Constrained Quantization on Symplectic Manifolds and Quantum Distribution Functions

TL;DR

This paper develops a geometric quantization scheme for constrained systems on symplectic manifolds, introducing quantum distribution functions and generalized coherent states with potential physical applications.

Contribution

It extends phase space quantization methods by incorporating constrained quantization techniques, generalizing Gupta-Bleuler conditions, and connecting to Berezin quantization.

Findings

Proposed a new quantization scheme for constrained systems.

Constructed generalized coherent states related to Berezin quantization.

Introduced quantum distribution functions with physical interpretation.

Abstract

A quantization scheme based on the extension of phase space with application of constrained quantization technic is considered. The obtained method is similar to the geometric quantization. For constrained systems the problem of scalar product on the reduced Hilbert space is investigated and possible solution of this problem is done. Generalization of the Gupta-Bleuler like conditions is done by the minimization of quadratic fluctuations of quantum constraints. The scheme for the construction of generalized coherent states is considered and relation with Berezin quantization is found. The quantum distribution functions are introduced and their physical interpretation is discussed.