Microscopic theory of the Casimir force at thermal equilibrium: large-separation asymptotics

TL;DR

This paper provides a microscopic derivation of the large-separation asymptotic behavior of the Casimir force between metallic plates at finite temperature, confirming the universal $d^{-3}$ decay and elucidating the role of screening effects.

Contribution

It offers a fully microscopic, assumption-free calculation of the Casimir force's asymptotics, validating the universal formula and clarifying the physical origin of this universality.

Findings

The Casimir force asymptotically behaves as $-\frac{\zeta(3) k_B T}{8\pi d^3}$ at large separations.

Transverse electric modes do not contribute to the force in this regime.

Universal behavior is rooted in perfect screening sum rules in conducting media.

Abstract

We present an entirely microscopic calculation of the Casimir force $f(d)$ between two metallic plates in the limit of large separation $d$. The models of metals consist of mobile quantum charges in thermal equilibrium with the photon field at positive temperature $T$. Fluctuations of all degrees of freedom, matter and field, are treated according to the principles of quantum electrodynamics and statistical physics without recourse to approximations or intermediate assumptions. Our main result is the correctness of the asymptotic universal formula $ f(d) \sim -\frac{\zeta(3) \kB T}{8\pi d^3}$, $d\to\infty$. This supports the fact that, in the framework of Lifshitz' theory of electromagnetic fluctuations, transverse electric modes do not contribute in this regime. Moreover the microscopic origin of universality is seen to rely on perfect screening sum rules that hold in great generality for conducting media.