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

TL;DR

This paper compares three different methods for calculating the Casimir thermodynamic quantities for a massive scalar field between parallel plates, highlighting their differences and equivalences, especially in the massless limit.

Contribution

It provides explicit solutions for each approach and demonstrates the conditions under which the methods yield equivalent results, clarifying their applicability and limitations.

Findings

Zeta function and Schlomilch summation approaches are equivalent.

Results differ between approaches for massive fields, but agree in the massless limit.

The first approach shows specific thermodynamic behaviors at high temperature and mass.

Abstract

We compute the Casimir thermodynamic quantities for a massive real scalar field between two parallel plates with the Dirichlet 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, and the zero temperature subtraction method. Within the analytic continuation approach, we use two distinct methods which are based on the utilization of the zeta function and the Schlomilch summation formula. 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 show that, as expected, the results based on the zeta function and the Schlomilch summation formula are equivalent. 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 equivalent results only in the massless case. In particular, we show that the Casimir energy calculated only by the first approach has all three properties of going to zero as the temperature, mass of the field or the distance between the plates increases. Moreover, we show that in this approach the Casimir entropy reaches a positive constant in the high temperature limit, which can explain the linear term in the Casimir free energy.