The Schwinger mechanism revisited
  The vacuum persistence probability, $P_{vac}(t)$, for a system of charged fermions in a fixed, external, and spatially homogeneous electric field, was derived long ago by Schwinger; $w = -log[P_{vac}(t)]/ (V t)$ has often been identified as the rate at which fermion-antifermion pairs are produced per unit volume due to the electric field. In this paper, we separately compute exact expressions for both $w$ and for the rate of fermion-antifermion pair production per unit volume, $\Gamma$, and show that they differ. While $w$ is given by the standard Schwinger mechanism result $w$, an infinite series, the pair production rate, $\Gamma$, is just the first term of that series. Our calculation is done for a system with periodic boundary conditions in the $A_0=0$ gauge but the result should hold for any consistent set of boundary conditions. We discuss, the physical reason why the rates $w$ and $\Gamma$ differ. 
Thomas D. Cohen, and David A. McGady