Path integral quantization of the relativistic Hopfield model

TL;DR

This paper applies path integral quantization to a relativistic Hopfield model, using the Faddeev-Jackiw approach to handle constraints, and demonstrates equivalence with Dirac quantization, providing a novel example of electromagnetic field quantization in this context.

Contribution

It introduces a covariant path integral quantization of the relativistic Hopfield model using Faddeev-Jackiw constraints, with explicit comparison to Dirac quantization methods.

Findings

Faddeev-Jackiw propagator matches Dirac quantization results

Provides a covariant gauge quantization example for electromagnetic and polarization fields

Enhances understanding of constrained quantization in mesoscopic light-matter interaction models

Abstract

The path integral quantization method is applied to a relativistically covariant version of the Hopfield model, which represents a very interesting mesoscopic framework for the description of the interaction between quantum light and dielectric quantum matter, with particular reference to the context of analogue gravity. In order to take into account the constraints occurring in the model, we adopt the Faddeev-Jackiw approach to constrained quantization in the path integral formalism. In particular we demonstrate that the propagator obtained with the Faddeev-Jackiw approach is equivalent to the one which, in the framework of Dirac canonical quantization for constrained systems, can be directly computed as the vacuum expectation value of the time ordered product of the fields. Our analysis also provides an explicit example of quantization of the electromagnetic field in a covariant gauge and coupled with the polarization field, which is a novel contribution to the literature on the Faddeev-Jackiw procedure.