A quantum kinetic equation for particle production in the Schwinger   mechanism
  A quantum kinetic equation is derived for the description of pair production in a time-dependent homogeneous electric field $E(t)$. As a source term, the Schwinger mechanism for particle creation is incorporated. Possible particle production due to collisions and collisional damping are neglected. The main result is a kinetic equation of non-Markovian character. In the low density approximation, the source term is reduced to the leading part of the well known Schwinger formula for the probability of pair creation. We discuss the momentum and time dependence of the derived source term and compare with other approaches. 
S. Schmidt, D. Blaschke, G. R\"opke, S.A. Smolyansky, A.V.   Prozorkevich and V.D. Toneev