Derivation of the magnetic Euler-Heisenberg energy

Philippe Gravejat (1) , Mathieu Lewin (2) , Eric Séré (2)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Domaines: Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]; Physique Quantique [quant-ph]

Philippe Gravejat, Mathieu Lewin, Eric Séré. Derivation of the magnetic Euler-Heisenberg energy. 2016. ⟨hal-01270637v2⟩