{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "1GJ6SRT6B7"
    },
    {
      "field": "uses",
      "target": "1J67Y01N4X"
    },
    {
      "field": "uses",
      "target": "1Y7AH9QTEQ"
    },
    {
      "field": "uses",
      "target": "2BDH4RQZQ9"
    },
    {
      "field": "uses",
      "target": "2CKZETC3QA"
    },
    {
      "field": "uses",
      "target": "3JE85VQR4G"
    },
    {
      "field": "uses",
      "target": "5E547TJ7PS"
    },
    {
      "field": "uses",
      "target": "5J603A6DFH"
    },
    {
      "field": "uses",
      "target": "6VC9QE39XH"
    },
    {
      "field": "uses",
      "target": "7PY20WMZDZ"
    },
    {
      "field": "uses",
      "target": "972X7QM34D"
    },
    {
      "field": "uses",
      "target": "C9DT4Y63CY"
    },
    {
      "field": "uses",
      "target": "CFDZJ55N95"
    },
    {
      "field": "uses",
      "target": "CTYQQWJPMB"
    },
    {
      "field": "uses",
      "target": "DH3PYQJXH0"
    },
    {
      "field": "uses",
      "target": "E3RQ0J3ACZ"
    },
    {
      "field": "uses",
      "target": "EHGRPCTM3J"
    },
    {
      "field": "uses",
      "target": "ETNDTECY97"
    },
    {
      "field": "uses",
      "target": "Z877XDBMQ3"
    },
    {
      "field": "about",
      "target": "6VC9QE39XH"
    },
    {
      "field": "about",
      "target": "CFDZJ55N95"
    },
    {
      "field": "about",
      "target": "DH3PYQJXH0"
    },
    {
      "field": "about",
      "target": "E3RQ0J3ACZ"
    },
    {
      "field": "specializes",
      "target": "9AB7HYGPEK"
    },
    {
      "field": "defines",
      "target": "DZF985Y137"
    }
  ],
  "body": "\n## Summary\n\nA unitary representation of a [[topological group|92700QAB5Y]] $G$ is a homomorphism from $G$ to the [[unitary operators|MM0EJ2GDE4]] on a [[Hilbert space|HM2HHE3J51]].\nIt must be continuous in the strong operator topology.\nIt is the kind of [[group representation|WYGRXCG2RN]] used for infinite groups in analysis and physics, because unitary operators preserve lengths and angles.\nHarmonic analysis on a locally compact group is largely the study of its unitary representations.\nIn quantum mechanics, symmetry groups act on state spaces in this way.\n\n## Explanations\n\n### L1 human\n\nA unitary operator is a rotation of a possibly infinite-dimensional space: it keeps all lengths and angles.\nA unitary representation assigns such a rotation to each element of a group, so that the product of two elements goes to the composition of their rotations.\nFor example, the real line acts on the space $L^2(\\R)$ of square-integrable functions by shifting: the number $t$ sends $f$ to the function $x \\mapsto f(x - t)$.\nShifting does not change the integral of $\\abs{f}^2$, so each shift is unitary, and shifting by a small amount changes $f$ only a little.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "22D10",
      "43A65"
    ],
    "name": "Unitary representation",
    "same_as": {
      "wikidata": "Q10359539",
      "wikipedia": "Unitary representation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Unitary_representation"
      }
    ],
    "specializes": [
      "WYGRXCG2RN"
    ],
    "tag": "74BPYJNZTM"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/74BPYJNZTM/",
  "links": [
    {
      "field": "specializes",
      "target": "WYGRXCG2RN"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/74BPYJNZTM.md",
  "msc": [
    "22D10",
    "43A65"
  ],
  "name": "Unitary representation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/74B/74BPYJNZTM.md",
  "status": null,
  "tag": "74BPYJNZTM",
  "uri": "m2://74BPYJNZTM",
  "url": "https://tamnd.github.io/math2/wiki/unitary-representation/",
  "writing": {
    "tag": "74BPYJNZTM",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}