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 equivalent to -log[P-vac(t)]/Vt 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 of w = (qE)(2)/4 pi(3) Sigma(infinity)(n=1) 1/n(2) exp (- n pi m(2)/qE), 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.