Worldline approach for numerical computation of electromagnetic Casimir energies. I. Scalar field coupled to magnetodielectric media

TL;DR

This paper introduces a worldline method for numerically computing Casimir energies of scalar fields in layered media, providing a new approach that can approximate electromagnetic Casimir energies in complex geometries.

Contribution

It develops a worldline path integral framework for scalar fields coupled to magnetodielectric media, applicable to arbitrary geometries and related to electromagnetic polarization modes.

Findings

Path integrals converge to known solutions in special cases

Monte Carlo methods effectively evaluate Casimir energies

Scalar approach approximates electromagnetic Casimir energies in complex geometries

Abstract

We present a worldline method for the calculation of Casimir energies for scalar fields coupled to magnetodielectric media. The scalar model we consider may be applied in arbitrary geometries, and it corresponds exactly to one polarization of the electromagnetic field in planar layered media. Starting from the field theory for electromagnetism, we work with the two decoupled polarizations in planar media and develop worldline path integrals, which represent the two polarizations separately, for computing both Casimir and Casimir-Polder potentials. We then show analytically that the path integrals for the transverse-electric (TE) polarization coupled to a dielectric medium converge to the proper solutions in certain special cases, including the Casimir-Polder potential of an atom near a planar interface, and the Casimir energy due to two planar interfaces. We also evaluate the path integrals numerically via Monte-Carlo path-averaging for these cases, studying the convergence and performance of the resulting computational techniques. While these scalar methods are only exact in particular geometries, they may serve as an approximation for Casimir energies for the vector electromagnetic field in other geometries.