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.

We give an explicit demonstration that the derivative expansion of the QED effective action is a divergent but Borel summable asymptotic series, for a particular inhomogeneous background magnetic field. A duality transformation B → iE gives a non-Borel-summable perturbative series for a time dependent background electric field, and Borel dispersion relations yield the non-perturbative imaginary part of the effective action, which determines the pair production probability. Resummations of leading Borel approximations exponentiate to give perturbative corrections to the exponents in the non-perturbative pair production rates. Comparison with a WKB analysis suggests that these divergence properties are general features of derivative expansions and effective actions.

We compute the exact QED 3+1 effective action for fermions in the presence of a family of static but spatially inhomogeneous magnetic field profiles. An asymptotic expansion of this exact effective action yields an all-orders derivative expansion, the first terms of which agree with independent derivative expansion computations. These results generalize analogous earlier results by Cangemi et al in QED 2+1 .The effective action plays a central role in quantum field theory. Here we consider the effective action in quantum electrodynamics (QED) for fermions in the presence of a background electromagnetic field. Using the proper-time technique [1], Schwinger showed that the QED effective action can be computed exactly for a constant (and for a plane wave) electromagnetic field. For general electromagnetic fields the effective action cannot be computed exactly, so one must resort to some sort of perturbative expansion. A common perturbative approach is known as the derivative expansion [2,3,4] in which one expands formally about the constant field case, assuming that the background is 'slowly varying'. However, even first-order derivative expansion calculations of the effective action are cumbersome, and somewhat difficult to interpret physically. A complementary approach is to seek other (i.e., inhomogeneous) background fields for which the effective action can be computed exactly, with the hope that this will lead to a better nonperturbative understanding of the derivative expansion. There are two technical impediments to such an exact computation of the effective action. First, the 1 background field must be such that the associated Dirac operator has a spectrum that is known exactly. Second, this spectrum will (in general) contain both discrete and continuum states, and so an efficient method is needed to trace over the entire spectrum. (Note that in the constant field case the spectrum is purely discrete so this trace is a simple sum). Cangemi et al [5] used a resolvent technique to obtain an exact answer for the effective action in 2 + 1-dimensional QED for massive fermions in the presence of static but spatially inhomogeneous magnetic fields of the form B(x, y) = B sech 2 (

We study the formation of vacuum condensates in (2+1)-dimensional QED in the presence of inhomogeneous background magnetic fields. For a large class of magnetic fields, the condensate is shown to be proportional to the inhomogeneous magnetic field, in the large flux limit. This may be viewed as a Iocal form of the integrated degeneracy-flux relation of Aharonov and Casher.

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