Gauge-invariant relativistic Wigner functions

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 x 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 (h --> 0) two relativistic Boltzmann-Vlasov equations result for the diagonal positon and negaton densities. Even though the Planck constant h is absent from the latter equations, the real part of the positon-negaton cross-density does not vanish.