Effective Action of QED in Electric Field Backgrounds II: Spatially Localized Fields

TL;DR

This paper derives the exact one-loop effective action for scalar and spinor QED in a specific spatially localized electric field, revealing duality with time-dependent fields and confirming the relation between vacuum persistence and pair production.

Contribution

It provides an exact calculation of the effective action in a spatially localized electric field and introduces an approximation method highlighting duality with time-dependent fields.

Findings

Exact effective action obtained for a sech^2 electric field.

Confirmed the relation between vacuum persistence and pair production.

Showed duality between space-dependent and time-dependent electric fields.

Abstract

We find the Bogoliubov coefficient from the tunneling boundary condition on a charged particle coupled to a static electric field $E_0 sech^2 (z/L)$ and, using the regularization scheme in Phys. Rev. D 78, 105013 (2008), obtain the exact one-loop effective action in scalar and spinor QED. It is shown that the effective action satisfies the general relation between the vacuum persistence and the mean number of produced pairs. We advance an approximation method for general electric fields and show the duality between the space-dependent and time-dependent electric fields of the same form at the leading order of the effective actions.