The Casimir Effect for Fermions in One Dimension

TL;DR

This paper investigates the Casimir effect for fermions in one dimension, analyzing how boundary conditions and background fields influence the Casimir energy, including cases with sharp spikes where divergences occur.

Contribution

It provides a detailed analysis of the Casimir energy for fermions with various background fields, especially the effects of sharp boundary conditions and divergences at spikes.

Findings

Casimir energy is finite for smooth backgrounds but diverges at Dirac spikes.

The energy density is finite except at the locations of sharp spikes.

Interaction energy and force between spikes depend on their strength and separation.

Abstract

We study the Casimir problem for a fermion coupled to a static background field in one space dimension. We examine the relationship between interactions and boundary conditions for the Dirac field. In the limit that the background becomes concentrated at a point (a ``Dirac spike'') and couples strongly, it implements a confining boundary condition. We compute the Casimir energy for a masslike background and show that it is finite for a stepwise continuous background field. However the total Casimir energy diverges for the Dirac spike. The divergence cannot be removed by standard renormalization methods. We compute the Casimir energy density of configurations where the background field consists of one or two sharp spikes and show that the energy density is finite except at the spikes. Finally we define and compute an interaction energy density and the force between two Dirac spikes as a function of the strength and separation of the spikes.