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Gauge-Invariant Wigner Equation for Electromagnetic Fields: Strong and Weak Formulation

Clemens Etl, Mauro Ballicchia, Mihail Nedjalkov, Hans Kosina

Published Jan 1, 2025• 0 citations
Publisher
DOI
10.2139/ssrn.5341947

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Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. 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, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.

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Gauge-invariant Wigner theory describes the quantum-mechanical evolution of charged particles in phase space, which is spanned by position and kinetic momentum. This approach uses the electromagnetic field variables instead of the electrodynamic potentials. 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, which simplifies existing formulations. First, a gauge-dependent equation is derived using Moyal’s formulation. A transformation of the Wigner function introduced by Stratonovich yields the strong form of the gauge-invariant equation. Expressing the pseudo-differential operators by integral operators gives the weak form of the gauge-invariant equation. An analysis of the different properties of the gauge-invariant equation is given, as well as the different requirements for the regularity and asymptotic behavior of the strong and weak forms.
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.
Gauge-invariant quantum kinetic equations for interacting electrons are deduced using the Keldysh diagrammatic technique. The Dyson equations are transformed using a special type of the Wigner representation that produces gauge-invariant Green functions. As a result, they depend on the variables having a meaning of position and kinetic momentum. The Wigner representation used makes it necessary to modify the diagrammatic technique in such a way that it will be able to account for the momentum-energy exchange between the system and the electromagnetic field. The formalism obtained makes it possible to carry out many-particle calculations for non-linear systems in arbitrary electromagnetic fields. Some particular simple cases are considered. A special discussion is given regarding the meaning of the detailed balance in such a formulation.