New partial resummation of the QED effective action

TL;DR

This paper proposes a new partial resummation technique for the QED effective action's proper-time series, potentially simplifying calculations involving electromagnetic field invariants and extending to gravity effects.

Contribution

It introduces a conjecture for summing all terms with field-strength invariants in the QED effective Lagrangian, offering a novel approach to handle the series expansion.

Findings

Partial summation encapsulated in a Heisenberg-Euler-like factor

Discussion of implications for quantum electrodynamics

Extension considerations in gravitational backgrounds

Abstract

We explain a conjecture which states that the proper-time series expansion of the one-loop effective Lagrangian of quantum electrodynamics can be partially summed in all terms containing the field-strength invariants $\mathcal{F} = \frac{1}{4} F_{\mu\nu}F^{\mu\nu} (x)$, $\mathcal{G}= \frac{1}{4} \tilde F_{\mu\nu}F^{\mu\nu}(x)$. This summation is encapsulated in a factor with the same form as the (spacetime-dependent) Heisenberg-Euler Lagrangian density. We also discuss some implications and a possible extension in presence of gravity. We will focus on the scalar field case.