The Pauli-Villars regularised free energy of Dirac's vacuum in purely magnetic fields

TL;DR

This paper rigorously derives the finite-temperature effective magnetic Lagrangian for the Dirac vacuum using Pauli-Villars regularisation, connecting it to the classical Euler-Heisenberg formula with thermal corrections.

Contribution

It provides the first rigorous derivation of the one-loop effective magnetic Lagrangian at positive temperature for the Dirac vacuum, including ultraviolet divergence removal.

Findings

Established the functional's properties before the slowly varying field limit.

Proved convergence to the Euler-Heisenberg formula with thermal corrections.

Extended the understanding of the Dirac vacuum's non-linear behavior at finite temperature.

Abstract

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 understood. 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 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 in 1979.