Oscillator representation

In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself.

The contraction operators, determined only up to a sign, have kernels that are Gaussian functions. On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and summarizes how the theory can be extended.