  We use the evolution operator method to find the one-loop effective action of
scalar and spinor QED in electric field backgrounds in terms of the Bogoliubov
coefficient between the ingoing and the outgoing vacua. We obtain the exact
one-loop effective action for a Sauter-type electric field, E_0 sech^2 (t/tau),
and show that the imaginary part correctly yields the vacuum persistence. The
renormalized effective action shows the general relation between the vacuum
persistence and the total mean number of created pairs for the constant and the
Sauter-type electric field.
