Heisenberg-Euler and the Quantum Dilogarithm

TL;DR

This paper presents a novel dispersion integral representation of the Heisenberg-Euler QED effective Lagrangian using Faddeev's quantum dilogarithm, revealing deep connections to electromagnetic duality and nonperturbative effects.

Contribution

It introduces a new integral representation involving the quantum dilogarithm, linking nonperturbative imaginary parts to this special function and highlighting electromagnetic duality.

Findings

Expresses the imaginary part of the Lagrangian as the quantum dilogarithm.

Rewrites the real part as a dispersion integral with the quantum dilogarithm and its dual.

Connects the effective Lagrangian to all one-loop QED scattering amplitudes.

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.