Four approaches to quantization of the relativistic particle

TL;DR

This paper compares four different methods of quantizing the relativistic particle, clarifying their relationships and emphasizing the importance of the full lapse range in path integral formulations.

Contribution

It systematically analyzes and connects four quantization approaches for the relativistic particle, providing insights into their equivalences and differences.

Findings

All four approaches are interconnected and can be related through the BFV path integral.

Using the full lapse range is crucial for accurate path integral quantization.

The relationships between reduced phase space, Dirac, BRST, and Fock quantizations are explicitly established.

Abstract

The connection between four different approaches to quantization of the relativistic particle is studied: reduced phase space quantization, Dirac quantization, BRST quantization, and (BRST)-Fock quantization are each carried out. The connection to the BFV path integral in phase space is provided. In particular, it is concluded that that the full range of the lapse should be used in such path integrals. The relationship between all these approaches is established.