A Uniqueness Theorem for Constraint Quantization

TL;DR

This paper proves a uniqueness theorem for the rigging map in the Dirac quantization of constrained systems, showing it is uniquely given by group averaging for finite-dimensional gauge groups.

Contribution

It establishes a condition ensuring the rigging map's uniqueness in the Dirac approach, linking it to group averaging methods for finite-dimensional gauge groups.

Findings

Rigging map uniqueness under specific conditions

Rigging map given by group averaging techniques

Applicable to finite-dimensional Lie gauge groups

Abstract

This work addresses certain ambiguities in the Dirac approach to constrained systems. Specifically, we investigate the space of so-called ``rigging maps'' associated with Refined Algebraic Quantization, a particular realization of the Dirac scheme. Our main result is to provide a condition under which the rigging map is unique, in which case we also show that it is given by group averaging techniques. Our results comprise all cases where the gauge group is a finite-dimensional Lie group.