Dirac fields in a Bohm-Aharonov background and spectral boundary conditions

TL;DR

This paper investigates the behavior of Dirac fields around an Aharonov-Bohm flux string, employing spectral boundary conditions to ensure self-adjointness and analyzing implications for vacuum fermionic number and Casimir energy.

Contribution

It introduces a method of imposing spectral boundary conditions at a finite radius to study Dirac fields in an Aharonov-Bohm background, ensuring mathematical consistency and physical insights.

Findings

Eigenfunctions compatible with self-adjointness

Invariance under flux translation

Consistent vacuum fermionic number and Casimir energy calculations

Abstract

We study the problem of a Dirac field in the background of an Aharonov-Bohm flux string. We exclude the origin by imposing spectral boundary conditions at a finite radius then shrinked to zero. Thus, we obtain a behaviour of the eigenfunctions which is compatible with the self-adjointness of the radial Hamiltonian and the invariance under integer translations of the reduced flux. After confining the theory to a finite region, we check the consistency with the index theorem, and discuss the vacuum fermionic number and Casimir energy.