$(\mathcal{F},\mathcal{G})$-summed form of the QED effective action

TL;DR

This paper proposes a conjecture that the proper-time series expansion of the one-loop QED effective Lagrangian can be summed to include all field-strength invariants and derivatives, generalizing the Heisenberg-Euler Lagrangian to arbitrary spacetime-dependent fields.

Contribution

The authors introduce a conjecture for summing the proper-time expansion of the QED effective action, extending the Heisenberg-Euler Lagrangian to include derivatives and spacetime-dependent fields, with proof up to sixth order.

Findings

Conjecture supported up to sixth order in proper time.

Derived solvable electromagnetic backgrounds.

Discussed implications for pair production in nonconstant fields.

Abstract

We conjecture that the proper-time series expansion of the one-loop effective Lagrangian of quantum electrodynamics can be summed in all terms containing the field-strength invariants $\mathcal{F} = \frac{1}{4} F_{\mu\nu}F^{\mu\nu} (x)$, $\mathcal{G}= \frac{1}{4} \tilde F_{\mu\nu}F^{\mu\nu}(x)$, including those also possessing derivatives of the electromagnetic field strength. This partial resummation is exactly encapsulated in a factor with the same form as the Heisenberg-Euler Lagrangian density, except that now the electric and magnetic fields can depend arbitrarily on spacetime coordinates. We provide strong evidence for this conjecture, which is proved to sixth order in the proper time. Furthermore, and as a byproduct, we generate some solvable electromagnetic backgrounds. We also discuss the implications for a generalization of the Schwinger formula for pair production induced by nonconstant electric fields. Finally, we briefly outline the extension of these results in the presence of gravity.