Fermion pair production in a strong electric field

The initial-value problem for the quantum back reaction in spinor QED is formulated and solved in the semiclassical mean-field approximation for a homogeneous but time-dependent electric field $E\left(t\right)\mathrm{}$. We apply the method of adiabatic regularization to the Dirac equation in order to renormalize the expectation value of the current and derive a finite coupled set of ordinary differential equations for the time evolution of the system. We solve this system in 1+1 dimensions numerically and compare the solution to a simple model based on a relativistic Boltzmann-Vlasov equation, with a particle production source term inferred from the Schwinger particle creation rate and a Pauliblocking factor. This model reproduces very well the time behavior of the electric field and the creation rate of $e^{+}{e}^{-}$ pairs of the semiclassical calculation.