Gauge-invariant Wigner equation for electromagnetic fields: Strong and weak formulation

TL;DR

This paper introduces a simplified, gauge-invariant Wigner equation for charged particles in electromagnetic fields, providing both strong and weak formulations and analyzing their properties and equivalence.

Contribution

A new, simplified differential operator-based gauge-invariant Wigner equation for electrons in electromagnetic fields, with derivation of strong and weak forms and analysis of their properties.

Findings

Derived a gauge-invariant Wigner equation using differential operators

Established the equivalence of strong and weak formulations

Analyzed properties and regularity requirements of the formulations

Abstract

Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in the presence of an electromagnetic field in phase space, which is spanned by position and kinetic momentum. This approach is independent of the chosen potentials, as it depends only on the electric and magnetic field variables. Several approaches to derive a gauge-invariant Wigner evolution equation have been reported, which are generally complex. This work presents a new formulation for a single electron in a general electromagnetic field based solely on differential operators that simplify existing formulations. A gauge-dependent equation is derived first using Moyal's equation. A transformation of the Wigner function, introduced by Stratonovich, is then used to make the equation gauge-invariant, which gives us a strong formulation of the problem. This equation can be transformed into its weak form, which proves that both formulations are equivalent. An analysis of the different properties of the gauge-dependent and gauge-invariant formulations is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak formulations.