Faddeev-Jackiw quantization and the path integral

TL;DR

This paper demonstrates how the Faddeev-Jackiw quantization method can be effectively integrated into the path integral framework, simplifying the quantization of constrained systems with applications to models like particles on surfaces and quantum fields.

Contribution

It provides a geometric implementation of the Faddeev-Jackiw method within the path integral approach, validating it through multiple models and clarifying misconceptions about its criticisms.

Findings

The method aligns with Dirac's approach for constrained systems.

Successful application to quantum particles on surfaces and quantum fields.

Clarification that previous criticisms of the method are unfounded.

Abstract

The method for quantization of constrained theories that was suggested originally by Faddeev and Jackiw along with later modifications is discussed. The particular emphasis of this paper is to show how it is simple to implement their method within the path integral framework using the natural geometric structure that their method utilizes. The procedure is exemplified with the analysis of two models: a quantum mechanical particle constrained to a surface (of which the hypersphere is a special case), and a quantized Schr\"odinger field interacting with a quantized vector field for both the massive and the massless cases. The results are shown to agree with what is found using the Dirac method for constrained path integrals. We comment on a previous path integral analysis of the Faddeev-Jackiw method. We also discuss why a previous criticism of the Faddeev-Jackiw method is unfounded and why suggested modifications of their method are unnecessary.