Quantization of constrained systems as Dirac first class versus second class: a toy model and its implications

TL;DR

This paper compares Dirac's First and Second Class constrained systems using a toy model, highlighting how converting First Class to Second Class systems can clarify quantization procedures and resolve inconsistencies.

Contribution

It demonstrates the advantages of converting First Class constraints into Second Class form before quantization, with implications for Dirac's quantization method.

Findings

Second Class systems yield well-defined reduced phase space.

Converting First Class to Second Class clarifies quantum constraint algebra.

The approach resolves inconsistencies in Dirac quantization.

Abstract

A toy model (suggested by Klauder) is analyzed from the perspective of First Class and Second Class Dirac constrained systems. The comparison is made by turning a First Class into a Second Class system with the introduction of suitable auxiliary conditions. The links between Dirac's system of constraints, the Faddeev-Popov canonical functional integral method and the Maskawa-Nakajima procedure to reduced phase space are explicitly illustrated. The model reveals stark contrasts and physically distinguishable results between First and Second class routes. Physically relevant systems such as the relativistic point particle and electrodynamics are briefly recapped. Besides its pedagogical value, the article also advocates the route of rendering First Class into Second Class systems prior to quantization. Second Class systems lead to well-defined reduced phase space and physical observables; absence of inconsistencies in the closure of quantum constraint algebra; and consistent promotion of fundamental Dirac brackets to quantum commutators. As First Class systems can be turned into well-defined Second Class ones, this has implications for the soundness of "Dirac quantization" of First Class constrained systems by simple promotion of Poisson, rather than Dirac brackets, to commutators without proceeding through Second Class procedures.