Photon Green's function and the Casimir energy in a medium

TL;DR

This paper develops a new expansion for the electromagnetic Green's function in media with arbitrary permittivity and permeability, leading to novel expressions for Casimir energy and potential applications to complex geometries.

Contribution

It introduces a new Born series expansion that includes both traditional and novel terms, simplifying calculations of Casimir energies in media with arbitrary properties.

Findings

Derived a new Green's function expansion with two interaction types.

Obtained simplified Casimir energy expressions for specific media.

Applied the method to a dielectric-diamagnetic sphere.

Abstract

A new expansion is established for the Green's function of the electromagnetic field in a medium with arbitrary $\epsilon$ and $\mu$. The obtained Born series are shown to consist of two types of interactions - the usual terms (denoted $\cal P$) that appear in the Lifshitz theory combined with a new kind of terms (which we denote by $\cal Q$) associated with the changes in the permeability of the medium. Within this framework the case of uniform velocity of light ($\epsilon\mu={\rm const}$) is studied. We obtain expressions for the Casimir energy density and the first non-vanishing contribution is manipulated to a simplified form. For (arbitrary) spherically symmetric $\mu$ we obtain a simple expression for the electromagnetic energy density, and as an example we obtain from it the Casimir energy of a dielectric-diamagnetic ball. It seems that the technique presented can be applied to a variety of problems directly, without expanding the eigenmodes of the problem and using boundary condition considerations.