Casimir interaction of arbitrarily shaped conductors

TL;DR

This paper reviews a practical method for calculating Casimir forces between arbitrarily shaped conductors, improving accuracy over previous approximations and demonstrating its effectiveness through various geometries, including complex and topologically interesting cases.

Contribution

It introduces a systematic implementation of the multiple scattering formalism for Casimir interactions, with a focus on the leading two-point scattering term and its extensions, applicable to complex geometries and finite temperatures.

Findings

Accurate calculation of Casimir forces for various geometries.

Validation of the two-point scattering approximation against exact results.

Extension of the method to finite temperature scenarios.

Abstract

We review a systematic practical implementation of the multiple scattering formalism due to Balian and Duplantier [R. Balian and B. Duplantier, Ann. Phys. (NY) \textbf{104}, 300 (1977); \textbf{112}, 165 (1978)] for the calculation of the Casimir interaction between arbitrarily shaped smooth conductors. The leading two-point scattering term of the expansion has a simple compact form, amenable to exact or accurate numerical evaluation. It is a general expression which improves upon the proximity force and pairwise summation approximations. We show that for many geometries it captures the bulk of the interaction effect. The inclusion of terms beyond the two-point approximation provides an accuracy check and explains screening. As an illustration of the power and versatility of the method we re-evaluate sphere-sphere and sphere-plane interactions and compared the results with previous findings that employed different methods. We also compute for the first time interaction of a hyperboloid (mimicking an atomic force microscope tip) and a plane. We also analyze the anomalous situations involving long cylindrical conductors where the two-point scattering approximation fails qualitatively. In such cases analytic summation of the entire scattering series is carried out and a topological argument is put forward as an explanation of the result. We give the extension of this theory to the case of finite temperatures where the two-point scattering approximation result has a simple compact form, also amenable to exact or accurate numerical evaluation.