Towards a heat kernel expansion for the electromagnetic field interacting with a dielectric body of arbitrary form

TL;DR

This paper reviews heat kernel expansion methods for electromagnetic fields interacting with arbitrary dielectric bodies, proposing a new multiple reflection expansion approach that relates coefficients to geometric invariants.

Contribution

It introduces a multiple reflection expansion method for electromagnetic fields with dielectric interfaces of arbitrary shape, linking heat kernel coefficients to geometric invariants.

Findings

The multiple reflection expansion method is promising for arbitrary dielectric interfaces.

Heat kernel coefficients can be expressed through geometric invariants.

The approach is demonstrated using a scalar photon toy model.

Abstract

The results on the heat kernel expansion for the electromagnetic field in the background of dielectric media are briefly reviewed. The common approaches to the calculation of the heat kernel coefficients are discussed from the viewpoint of their applicability to the electromagnetic field interacting with dielectric body of arbitrary form. Using the toy-model of scalar photons we develop multiple reflection expansion method which seems the most promising one when the field obeys dielectric-like matching conditions on an arbitrary interface and show that the heat kernel coefficients are expressible through geometric invariants of the latter.