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The path integral quantization method is applied to a relativisticallycovariant version of the Hopfield model, which represents a very interestingmesoscopic framework for the description of the interaction between quantumlight and dielectric quantum matter, with particular reference to the contextof analogue gravity. In order to take into account the constraints occurring inthe model, we adopt the Faddeev-Jackiw approach to constrained quantization inthe path integral formalism. In particular we demonstrate that the propagatorobtained with the Faddeev-Jackiw approach is equivalent to the one which, inthe framework of Dirac canonical quantization for constrained systems, can bedirectly computed as the vacuum expectation value of the time ordered productof the fields. Our analysis also provides an explicit example of quantizationof the electromagnetic field in a covariant gauge and coupled with thepolarization field, which is a novel contribution to the literature on theFaddeev-Jackiw procedure.

Path integral quantization of the relativistic Hopfield model