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  "body": "\n## Summary\n\nThe oscillator representation is a unitary representation of the [[metaplectic group|8B0JM2YDE2]], the double cover of the real symplectic group, on $L^2(\\R^n)$.\nIt exists because symplectic linear maps preserve the canonical commutation relations, and the [[Stone–von Neumann theorem|Z877XDBMQ3]] then gives an intertwining operator for each of them.\nIt is the real case of the Weil representation, and it underlies the theory of theta functions and the [[theta correspondence|ZBGT50BW0Z]].\nIts name comes from the quantum harmonic oscillator, whose Hamiltonian generates the action of a compact one-parameter subgroup.\n\n## Explanations\n\n### L1 human\n\nLinear changes of the coordinates $(x, \\xi)$ of position and momentum that keep the symplectic area form are the symplectic maps.\nQuantum mechanics replaces $(x, \\xi)$ by operators on wave functions $\\psi(x)$.\nEach symplectic change of coordinates then corresponds to a unitary operator on wave functions, defined up to a sign.\nFor example, a quarter turn in the $(x, \\xi)$ plane corresponds, up to a constant, to the Fourier transform, which swaps position and momentum.\nThese operators together form the oscillator representation.\n",
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