Quantization of Constrained Systems as Dirac First Class versus Second Class: A Toy Model and Its Implications

A toy model (suggested by Klauder) was analyzed from the perspective of first-class and second-class Dirac constrained systems. First-class constraints are often associated with the existence of important gauge symmetries in a system. A comparison was made by turning a first-class system 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 for reducing the 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 systems into second-class systems prior to quantization. Second-class systems lead to a well-defined reduced phase space and physical observables; an absence of inconsistencies in the closure of quantum constraint algebra; and the 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 the "Dirac quantization" of first-class constrained systems by the simple promotion of Poisson brackets, rather than Dirac brackets, to commutators without proceeding through second-class procedures.