Gauge-Invariant Relativistic Wigner Functions in External Electromagnetic Fields

Abstract

On the basis of the Hamiltonian form of the Klein-Gordon equation of a charged scalar particle field introduced by Feshbach and Villars, the gaugeinvariant 2×2 Wigner matrix has been constructed whose diagonal elements describe positive and negative charge densities and the off-diagonal elements correspond to cross-densities in phase-space. The system of coupled transport equations has been derived in 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 general relativistic quantum Boltzmann-Vlasov equation (RQBVE) is discussed. In the non-relativistic quantum Boltzmann-Vlasov equation (RQBVE) is discussed. In the non-relativistic limit (p/mc→0) the set of equations of motion decouple to two independent quantum transport equations describing the dynamics of oppositely charged position and negaton densities. In the classical limit (þ→0) two relativistic Boltzmann-Vlasov equations result for the diagonal position and negaton densities. Even though the Planck constant þ is absent in the latter equations, the real part of the positon-negaton cross density does not vanish.