On the Generality of Refined Algebraic Quantization

TL;DR

This paper investigates the scope of refined algebraic quantization as a method for quantizing constrained systems, establishing conditions under which it aligns with the traditional Dirac approach for systems with Lie algebra constraints.

Contribution

It identifies technical conditions where refined algebraic quantization reproduces the Dirac scheme, linking inner product choices to rigging maps.

Findings

Refined algebraic quantization can replicate Dirac quantization under certain conditions.

Inner product choices are equivalent to rigging maps in the quantization process.

The approach applies to systems with constraints forming a Lie algebra.

Abstract

The Dirac quantization `procedure' for constrained systems is well known to have many subtleties and ambiguities. Within this ill-defined framework, we explore the generality of a particular interpretation of the Dirac procedure known as refined algebraic quantization. We find technical conditions under which refined algebraic quantization can reproduce the general implementation of the Dirac scheme for systems whose constraints form a Lie algebra with structure constants. The main result is that, under appropriate conditions, the choice of an inner product on the physical states is equivalent to the choice of a ``rigging map'' in refined algebraic quantization.