Casimir Energies and Pressures for $\delta$-function Potentials

TL;DR

This paper calculates Casimir energies and pressures for scalar fields with delta-function potentials in various dimensions, revealing finite results, divergences at weak coupling, and clarifying their relation to boundary conditions and electromagnetic analogs.

Contribution

It provides explicit calculations of Casimir energies for delta-function potentials, including spherical shells, and clarifies the role of surface terms, divergences, and the connection to electromagnetic TM and TE modes.

Findings

Finite Casimir pressures for parallel planes match Dirichlet limits.

Logarithmic divergence in weak-coupling Casimir energy for spherical shells.

Divergences can cancel between delta and derivative delta potentials.

Abstract

The Casimir energies and pressures for a massless scalar field associated with $\delta$-function potentials in 1+1 and 3+1 dimensions are calculated. For parallel plane surfaces, the results are finite, coincide with the pressures associated with Dirichlet planes in the limit of strong coupling, and for weak coupling do not possess a power-series expansion in 1+1 dimension. The relation between Casimir energies and Casimir pressures is clarified,and the former are shown to involve surface terms. The Casimir energy for a $\delta$-function spherical shell in 3+1 dimensions has an expression that reduces to the familiar result for a Dirichlet shell in the strong-coupling limit. However, the Casimir energy for finite coupling possesses a logarithmic divergence first appearing in third order in the weak-coupling expansion, which seems unremovable. The corresponding energies and pressures for a derivative of a $\delta$-function potential for the same spherical geometry generalizes the TM contributions of electrodynamics. Cancellation of divergences can occur between the TE ($\delta$-function) and TM (derivative of $\delta$-function) Casimir energies. These results clarify recent discussions in the literature.