Derivation of the magnetic Euler-Heisenberg energy

TL;DR

This paper rigorously derives the magnetic Euler-Heisenberg energy from quantum field theory, showing how the polarized vacuum energy converges to a nonlinear functional in a semi-classical limit with regularization and renormalization.

Contribution

It provides the first rigorous derivation of the magnetic Euler-Heisenberg energy from quantum electrodynamics in a semi-classical regime.

Findings

Convergence of Dirac vacuum energy to the Euler-Heisenberg functional.

Regularization of ultraviolet divergences via Pauli-Villars method.

Discussion of charge renormalization to remove regularization effects.

Abstract

In quantum field theory, the vacuum is a fluctuating medium which behaves as a nonlinear polarizable material. In this article, we perform the first rigorous derivation of the magnetic Euler-Heisenberg effective energy, a nonlinear functional that describes the effective fluctuations of the quantum vacuum in a classical magnetic field. We start from a classical magnetic field in interaction with a quan-tized Dirac field in its ground state, and we study a limit in which the classical magnetic field is slowly varying. After a change of scales, this is equivalent to a semi-classical limit $\hbar\to0$, with a strong magnetic field of order $1/\hbar$. In this regime, we prove that the energy of Dirac's polarized vacuum converges to the Euler-Heisenberg functional. The model has ultraviolet divergences, which we regularize using the Pauli-Villars method. We also discuss how to remove the regularization of the Euler-Heisenberg effective Lagrangian, using charge renormaliza-tion, perturbatively to any order of the coupling constant.