Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique

TL;DR

This paper rigorously derives formulas for the finite temperature Casimir free energy in rectangular cavities for scalar and electromagnetic fields using zeta regularization, analyzing temperature effects and transition points.

Contribution

It provides explicit formulas for Casimir free energy at finite temperature in rectangular cavities, including low/high temperature expansions and transition temperature analysis, using a rigorous zeta function approach.

Findings

Free energy decreases with temperature.

Existence of a unique transition temperature where Casimir energy changes sign.

Results for non-closed cavities match traditional methods.

Abstract

We derive rigorously explicit formulas of the Casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by zeta regularization method. We study both the low and high temperature expansions of the free energy. In each case, we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms. We show that the free energy is always a decreasing function of temperature. In the cases of massless scalar field with Dirichlet boundary condition and electromagnetic field, the zero temperature Casimir free energy might be positive. In each of these cases, there is a unique transition temperature (as a function of the side lengths of the cavity) where the Casimir energy change from positive to negative. When the space dimension is equal to two and three, we show graphically the dependence of this transition temperature on the side lengths of the cavity. Finally we also show that we can obtain the results for a non-closed rectangular cavity by letting the size of some directions of a closed cavity going to infinity, and we find that these results agree with the usual integration prescription adopted by other authors.