The Pauli–Villars Regularised Free Energy of Dirac’s Vacuum in Purely Magnetic Fields

Abstract

The Dirac vacuum is a nonlinear polarisable medium rather than an empty space. This nonlinear behaviour starts to be significant for extremely large electromagnetic fields such as the magnetic field on the surface of certain neutron stars. Even though the null temperature case was deeply studied in the past decades, the problem at nonzero temperature needs to be better understood. In this work, we present the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature, a nonlinear functional describing the free energy of the Dirac vacuum in a classical magnetic field. After introducing our model, we properly define the free energy functional using the Pauli–Villars regularisation technique in order to remove the worst ultraviolet divergences, which represent a well-known issue of the theory. The study of the properties of this functional is addressed before focusing on the limit of slowly varying classical magnetic fields. In this regime, we prove the convergence of this functional to the Euler–Heisenberg formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich (Phys Rev D 19:2385–2390, 1979).