  The QED vacuum is a non-linear polarisable medium rather than an empty space.
In this work, we present the first rigorous derivation of the one-loop
effective magnetic Lagrangian at positive temperature, a non-linear functional
describing the free energy of quantum vacuum in a classical magnetic field. We
start by properly defining the free energy functional using the Pauli-Villars
regularisation technique in order to remove the worst ultraviolet divergence
which represents a well known issue of the theory. Then, we study 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, by recovering the effective Lagrangian first derived by Dittrich
in 1979.
