Finite Temperature Casimir Effect of Scalar Field

TL;DR

This paper analytically derives the finite-temperature Casimir effect for scalar fields, addressing the negative entropy issue and predicting a transition to repulsive forces at high temperatures.

Contribution

It provides new analytic expressions for Helmholtz free energy, Casimir force, and entropy at finite temperature for scalar fields with Dirichlet boundary conditions.

Findings

Negative Casimir entropy depends on regularization choices.

Thermal Casimir force can become repulsive at high temperatures.

Argues against adding counterterms for thermal corrections.

Abstract

We derive analytic expressions for the Helmholtz free energy, Casimir force, and Casimir entropy for both one-dimensional and three-dimensional scalar fields with Dirichlet boundary conditions at finite temperature. We investigate the negative Casimir entropy problem in these systems, as well as for a scalar field in the bulk of a three-dimensional sphere, and find that this issue arises under different regularization prescriptions with differing counterterms. We argue against introducing any counterterms for the thermal corrections to the Casimir effect and predict that the thermal-fluctuation-induced Casimir force becomes repulsive in the high-temperature regime -- for instance, when $aT/(\pi\hbar{v})>0.2419$ for three-dimensional scalar fields.