QED Effective Action Revisited

TL;DR

This work refines a convergent series representation for the QED effective action, casting the Heisenberg–Euler correction $\Delta {\cal L}$ into a unified real/imaginary part expansion using a special-function framework with invariants ${\cal F}$ and ${\cal G}$ and secular variables $a$ and $b$. It provides a corrected mathematical identity for the key product $a b \coth(a z) \cot(b z)$, derives a complete partial-fraction decomposition, and discusses the implications for electric–magnetic duality, including the roles of contour choices and invariant quantities ${\cal F}$ and ${\cal G}$. A central contribution is the demonstration of a powerful convergence acceleration strategy, CNCT, which combines Van Wijngaarden condensation with a delta transform to sum slowly convergent monotonic series, enabling reliable numerical evaluation even at strong fields. The results have practical impact for high-precision QED calculations in strong-field contexts, informing predictions for light-by-light scattering, vacuum birefringence, and pair production in astrophysical or laser–driven environments.

Abstract

The derivation of a convergent series representation for the quantum electrodynamic effective action obtained by two of us (S.R.V. and D.R.L.) in [Can. J. Phys. vol. 71, p. 389 (1993)] is reexamined. We present more details of our original derivation. Moreover, we discuss the relation of the electric-magnetic duality to the integral representation for the effective action, and we consider the application of nonlinear convergence acceleration techniques which permit the efficient and reliable numerical evaluation of the quantum correction to the Maxwell Lagrangian.