Deformation quantization of constrained systems: a group averaging approach

TL;DR

This paper introduces a phase space approach to Dirac quantization for constrained systems using a star group averaging method, enabling explicit solutions to constraints via a physically motivated Wigner distribution.

Contribution

It proposes a novel phase space implementation of Dirac quantization with a star group averaging technique, extending the Refined Algebraic Quantization formalism.

Findings

The method explicitly solves constraints using generalized functions on phase space.

Test cases recover standard results, validating the approach.

Abstract

Motivated by certain concepts introduced by the Refined Algebraic Quantization formalism for constrained systems which has been successfully applied within the context of Loop Quantum Gravity, in this paper we propose a phase space implementation of the Dirac quantization formalism to appropriately include systems with constraints. In particular, we propose a physically prescribed Wigner distribution which allows the definition of a well-defined inner product by judiciously introducing a star version of the group averaging of the constraints. This star group averaging procedure is obtained by considering the star-exponential of the constraints and then integrating with respect to a suitable measure. In this manner, the proposed physical Wigner distribution explicitly solves the constraints by including generalized functions on phase space. Finally, in order to illustrate our approach, our proposal is tested in a couple of simple examples which recover standard results found in the literature.