Effective action of QED in electric field backgrounds

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, E₀sech²(t/τ), 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.