Casimir effect at finite temperature in a real scalar field theory

TL;DR

This paper evaluates the Casimir free energy for a self-interacting scalar field at finite temperature, employing perturbative and non-perturbative methods to incorporate thermal effects and boundary conditions.

Contribution

It introduces two approaches for calculating the finite-temperature Casimir effect in an interacting scalar field, including a non-perturbative method based on the exact generating functional.

Findings

Exact results for free energy in special cases

Low and high temperature expansions via duality

Inclusion of non-perturbative thermal corrections

Abstract

We use a functional approach to evaluate the Casimir free energy for a self-interacting scalar field in $d+1$ dimensions, satisfying Dirichlet boundary conditions on two parallel planes. When the interaction is turned off, exact results for the free energy in some particular cases may be found, as well as low and high temperature expansions based on a duality relation that involves the inverse temperature $\beta$ and the distance between the mirrors, $a$. For the interacting theory, we derive and implement two different approaches. The first one is a perturbative expansion built with a thermal propagator that satisfies Dirichlet boundary conditions on the mirrors. The second approach uses the exact finite-temperature generating functional as a starting point. In this sense, it allows one to include, for example, non-perturbative thermal corrections into the Casimir calculation, in a controlled way. We present results for calculations performed using those two approaches.