On the QED Effective Action in Time Dependent Electric Backgrounds

TL;DR

The paper develops and applies a resolvent-based framework to compute the QED effective action in a time-dependent electric background, obtaining an exact integral representation for $S_{eff}$ in $E(t)=E\,sech^2(t/\tau)$. It demonstrates dispersion relations that connect the nonperturbative imaginary part (pair production) to the perturbative real part, and validates these results against the derivative expansion and a uniform semiclassical approximation. The work also provides an all-orders perspective, showing the derivative expansion is asymptotic but consistent with the exact solution in the solvable case, and extends the approach to a general time dependence via a semiclassical formulation. Together, these results offer a robust toolkit for estimating pair production in realistic, slowly varying fields and clarifying the interplay between perturbative and nonperturbative QED dynamics.

Abstract

We apply the resolvent technique to the computation of the QED effective action in time dependent electric field backgrounds. The effective action has both real and imaginary parts, and the imaginary part is related to the pair production probability in such a background. The resolvent technique has been applied previously to spatially inhomogeneous magnetic backgrounds, for which the effective action is real. We explain how dispersion relations connect these two cases, the magnetic case which is essentially perturbative in nature, and the electric case where the imaginary part is nonperturbative. Finally, we use a uniform semiclassical approximation to find an expression for very general time dependence for the background field. This expression is remarkably similar in form to Schwinger's classic result for the constant electric background.