The Casimir effect as scattering problem

TL;DR

This paper presents a novel scattering approach to calculating Casimir forces between obstacles, using a billiard analogy and a trace formula to obtain exact energies applicable across various configurations.

Contribution

It introduces a self-regulating formalism that maps Casimir energy calculations onto quantum billiard problems, enabling exact and versatile computations for multiple obstacles.

Findings

Exact Casimir energy calculations for two spheres and sphere-plate configurations.

Applicable to various boundary conditions like Dirichlet and Neumann.

Formalism extends to multiple obstacles and higher dimensions.

Abstract

We show that Casimir-force calculations for a finite number of non-overlapping obstacles can be mapped onto quantum-mechanical billiard-type problems which are characterized by the scattering of a fictitious point particle off the very same obstacles. With the help of a modified Krein trace formula the genuine/finite part of the Casimir energy is determined as the energy-weighted integral over the log-determinant of the multi-scattering matrix of the analog billiard problem. The formalism is self-regulating and inherently shows that the Casimir energy is governed by the infrared end of the multi-scattering phase shifts or spectrum of the fluctuating field. The calculation is exact and in principle applicable for any separation(s) between the obstacles. In practice, it is more suited for large- to medium-range separations. We report especially about the Casimir energy of a fluctuating massless scalar field between two spheres or a sphere and a plate under Dirichlet and Neumann boundary conditions. But the formalism can easily be extended to any number of spheres and/or planes in three or arbitrary dimensions, with a variety of boundary conditions or non-overlapping potentials/non-ideal reflectors.