Derivation of the magnetic Euler–Heisenberg energy

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 quantized 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 → 0, with a strong magnetic field of order 1/ . 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 renormalization, perturbatively to any order of the coupling constant.

Shadow of a noncommutative inspired Einstein-Euler-Heisenberg black hole

“…Several non-linear electrodynamics are known [3][4][5][6][7]. Perhaps, one of the simplest is Euler-Heisenberg nonlinear electrodynamics [8]; it arises quite naturally from calculations in Quantum Electrodynamics (QED) at one loop as the effective description of the vacuum as a fluctuating medium [9]. It is natural to study Euler-Heisenberg (EH) non-linear electrodynamics when extending Maxwell electrodynamics due to its sound physical basis and experimental verification; it is relevant, for example, to describe magnetars [10,11].…”