Path-integral quantization of the electromagnetic field in the Hopfield dielectric beyond dipole approximation

Abstract

The paper contains an application of the path-integration method to the quantization of the electromagnetic field in a linear material medium modelled as the Hopfield dielectric. Except for the frequency dispersion of the medium, non-dipole effects are included leading to the wave vector dependence of the dielectric function. Starting from a local microscopic Lagrangian, the elimination of the matter degrees of freedom leads to the effective action describing dynamics of the classical field in the medium, from which the classical constitutive relation, non-local, both in time and space variables, could be determined. Full quantization of the model is achieved by integration over all fields with source terms included into the Lagrangian, and taking into account the constraint and gauge fixing term, characteristic for a quantized gauge theory. This gives the generating functional from which quantum propagators could be constructed. Operators of effective quantum electromagnetic and polarization fields were further retrieved from the propagators. The quantum constitutive equation containing the absorption noise term was also determined. The canonical equal time commutation rules are examined using both the BJL-limiting procedure and using explicit expressions for the field operators—the results in both cases are the same.