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

• Definition of a gauge-independent density matrix and Wigner function according to Stratonovic. • Derivation of a Wigner equation that fully includes the electromagnetic fields E and B. • Strong formulation of the gauge-independent Wigner equation using pseudo-differential operators. • Weak formulation of the gauge-independent Wigner equation using integral operators. • Quantum mechanics: standard formulation requires the use of electrodynamic potentials; in the phase space formulation, the fields E and B can be used. 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.