The Casimir Effect for Parallel Plates Revisited

TL;DR

This paper revisits the Casimir effect for parallel plates within the local quantum field theory framework, highlighting the importance of boundary conditions and proposing that periodic boundary conditions yield finite energy densities relevant for cosmological dark energy models.

Contribution

It introduces a refined analysis of boundary conditions in the Casimir effect using local quantum field theory, emphasizing the significance of intensive variables like energy per unit area.

Findings

Periodic boundary conditions lead to finite energy per unit area.

Dirichlet boundary conditions cause divergence in energy density.

Periodic b.c. may have implications for dark energy in cosmology.

Abstract

The Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (b.c.) on infinite parallel plates is revisited in the local quantum field theory (lqft) framework introduced by B.Kay. The model displays a number of more realistic features than the ones he treated. In addition to local observables, as the energy density, we propose to consider intensive variables, such as the energy per unit area $\epsilon$, as fundamental observables. Adopting this view, lqft rejects Dirichlet (the same result may be proved for Neumann or mixed) b.c., and accepts periodic b.c.: in the former case $\epsilon$ diverges, in the latter it is finite, as is shown by an expression for the local energy density obtained from lqft through the use of the Poisson summation formula. Another way to see this uses methods from the Euler summation formula: in the proof of regularization independence of the energy per unit area, a regularization-dependent surface term arises upon use of Dirichlet b.c. but not periodic b.c.. For the conformally invariant scalar quantum field, this surface term is absent, due to the condition of zero trace of the energy momentum tensor, as remarked by B.De Witt. The latter property does not hold in tha application to the dark energy problem in Cosmology, in which we argue that periodic b.c. might play a distinguished role.