QED effective actions in inhomogeneous backgrounds: Summing the derivative expansion

Abstract

Abstract. The QED effective action encodes nonlinear interactions due to quantum vacuum polarization effects. While much is known for the special case of electrons in a constant electromagnetic field (the Euler-Heisenberg case), much less is known for inhomogeneous backgrounds. Such backgrounds are more relevant to experimental situations. One way to treat inhomogeneous backgrounds is the "derivative expansion", in which one formally expands around the soluble constant-field case. In this talk I use some recent exactly soluble inhomogeneous backgrounds to perform precision tests on the derivative expansion, to learn in what sense it converges or diverges. A closely related question is to find the exponential correction to Schwinger's pair-production formula for a constant electric field, when the electric background is inhomogeneous.This talk is concerned with the one-loop QED effective action [1,2]:Here D µ = ∂ µ + ieA µ is the covariant derivative, and so S[A] is a functional of the classical background field A µ (x). The effective action is the generating functional for one-fermion-loop Green's functions, which describe the nonlinear QED effects to this order. Ideally, one would like to know S[A] for any field A µ (x), but this is not feasible. However, for the special case where the field strength tensor, F µν = ∂ µ A ν − ∂ ν A µ , is uniform, the effective action can be evaluated in closed form [3][4][5]:Here the first term is the familiar classical Maxwell action, while the next term gives the leading quantum correction, which is quartic in the field strengths. The fine structure constant α = e 2 4π in these units. All higher terms in the expansion (2) are known explicitly [3][4][5]. It can also be shown that when the background field is a constant electric field of strength E, the effective action has an exponentially small imaginary part