Star Product for Second Class Constraint Systems from a BRST Theory

TL;DR

This paper develops a covariant deformation quantization method for second-class constrained systems using BRST theory, generalizing Fedosov quantization and explicitly constructing the star product for Dirac brackets.

Contribution

It introduces a novel explicit construction of the star product for second-class systems via an effective gauge system and BRST quantization, extending Fedosov's approach.

Findings

Constructed the star product for Dirac brackets explicitly.

Introduced the Dirac connection as a counterpart to the symplectic connection.

Showed the star product reduces to Fedosov's on the constraint surface.

Abstract

We propose an explicit construction of the deformation quantization of the general second-class constrained system, which is covariant with respect to local coordinates on the phase space. The approach is based on constructing the effective first-class constraint (gauge) system equivalent to the original second-class one and can also be understood as a far-going generalization of the Fedosov quantization. The effective gauge system is quantized by the BFV-BRST procedure. The star product for the Dirac bracket is explicitly constructed as the quantum multiplication of BRST observables. We introduce and explicitly construct a Dirac bracket counterpart of the symplectic connection, called the Dirac connection. We identify a particular star product associated with the Dirac connection for which the constraints are in the center of the respective star-commutator algebra. It is shown that when reduced to the constraint surface, this star product is a Fedosov star product on the constraint surface considered as a symplectic manifold.