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Coherent states in geometric quantization

Authors
Journal
Journal of Geometry and Physics
0393-0440
Publisher
Elsevier
Publication Date
Volume
57
Issue
2
Identifiers
DOI: 10.1016/j.geomphys.2006.04.007
Keywords
  • Geometric Quantization
  • Coherent State
  • Generalized Bergman Kernel
Disciplines
  • Mathematics
  • Physics

Abstract

Abstract In this paper we study overcomplete systems of coherent states associated to compact integral symplectic manifolds by geometric quantization. Our main goals are to give a systematic treatment of the construction of such systems and to collect some recent results. We begin by recalling the basic constructions of geometric quantization in both the Kähler and non-Kähler cases. We then study the reproducing kernels associated to the quantum Hilbert spaces and use them to define symplectic coherent states. The rest of the paper is dedicated to the properties of symplectic coherent states and the corresponding Berezin–Toeplitz quantization. Specifically, we study overcompleteness, symplectic analogues of the basic properties of Bargmann’s weighted analytic function spaces, and the ‘maximally classical’ behavior of symplectic coherent states. We also find explicit formulas for symplectic coherent states on compact Riemann surfaces.

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