The Segal–Bargmann transform and its generalizations

Abstract

This paper gives an introduction to the Segal–Bargmann transform and its generalizations. The classical Segal–Bargmann transform is a unitary transform between the Schr˝odinger and Fock representations of quantum mechanics on Euclidean space. Hall has generalized this transform to include the case of compact Lie groups. His transform consists of two steps: First take the convolution with the heat kernel and then take the analytic continuation of the resulting function. We will prove the unitarity of this transform by showing that both of these steps are unitary. In the last part we will try to generalize this method to include the case of noncompact symmetric spaces.