Coherent states on spheres

We describe a family of coherent states and an associated resolution of the identity for a quantum particle whose classical configuration space is the d-dimensional sphere Sd. The coherent states are labeled by points in the associated phase space T*(Sd). These coherent states are not of Perelomov type but rather are constructed as the eigenvectors of suitably defined annihilation operators. We describe as well the Segal-Bargmann representation for the system, the associated unitary Segal-Bargmann transform, and a natural inversion formula. Although many of these results are in principle special cases of the results of Hall and Stenzel, we give here a substantially different description based on ideas of Thiemann and of Kowalski and Rembielinski. All of these results can be generalized to a system whose configuration space is an arbitrary compact symmetric space. We focus on the sphere case in order to carry out the calculations in a self-contained and explicit way