Non-locality of energy separating transformations for Dirac electrons in a magnetic field

TL;DR

This paper analyzes the non-locality of the Moss-Okninski transformation used to separate energy states in the Dirac equation for electrons in magnetic fields, revealing dependencies on physical length scales like the Compton wavelength and magnetic radius.

Contribution

It provides a detailed characterization of the non-locality of the energy-separating transformation in magnetic fields, including for 2+1 Dirac equations, based on kernel analysis and variance calculations.

Findings

Non-locality along magnetic field is characterized by the Compton wavelength.

Transverse non-locality depends on magnetic radius and Compton wavelength.

The study extends to the 2+1 Dirac equation case.

Abstract

We investigate a non-locality of Moss-Okninski transformation (MOT) used to separate positive and negative energy states in the 3+1 Dirac equation for relativistic electrons in the presence of a magnetic field. Properties of functional kernels generated by the MOT are analyzed and kernel non-localities are characterized by calculating their second moments parallel and perpendicular to the magnetic field. Transformed functions are described and investigated by computing their variances. It is shown that the non-locality of the energy-separating transformation in the direction parallel to the magnetic field is characterized by the Compton wavelength $\lambda_c=\hbar/mc$. In the plane transverse to magnetic field the non-locality depends both on magnetic radius $L=(\hbar/eB)^{1/2}$ and $\lambda_c$. The non-locality of MO transformation for the 2+1 Dirac equation is also considered.