Dirac fields in the background of a magnetic flux string and spectral boundary conditions

TL;DR

This paper investigates the behavior of Dirac fields around an Aharonov-Bohm flux string, applying spectral boundary conditions to ensure self-adjointness and analyzing physical quantities like 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 flux string backgrounds, ensuring mathematical consistency and physical insight.

Findings

Eigenfunctions compatible with self-adjointness and flux invariance.

Consistent evaluation of vacuum fermionic number.

Calculation of Casimir energy in the flux background.

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 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 evaluate its vacuum fermionic number and Casimir energy.