Coherent states on spheres
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.
- Publication:
-
Journal of Mathematical Physics
- Pub Date:
- March 2002
- DOI:
- 10.1063/1.1446664
- arXiv:
- arXiv:quant-ph/0109086
- Bibcode:
- 2002JMP....43.1211H
- Keywords:
-
- 03.65.Ge;
- 03.65.Db;
- 02.30.Uu;
- Solutions of wave equations: bound states;
- Functional analytical methods;
- Integral transforms;
- Quantum Physics;
- General Relativity and Quantum Cosmology;
- High Energy Physics - Theory;
- Mathematical Physics
- E-Print:
- Revised version. Submitted to J. Mathematical Physics