Schwinger Mechanism for Fermion Pair Production in the Presence of Arbitrary Time Dependent Background Electric Field

We study the Schwinger mechanism for the pair production of fermions in the presence of an arbitrary time-dependent background electric field E(t) by directly evaluating the path integral. We obtain an exact non-perturbative result for the probability of fermion-antifermion pair production per unit time per unit volume per unit transverse momentum (of the fermion or antifermion) from the arbitrary time dependent electric field E(t) via Schwinger mechanism. We find that the exact non-perturbative result is independent of all the time derivatives d^nE(t)/dt^n, where n=1,2,....\infty. This result has the same functional dependence on E as the Schwinger's constant electric field E result with the replacement: E -> E(t).