An algebraic extension of Dirac quantization: Examples

TLDR: In this article, the key conceptual and technical aspects of the algebraic program are illustrated through a number of finite dimensional examples and some of the analysis is motivated by certain peculiar problems endemic to quantum gravity.

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 nonlinear, diffeomorphism invariant theories such as general relativity Recently, an extension of the required type was proposed 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