A Uniqueness Theorem for Constraint Quantization

TL;DR

This paper addresses ambiguities in Dirac-style constrained quantization by refining the RAQ framework and focusing on the rigging map $\eta$. It introduces a group-algebra approach with $L^1$-based structures and shows that, whenever the group averaging integral converges rapidly enough and yields a nontrivial result, the rigging map is unique up to a scalar and coincides with the group-averaging construction. Crucially, the results apply to any locally compact Lie gauge group, including non-unimodular and non-amenable cases, by leveraging the symmetric measure ${d_0}g$ and a unimodularisation strategy. The work also defines ultraweak containment as a related topology that clarifies when RAQ and group averaging can reproduce a trivial gauge-invariant sector, offering a robust criterion for constructing the physical Hilbert space in constrained quantum systems with finite-dimensional 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.