Finite temperature Casimir effect for massive scalar field in spacetime with extra dimensions

TL;DR

This paper calculates the finite temperature Casimir energy and force for a massive scalar field in a higher-dimensional spacetime with boundaries, revealing how mass and boundary conditions influence the force's magnitude and nature.

Contribution

It provides explicit formulas and asymptotic behaviors for the Casimir force in a higher-dimensional setting with massive fields, extending previous massless analyses.

Findings

Casimir force decreases with increasing mass.

Transition from massless to massive changes force from long-range to short-range.

Dirichlet-Neumann boundary condition combination yields a repulsive force.

Abstract

We compute the finite temperature Casimir energy for massive scalar field with general curvature coupling subject to Dirichlet or Neumann boundary conditions on the walls of a closed cylinder with arbitrary cross section, located in a background spacetime of the form $M^{d_1+1}\times \mathcal{N}^n$, where $M^{d_1+1}$ is the $(d_1+1)$-dimensional Minkowski spacetime and $\mathcal{N}^n$ is an $n$-dimensional internal manifold. The Casimir energy is regularized using the criteria that it should vanish in the infinite mass limit. The Casimir force acting on a piston moving freely inside the closed cylinder is derived and it is shown that it is independent of the regularization procedure. By letting one of the chambers of the cylinder divided by the piston to be infinitely long, we obtain the Casimir force acting on two parallel plates embedded in the cylinder. It is shown that if both the plates assume Dirichlet or Neumann boundary conditions, the strength of the Casimir force is reduced by the increase in mass. Under certain conditions, the passage from massless to massive will change the nature of the force from long range to short range. Other properties of the Casimir force such as its sign, its behavior at low and high temperature, and its behavior at small and large plate separations, are found to be similar to the massless case. Explicit exact formulas and asymptotic behaviors of the Casimir force at different limits are derived. The Casimir force when one plate assumes Dirichlet boundary condition and one plate assumes Neumann boundary condition is also derived and shown to be repulsive.