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Acta Physica Hungarica B) Quantum Electronics

Gauge-Invariant Relativistic Wigner Functions in External Electromagnetic Fields

Sándor Varró, Juha Javanainen

Published Jan 1, 2004• 1 citation
Publisher
DOI
10.1556/aph.20.2004.1-2.5

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Physics Letters A•Nov 6, 2025•0 citations
On the basis of the Hamiltonian form of the Klein–Gordon equation for a charged scalar particle field introduced by Feshbach and Villars, the gauge-invariant 2 × 2 Wigner matrix has been constructed whose diagonal elements describe positive and negative charge densities and whose off-diagonal elements correspond to cross-densities in phase space. The system of coupled transport equations has been derived in the case of interaction with an arbitrary external electromagnetic field. A gauge-independent generalization of the free particle representation due to Feshbach and Villars is given, and on the basis of it both the nonrelativistic and the classical limits of the relativistic quantum Boltzmann–Vlasov equation are discussed. In the nonrelativistic limit ( p / mc → 0) the set of equations of motion decouple to two independent quantum transport equations describing the dynamics of oppositely charged positon and negaton densities. In the classical limit (ℏ → 0) two relativistic Boltzmann–Vlasov equations result for the diagonal positon and negaton densities. Even though the Planck constant ℏ is absent from the latter equations, the real part of the positon–negaton cross-density does not vanish.
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.