The One Loop Effective Action of QED for a General Class of Electric Fields

TL;DR

This paper calculates the one-loop effective action in QED for a broad class of electric fields varying with light cone time, extending Schwinger's formula to non-constant fields.

Contribution

It generalizes Schwinger's formula to arbitrary electric fields depending on light cone time, providing a more comprehensive understanding of vacuum polarization effects.

Findings

Derived a closed-form expression for the effective action with time-dependent electric fields.

Extended the applicability of Schwinger's formula beyond constant fields.

Provided insights into vacuum stability under dynamic electric fields.

Abstract

We compute the effective action of QED at one loop order for an electric field which points in the $\hat{z}$ direction and depends arbitrarily upon the light cone time coordinate, $x^+ = (x^0 + x^3)/\sqrt{2}$. This calculation generalizes Schwinger's formula for the vacuum persistence probability in the presence of a constant electric field.