Brian C. Hall, Jeffrey J. Mitchell
Coherent states on spheres
(104K, Latex)
ABSTRACT. 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 S^d. The coherent states are
labeled by points in the associated phase space T*(S^d). 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 B. Hall and M. Stenzel, we give here
a substantially different description based on ideas of T. Thiemann
and of K. Kowalski and J. 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 be able to carry out the calculations in a self-contained and
explicit way.