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The Pauli-Villars-regularized Dirac vacuum in electromagnetic fields at positive temperature

William Borrelli, Umberto Morellini

TL;DR

This work rigorously defines the free energy of a quantised Dirac field in external electromagnetic fields at positive temperature using Pauli–Villars regularization, extending prior purely magnetic analyses to include electrostatic potentials. The authors formulate the PV energy functional $ ext{F}_{ ext{PV}}$ and prove its well-posedness on an energy space, decomposing it into a quadratic, linear-response part $ ext{F}_2$ and a controlled remainder $ ext{R}$, with explicit Fourier multipliers $M^0(k)$, $M^T(k,eta)$ and a temperature-dependent term $oldsymbol{ } abla V$-dependent $oldsymbol{ } ext{Γ}(k,eta)$. They provide detailed analyses of the second-order term, including the split into zero-temperature and thermal contributions and the appearance of a potential $k o 0$ singularity in $ ext{Γ}(k,eta)$ that hints at Debye screening in the presence of an electrostatic field. The approach preserves gauge symmetry and yields gauge-invariant estimates, enabling a robust treatment of vacuum polarization at finite temperature with a rigorous mathematical foundation that informs both theoretical QED and potential applications to polarisation phenomena in strong-field contexts.

Abstract

In this paper we consider a model of the Dirac vacuum in classical electromagnetic fields at positive temperature. We adopt the Pauli-Villars regularisation technique in order to properly define the free energy of the vacuum, extending the previous work by the second named author on the purely magnetic case. This work is intended as a first step in understanding polarisation effects in the vacuum at positive temperature, in presence of both electrostatic and magnetic potentials.

The Pauli-Villars-regularized Dirac vacuum in electromagnetic fields at positive temperature

TL;DR

This work rigorously defines the free energy of a quantised Dirac field in external electromagnetic fields at positive temperature using Pauli–Villars regularization, extending prior purely magnetic analyses to include electrostatic potentials. The authors formulate the PV energy functional and prove its well-posedness on an energy space, decomposing it into a quadratic, linear-response part and a controlled remainder , with explicit Fourier multipliers , and a temperature-dependent term -dependent . They provide detailed analyses of the second-order term, including the split into zero-temperature and thermal contributions and the appearance of a potential singularity in that hints at Debye screening in the presence of an electrostatic field. The approach preserves gauge symmetry and yields gauge-invariant estimates, enabling a robust treatment of vacuum polarization at finite temperature with a rigorous mathematical foundation that informs both theoretical QED and potential applications to polarisation phenomena in strong-field contexts.

Abstract

In this paper we consider a model of the Dirac vacuum in classical electromagnetic fields at positive temperature. We adopt the Pauli-Villars regularisation technique in order to properly define the free energy of the vacuum, extending the previous work by the second named author on the purely magnetic case. This work is intended as a first step in understanding polarisation effects in the vacuum at positive temperature, in presence of both electrostatic and magnetic potentials.
Paper Structure (11 sections, 5 theorems, 150 equations)

This paper contains 11 sections, 5 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Derivation of the model
  3. Main theorem
  4. Rigorous definition of the free energy
  5. Integrable four-potentials: proof of item i) in \ref{['thm:main']}
  6. Gauge-invariant estimates
  7. The quadratic term in \ref{['eq:FPV-decomposition']}
  8. Analysis of $\Gamma_1(k,\beta)$
  9. Analysis of $\Gamma_2(k,\beta)$
  10. Analysis of $\Gamma_3(k,\beta)$
  11. End of the proof of \ref{['thm:main']}

Key Result

Lemma 1.1

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

Theorems & Definitions (11)

  • Lemma 1.1
  • Remark 1.2
  • Theorem 1.3: Rigorous definition of $\mathcal{F}_{\mathrm{PV}}$
  • Remark 1.4
  • proof : Proof of item i) in \ref{['thm:main']}
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • ...and 1 more