Multiple Scattering Methods in Casimir Calculations

TL;DR

This paper reviews multiple scattering methods for Casimir effect calculations, deriving explicit formulas for various geometries and analyzing the accuracy of approximation methods like PFA.

Contribution

It provides new explicit closed-form expressions for Casimir energies in semitransparent bodies and evaluates the limitations of the proximity force approximation.

Findings

Exact summation of weak-coupling expansions for Casimir energies.

Explicit formulas for Casimir interactions between spheres, cylinders, and planes.

PFA is inaccurate at finite separations.

Abstract

Multiple scattering formulations have been employed for more than 30 years as a method of studying the quantum vacuum or Casimir interactions between distinct bodies. Here we review the method in the simple context of $\delta$-function potentials, so-called semitransparent bodies. (In the limit of strong coupling, a semitransparent boundary becomes a Dirichlet one.) After applying the method to rederive the Casimir force between two semitransparent plates and the Casimir self-stress on a semitransparent sphere, we obtain expressions for the Casimir energies between disjoint parallel semitransparent cylinders and between disjoint semitransparent spheres. Simplifications occur for weak and strong coupling. In particular, after performing a power series expansion in the ratio of the radii of the objects to the separation between them, we are able to sum the weak-coupling expansions exactly to obtain explicit closed forms for the Casimir interaction energy. The same can be done for the interaction of a weak-coupling sphere or cylinder with a Dirichlet plane. We show that the proximity force approximation (PFA), which becomes the proximity force theorem when the objects are almost touching, is very poor for finite separations.