Thermal Casimir effect in ideal metal rectangular boxes

TL;DR

This paper develops a zeta regularization method to compute the thermal Casimir effect in ideal metal rectangular boxes, providing finite expressions that match known results and reveal temperature-dependent force behaviors.

Contribution

It introduces a new renormalization procedure for the thermal Casimir effect in rectangular geometries, enabling finite calculations and analysis of temperature effects on Casimir forces.

Findings

The derived expressions reproduce known results for parallel plates.

Temperature-dependent Casimir force can be positive or negative depending on box dimensions.

Numerical results for cubes illustrate the temperature influence on Casimir free energy and force.

Abstract

The thermal Casimir effect in ideal metal rectangular boxes is considered using the method of zeta functional regularization. The renormalization procedure is suggested which provides the finite expression for the Casimir free energy in any restricted quantization volume. This expression satisfies the classical limit at high temperature and leads to zero thermal Casimir force for systems with infinite characteristic dimensions. In the case of two parallel ideal metal planes the results, as derived previously using thermal quantum field theory in Matsubara formulation and other methods, are reproduced starting from the obtained expression. It is shown that for rectangular boxes the temperature-dependent contribution to the electromagnetic Casimir force can be both positive and negative depending on side lengths. The numerical computations of the scalar and electromagnetic Casimir free energy and force are performed for cubes