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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $W = \\R^{2n}$ with the standard [[symplectic form|PQ7JJY0VPH]] $\\omega$.\nLet $\\mathbb H = W \\times \\R$ be the [[Heisenberg group|ZZYCV5EX9J]] with product $(w, s)(w', s') = (w + w', s + s' + \\tfrac12 \\omega(w, w'))$.\nLet $\\rho$ be the Schrödinger representation of $\\mathbb H$ on $L^2(\\R^n)$ with central character $(0, s) \\mapsto e^{is}$.\nLet $\\mathrm{Sp}(W) = \\mathrm{Sp}_{2n}(\\R)$ be the [[symplectic group|X12FS2JAP0]], which acts on $\\mathbb H$ by automorphisms $g \\cdot (w, s) = (gw, s)$.\nLet $p \\colon \\mathrm{Mp}(W) \\to \\mathrm{Sp}(W)$ be the [[metaplectic group|8B0JM2YDE2]], the connected double cover.\nThe *oscillator representation* is the unitary representation $\\omega_\\psi$ of $\\mathrm{Mp}(W)$ on $L^2(\\R^n)$ such that\n$$\\omega_\\psi(\\tilde g) \\, \\rho(w, s) \\, \\omega_\\psi(\\tilde g)^{-1} = \\rho(p(\\tilde g) w, s)$$\nfor all $\\tilde g \\in \\mathrm{Mp}(W)$ and $(w, s) \\in \\mathbb H$.\n\n## Remarks\n\nSuch a representation exists and is unique.\nFor each $g \\in \\mathrm{Sp}(W)$, the representation $(w, s) \\mapsto \\rho(gw, s)$ has the same central character as $\\rho$.\nSo by the [[Stone–von Neumann theorem|Z877XDBMQ3]] there is a unitary $T_g$ with $T_g \\rho(w, s) T_g^{-1} = \\rho(gw, s)$, unique up to a scalar of modulus $1$.\nThe map $g \\mapsto T_g$ is a projective representation of $\\mathrm{Sp}(W)$, and the scalars can be chosen so that it lifts to a genuine representation of $\\mathrm{Mp}(W)$.\nTwo lifts differ by a character of $\\mathrm{Mp}(W)$, which is trivial since $\\mathrm{Mp}(W)$ is connected with semisimple Lie algebra, hence equal to its commutator subgroup.\n\nExample: for $n = 1$, the dilation $g = \\operatorname{diag}(a, a^{-1})$ with $a \u003e 0$ lifts to the operator $\\psi(x) \\mapsto a^{-1/2} \\psi(x / a)$, up to sign.\nIt is unitary on $L^2(\\R)$.\nConjugation by it turns the position operator $x$ into $x / a$ and the momentum operator $-i \\, d/dx$ into $a \\cdot (-i \\, d/dx)$, which is the action of a diagonal symplectic map on position and momentum.\n\nNon-example: the operators $T_g$ cannot be normalised to give a representation of $\\mathrm{Sp}(W)$ itself.\nOnly the double cover carries a genuine representation, so the condition on $\\mathrm{Mp}(W)$ in the definition cannot be replaced by one on $\\mathrm{Sp}(W)$.\n\nThe oscillator representation splits into two irreducible pieces, on the even and on the odd functions.\n",
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