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

Umberto Morellini

TL;DR

This work provides a rigorous, temperature-dependent treatment of Dirac vacuum energy in the presence of a purely magnetic external field using Pauli–Villars regularisation. It defines a well-posed PV-regularised vacuum free energy $\mathcal{F}_{PV}$ at positive temperature, proves its gauge-invariant structure, and decomposes it into a finite quadratic part plus a smooth remainder. In the slowly varying magnetic-field regime, the authors derive a rigorous Euler–Heisenberg-type limit for the finite-temperature vacuum energy, obtaining explicit local energy densities $f_{PV}^0(|B|)$ and $f_{PV}^T(|B|,\beta)$ that recover the zero-temperature Euler–Heisenberg result with thermal corrections. The results address UV divergences inherent to QED, relate finite-temperature corrections to the dielectric response of Dirac’s vacuum, and provide a mathematically controlled bridge between microscopic quantum field theory and macroscopic effective Lagrangians applicable in extreme-field astrophysical contexts.

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.

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

TL;DR

This work provides a rigorous, temperature-dependent treatment of Dirac vacuum energy in the presence of a purely magnetic external field using Pauli–Villars regularisation. It defines a well-posed PV-regularised vacuum free energy at positive temperature, proves its gauge-invariant structure, and decomposes it into a finite quadratic part plus a smooth remainder. In the slowly varying magnetic-field regime, the authors derive a rigorous Euler–Heisenberg-type limit for the finite-temperature vacuum energy, obtaining explicit local energy densities and that recover the zero-temperature Euler–Heisenberg result with thermal corrections. The results address UV divergences inherent to QED, relate finite-temperature corrections to the dielectric response of Dirac’s vacuum, and provide a mathematically controlled bridge between microscopic quantum field theory and macroscopic effective Lagrangians applicable in extreme-field astrophysical contexts.

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.
Paper Structure (8 sections, 9 theorems, 160 equations)

This paper contains 8 sections, 9 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model and main results
  3. Rigorous definition of the PV-regularised free energy functional
  4. Integrable magnetic potentials: proof of the first part of \ref{['stat:main-thm-1']}
  5. Gauge-invariant estimates
  6. Proof of the second part of \ref{['stat:main-thm-1']}
  7. The regime of slowly varying magnetic fields: proof of \ref{['stat:main-thm-2']}
  8. The non-zero temperature regularisation in the ultraviolet regime

Key Result

Lemma 2.1

Let $m>0$ and $\boldsymbol{A}\in\dot{H}^1_\mathrm{div}\left(\mathbb{R}^3\right)$.

Theorems & Definitions (16)

  • Lemma 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Remark 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Remark 3.4
  • Remark 3.5
  • Remark 3.6
  • ...and 6 more