Group Averaging and Refined Algebraic Quantization: Where are we now?

TL;DR

The paper addresses the quantization of constrained systems in gravity contexts by focusing on refined algebraic quantization and group averaging as constructive routes to a physical inner product and observables. It explains the formalism through a kinematical space, a rigging map, and a gauge-group averaging procedure that yields the physical Hilbert space, clarifying differences between unimodular and non-unimodular groups and the role of structure functions. It surveys current progress and open questions across comparisons with other quantization methods, the treatment of structure functions, convergence issues, concrete examples, and semiclassical techniques, highlighting areas where further theoretical and computational work is needed. The work underscores the significance of resolving these issues for applying robust canonical quantization to quantum gravity and for connecting RAQ/group averaging to path-integral and geometric quantization frameworks.

Abstract

Refined Algebraic Quantization and Group Averaging are powerful methods for quantizing constrained systems. They give constructive algorithms for generating observables and the physical inner product. This work outlines the current status of these ideas with an eye toward quantum gravity. The main goal is provide a description of outstanding problems and possible research topics in the field.