An algebraic extension of Dirac quantization: Examples

TL;DR

This paper extends Dirac's quantization method for constrained systems using algebraic techniques, illustrated through finite-dimensional examples relevant to quantum gravity, addressing issues in non-linear, diffeomorphism-invariant theories.

Contribution

It introduces an algebraic extension of Dirac quantization, demonstrating its effectiveness through examples and resolving technical issues in quantizing constrained systems.

Findings

Algebraic extension addresses open issues in Dirac quantization.

Examples relevant to quantum gravity illustrate the method.

Resolves technical problems in reduced phase space quantization.

Abstract

An extension of the Dirac procedure for the quantization of constrained systems is necessary to address certain issues that are left open in Dirac's original proposal. These issues play an important role especially in the context of non-linear, diffeomorphism invariant theories such as general relativity. Recently, an extension of the required type was proposed by one of us using algebraic quantization methods. In this paper, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples. The choice of examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity. However, prior knowledge of general relativity is not assumed in the main discussion. Indeed, the methods introduced and conclusions arrived at are applicable to any system with first class constraints. In particular, they resolve certain technical issues which are present also in the reduced phase space approach to quantization of these systems.