Casimir stress in an inhomogeneous medium

TL;DR

This paper investigates the Casimir effect in an inhomogeneous dielectric medium, revealing divergences in stress and force at boundaries and exploring regularization methods to address these infinities, thus challenging current understanding.

Contribution

It introduces a detailed analysis of Casimir stress in inhomogeneous dielectrics and examines regularization techniques to handle boundary divergences, advancing theoretical understanding.

Findings

Casimir stress is infinite inside the inhomogeneous region.

Divergences occur at boundaries between inhomogeneous and homogeneous dielectrics.

Regularization can produce finite stress inside the medium but not at boundaries.

Abstract

The Casimir effect in an inhomogeneous dielectric is investigated using Lifshitz's theory of electromagnetic vacuum energy. A permittivity function that depends continuously on one Cartesian coordinate is chosen, bounded on each side by homogeneous dielectrics. The result for the Casimir stress is infinite everywhere inside the inhomogeneous region, a divergence that does not occur for piece-wise homogeneous dielectrics with planar boundaries. A Casimir force per unit volume can be extracted from the infinite stress but it diverges on the boundaries between the inhomogeneous medium and the homogeneous dielectrics. An alternative regularization of the vacuum stress is considered that removes the contribution of the inhomogeneity over small distances, where macroscopic electromagnetism is invalid. The alternative regularization yields a finite Casimir stress inside the inhomogeneous region, but the stress and force per unit volume diverge on the boundaries with the homogeneous dielectrics. The case of inhomogeneous dielectrics with planar boundaries thus falls outside the current understanding of the Casimir effect.