{
  "aliases": null,
  "backlinks": [
    {
      "field": "specializes",
      "target": "28VV812XWQ"
    },
    {
      "field": "defines",
      "target": "BRXKAGYEFQ"
    }
  ],
  "body": "\n## Summary\n\nA linear canonical transformation is an integral transform of functions on $\\R$ attached to a real $2 \\times 2$ matrix of determinant $1$.\nIt includes the [[Fourier transform|DPVM34ADZV]], the [[fractional Fourier transform|28VV812XWQ]], the Fresnel transform, scalings and multiplication by chirps as special cases.\nComposing two such transforms gives, up to a sign, the transform of the product matrix.\nSo they form a projective unitary representation of $SL(2, \\R)$, which is a genuine representation of the [[metaplectic group|8B0JM2YDE2]].\nIn optics it describes the propagation of light through lenses and free space in the paraxial approximation.\n\n## Explanations\n\n### L1 human\n\nA signal can be pictured in a plane with time along one axis and frequency along the other.\nLinear maps of this plane that keep areas and orientation are given by matrices $\\begin{pmatrix} a \u0026 b \\\\ c \u0026 d \\end{pmatrix}$ with $ad - bc = 1$.\nFor each such map there is a transform of signals that moves their picture in the plane accordingly.\nThe Fourier transform is the quarter-turn rotation, a scaling of time is a squeeze of the plane, and multiplying by $e^{i\\pi c t^2}$ is a shear.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "kind": "concept",
    "msc": [
      "44A15",
      "42A38",
      "81S30"
    ],
    "name": "Linear canonical transformation",
    "same_as": {
      "wikidata": "Q6553419",
      "wikipedia": "Linear canonical transformation"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Linear_canonical_transformation"
      }
    ],
    "specializes": [
      "YCK709QB06"
    ],
    "tag": "74BXAPVSW8"
  },
  "kind": "concept",
  "link": "https://tamnd.github.io/math2/node/74BXAPVSW8/",
  "links": [
    {
      "field": "specializes",
      "target": "YCK709QB06"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/74BXAPVSW8.md",
  "msc": [
    "44A15",
    "42A38",
    "81S30"
  ],
  "name": "Linear canonical transformation",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/74B/74BXAPVSW8.md",
  "status": null,
  "tag": "74BXAPVSW8",
  "uri": "m2://74BXAPVSW8",
  "url": "https://tamnd.github.io/math2/wiki/linear-canonical-transformation/",
  "writing": {
    "tag": "74BXAPVSW8",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}