Heisenberg-Euler and the Quantum Dilogarithm

TL;DR

The paper derives dispersive representations of the Heisenberg–Euler one-loop Lagrangian in constant electromagnetic fields using quantum dilogarithms. The imaginary part is given by the (compact or non-compact) quantum dilogarithm, while the real part emerges as a dispersion transform of this imaginary part and its modular dual, reflecting electromagnetic duality. It establishes explicit connections between spinor and scalar QED through dilogarithm identities, providing a unified, duality-aware framework for all one-loop, constant-field amplitudes and suggesting extensions to inhomogeneous fields and higher loops. This work reframes QED dispersion theory in terms of the quantum dilogarithm as a generator of amplitudes with many external legs.

Abstract

A dispersion integral representation of the Heisenberg-Euler QED effective lagrangian is derived, with Faddeev's quantum dilogarithm as a generalized Borel kernel. The nonperturbative imaginary part of the effective lagrangian is expressed as the quantum dilogarithm, while the real part has the form of a dispersion integral involving both the quantum dilogarithm and its modular dual, a manifestation of electromagnetic duality. The Heisenberg-Euler effective lagrangian generates all one-loop QED scattering amplitudes in a constant external field, with the Lorentz invariants of the constant background electromagnetic field playing the role of the Mandelstam variables in conventional QED dispersion theory.