Time evolution of the QED vacuum in a uniform electric Field: Complete   analytic solution by spinorial decomposition
  Exact analytical solutions are presented for the time evolution of the density of pairs produced in the QED vacuum by a time-independent, uniform electric field. The mathematical tool used here to describe the pair production is the Dirac-Heisenberg-Wigner function introduced before [Phys. Rev. D 44, 1825 (1991)]. The initial value problem for this function is solved by decomposing the solution into a product of spinors. The equations for spinors are much simpler and are solved analytically. These calculations are nonperturbative since pair production is due to quantum-mechanical tunneling and the explicit solutions clearly exhibit their nonanalytic behavior. 
Iwo Bialynicki-Birula and {\L}ukasz Rudnicki
hep-th math-ph math.MP quant-ph