Casimir free energy for massive fermions: a comparative study of various approaches

TL;DR

This paper compares three different methods for calculating the Casimir thermodynamic quantities for a massive fermion field between plates, revealing that the approaches generally produce different results and highlighting the unique properties of the first method.

Contribution

It provides a detailed comparison of three approaches to compute Casimir thermodynamic quantities for massive fermions, including explicit solutions and analysis of their differences.

Findings

Different approaches yield non-equivalent results.

The first approach's results tend to zero with increasing temperature, mass, or distance.

Explicit solutions for each approach are provided.

Abstract

We compute the Casimir thermodynamic quantities for a massive fermion field between two parallel plates with the MIT boundary conditions, using three different general approaches and present explicit solutions for each. The Casimir thermodynamic quantities include the Casimir Helmholtz free energy, pressure, energy and entropy. The three general approaches that we use are based on the fundamental definition of Casimir thermodynamic quantities, the analytic continuation method represented by the zeta function method, and the zero temperature subtraction method. We include the renormalized versions of the latter two approaches as well, whereas the first approach does not require one. Within each general approach, we obtain the same results in a few different ways to ascertain the selected cancellations of infinities have been done correctly. We then do a comparative study of the three different general approaches and their results, and show that they are in principle not equivalent to each other and they yield, in general, different results. In particular, we show that the Casimir thermodynamic quantities calculated only by the first approach have all three properties of going to zero as the temperature, the mass of the field, or the distance between the plates increases.