Exact quantisation of the relativistic Hopfield model

TL;DR

This paper presents a comprehensive covariant quantisation of the relativistic Hopfield model, addressing gauge issues, Lorentz invariance breaking, and ensuring causality in the quantum description of dielectric media.

Contribution

It provides the first exact covariant quantisation of the relativistic Hopfield model, including handling of gauge unphysical modes and Lorentz invariance considerations.

Findings

Successful covariant quantisation of the model

Implementation of a generalized Lautrup trick to handle dipole ghosts

Discussion of causality and relation to Wightman axioms

Abstract

We investigate the quantisation in the Heisenberg representation of a relativistically covariant version of the Hopfield model for dielectric media, which entails the interaction of the quantum electromagnetic field with the matter dipole fields. The matter fields are represented by a mesoscopic polarization field. A full quantisation of the model is provided in a covariant gauge, with the aim of maintaining explicit relativistic covariance. Breaking of the Lorentz invariance due to the intrinsic presence in the model of a preferred reference frame is also taken into account. Relativistic covariance forces us to deal with the unphysical (scalar and longitudinal) components of the fields, furthermore it introduces, in a more tricky form, the well-known dipole ghost of standard QED in a covariant gauge. In order to correctly dispose of this contribution, we implement a generalized Lautrup trick. Furthermore, causality and the relation of the model with the Wightman axioms are also discussed.