The
Dirac vacuum is a non-linear polarisable medium rather than an empty space. This
non-linear 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 non-zero temperature needs to be better under-
stood.
In this talk, we will present the first rigorous derivation of the one-loop effective magnetic La-
grangian at positive temperature, a non-linear functional describing the free energy of quantum
vacuum in a classical magnetic field. After introducing our model, we will properly define the free
energy functional using the Pauli-Villars regularisation technique in order to remove the worst ul-
traviolet divergences, which represent a well known issue of the theory. The study of the properties
of this functional will be addressed before focusing on the limit of slowly varying classical magnetic
fields. In this regime, one can prove the convergence of this functional to the Euler-Heisenberg
formula with thermal corrections, recovering the effective Lagrangian first derived by Dittrich in
1979. The talk is based on the work available at arXiv:2404.12733.