
      "review_status": "machine-draft-not-reviewed",
      "role": "occurrence",
      "role_basis": {
        "kind": "explicit-source-uuid-marker"
      },
      "label": "unitary representations",
      "context": {
        "range": [
          8281,
          9615
        ],
        "quote": "This method turns unitary symmetries of classical phase space into [<u class=\"concept-term\" data-concept-reference=\"concepts/representation-theory/unitary-representation\" data-concept-identity=\"urn:uuid:74059dab-b498-4887-9978-fad6081fc63d\" data-concept-route=\"/concept-identities#concept-74059dab-b498-4887-9978-fad6081fc63d\" id=\"first-79b3af86386f7fc2ae7b-en\">unitary representations</u>](../../../../../concepts/representation-theory/unitary-representation) of symmetry groups on quantum state spaces. It is explained in detail here because it applies to simple systems such as the $d$-dimensional harmonic oscillator and will also be fundamental to our later study of quantum field theory. In that setting, the observables of interest are operators in a [<u class=\"concept-term\" data-concept-reference=\"concepts/lie-theory/lie-algebra-representation\" data-concept-identity=\"urn:uuid:5bad1df2-281a-4c11-8362-128d54fb523a\" data-concept-route=\"/concept-identities#concept-5bad1df2-281a-4c11-8362-128d54fb523a\">Lie algebra representation</u>](../../../../../concepts/lie-theory/lie-algebra-representation), built from quadratic combinations of annihilation and creation operators. They arise from the constructions in this chapter applied to unitary group actions on phase space, which is infinite-dimensional in quantum field theory.",
        "quote_sha256": "cec01481beb36576c207dbbb6d0e26b7f57a703b143a60e88b434e38bfeceb43",
        "environment": null,
        "excerpt_omitted": false
      },
      "parallel_readings": [
        {
          "route": "books/woit-qtgr/2025-10-20/zh/chapter-25/index",
          "lang": "zh-CN",
          "source_category": "machine_draft",
          "basis": "registered-book-edition-section-language-context; not-concept-equivalence"
        }
      ],
      "id": "occ-f77ffff6d39cdab412d5a93a"
    }
  ],
  "registered_positions": [
    {
      "route": "books/woit-qtgr/2025-10-20/en/chapter-25/index",
      "lang": "en",
      "section": "25",
      "signal": "term-mark",
      "label": "unitary representations",
      "english": "unitary representations",
      "offset": 1546
    }
  ],
  "statements": [],
  "resolved_references": [],
  "reading_order": [
    "occ-f77ffff6d39cdab412d5a93a"
  ],
  "recorded_dependencies": [],
  "reading_paths": [],
  "coverage": {
    "registered_positions": 1,
    "located_occurrences": 1,
    "whole_book_search_complete": false,
    "registered_positions_truncated": false,
    "independent_human_review_implied": false
  },
  "unresolved": [],
  "base_revision": "9d2cca01599439563f146ee63083f2a977c030e9ed8bad255ee402c711de79ae",
  "optional_enrichments": []
}