The Hopfield model revisited: Covariance and Quantization

TL;DR

This paper extends the Hopfield model to include space-time varying dielectric properties, providing a covariant quantization scheme suitable for describing nonlinear optical effects like the Kerr effect in dielectric media.

Contribution

It introduces a covariant, phenomenological extension of the Hopfield model with a quantization scheme for space-time dependent dielectric parameters.

Findings

Developed a Lorentz-invariant Lagrangian for the extended model

Derived a covariant scalar product for quantization

Outlined applications to nonlinear dielectric phenomena

Abstract

There are several possible applications of quantum electrodynamics in dielectric media which require a quantum description for the electromagnetic field interacting with matter fields. The associated quantum models can refer to macroscopic electromagnetic fields or, in alternative, to mesoscopic fields (polarization fields) describing an effective interaction between electromagnetic field and matter fields. We adopt the latter approach, and focus on the Hopfield model for the electromagnetic field in a dielectric dispersive medium in a framework in which space-time dependent mesoscopic parameters occur, like susceptibility, matter resonance frequency, and also coupling between electromagnetic field and polarization field. Our most direct goal is to describe in a phenomenological way a space-time varying dielectric perturbation induced by means of the Kerr effect in nonlinear dielectric media. This extension of the model is implemented by means of a Lorentz-invariant Lagrangian which, for constant microscopic parameters, and in the rest frame, coincides with the standard one. Moreover, we deduce a covariant scalar product and provide a covariant quantization scheme which keeps into account the constraints implicit in the model. Examples of viable applications are indicated.