Finite temperature Casimir effect in piston geometry and its classical limit

TL;DR

This paper analytically investigates the Casimir force in a d-dimensional piston geometry at any temperature, revealing its always attractive nature, explicit force expressions, and a classical high-temperature limit where quantum effects vanish.

Contribution

It provides exact formulas for the Casimir force in piston geometries under various boundary conditions and uncovers a nontrivial classical limit at high temperatures.

Findings

Casimir force is always attractive in piston geometry.

Explicit formulas for small/large separations and low/high temperatures.

High temperature Casimir force scales linearly with temperature, indicating a classical limit.

Abstract

We consider the Casimir force acting on a $d$-dimensional rectangular piston due to massless scalar field with periodic, Dirichlet and Neumann boundary conditions and electromagnetic field with perfect electric conductor and perfect magnetic conductor boundary conditions. It is verified analytically that at any temperature, the Casimir force acting on the piston is always an attractive force pulling the piston towards the interior region, and the magnitude of the force gets larger as the separation $a$ gets smaller. Explicit exact expressions for the Casimir force for small and large plate separations and for low and high temperatures are computed. The limits of the Casimir force acting on the piston when some pairs of transversal plates are large are also derived. An interesting result regarding the influence of temperature is that in contrast to the conventional result that the leading term of the Casimir force acting on a wall of a rectangular cavity at high temperature is the Stefan--Boltzmann (or black body radiation) term which is of order $T^{d+1}$, it is found that the contributions of this term from the interior and exterior regions cancel with each other in the case of piston. The high temperature leading order term of the Casimir force acting on the piston is of order $T$, which shows that the Casimir force has a nontrivial classical $\hbar\to 0$ limit.