Derivation of the magnetic Euler–Heisenberg energy

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 ħ→0, with a strong magnetic field of order 1/ħ. 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.