Phase-space structure of the Dirac vacuum

We study the phase-space correlation function for the Dirac vacuum in the presence of simple field configurations. Our formalism rests on the Wigner transform of the Dirac-Heisenberg correlation function of the Dirac field coupled to the electromagnetic field. A self-consistent set of equations obeyed by the 16 components of the phase-space correlation function and by the electric and magnetic field is derived. Our approach is manifestly gauge invariant. A closed system of integro-differential equations is obtained neglecting the quantum fluctuations of the electromagnetic field as should be appropriate for strong fields. These equations are an extension of the Vlasov equations used in the description of plasma. Our first applications address the production of particles in strong external fields. We set a framework for the inclusion of the back reaction of produced particles and for the description of the local changes of the vacuum state.