Quantum vacuum energies and Casimir forces between partially transparent $\delta$-function plates

TL;DR

This paper investigates the quantum vacuum energies and Casimir forces between two partially transparent delta-function plates, revealing conditions for attraction or repulsion and the impact of boundary conditions on massless fields.

Contribution

It provides a detailed analysis of vacuum energies and Casimir forces for scalar fields interacting with delta-function plates, including conditions for attraction, repulsion, and unitarity constraints.

Findings

Attractive or repulsive Casimir forces depend on the sign of delta potentials.

Negative delta potentials require a minimum scalar field mass for unitarity.

Dirichlet boundary conditions are equivalent to infinite delta potentials.

Abstract

In this paper the quantum vacuum energies induced by massive fluctuations of one real scalar field on a configuration of two partially transparent plates are analysed. The physical properties of the infinitely thin plates are characterized by two Dirac-$\delta$ potentials. We find that an attractive/repulsive Casimir force arises between the plates when the weights of the $\delta$'s have equal/different sign. If some of the plates absorbs fluctuations below some threshold of energy (the corresponding weight is negative) there is the need to set a minimum mass to the scalar field fluctuations to preserve unitarity in the corresponding quantum field theory. Two repulsive $\delta$-interactions are compatible with massless fluctuations. The effect of Dirichlet boundary conditions at the endpoints of the interval $(-a,a)$ on a massless scalar quantum field theory defined on this interval is tantamount letting the weights of the repulsive $\delta$-interactions to $+\infty$.