  We derive a system of coupled partial differential equations for the
equal-time Wigner function in an arbitrary strong electromagnetic field using
the Dirac-Heisenberg-Wigner formalism. In the electrostatic limit, we present a
3+1-system of four coupled partial differential equations, which are completed
by Ampere's law. This electrostatic system is further studied for two different
cases. In the first case, we consider linearized wave propagation in plasma
accounting for the nonzero vacuum expectation values. We then derive the
dispersion relation and compare it with well-known limiting cases. In the
second case, we consider Schwinger pair production using the local density
approximation to allow for analytical treatment. The dependence of the pair
production rate on the perpendicular momentum is investigated and it turns out
that the spread of the produced pairs along with perpendicular momentum depends
on the strength of the applied electric field.
