Oscillator representation

Summary

The oscillator representation is a unitary representation of the metaplectic group, the double cover of the real symplectic group, on $L^2(\R^n)$. It exists because symplectic linear maps preserve the canonical commutation relations, and the Stone–von Neumann theorem then gives an intertwining operator for each of them. It is the real case of the Weil representation, and it underlies the theory of theta functions and the theta correspondence. Its name comes from the quantum harmonic oscillator, whose Hamiltonian generates the action of a compact one-parameter subgroup.