Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields

TL;DR

This paper constructs solutions to Maxwell's equations in Dirac's vacuum using Pauli-Villars regularization, revealing a nonlinear, nonlocal quantum vacuum that behaves as a polarizable medium, aligning with quantum electrodynamics at first order.

Contribution

It introduces a method to model the quantum vacuum as a nonlinear polarizable medium using Pauli-Villars regularization and convex analysis, bridging classical and quantum descriptions.

Findings

Vacuum behaves as a nonlinear polarizable medium

Quantum corrections are nonlinear and nonlocal

First-order polarization matches quantum electrodynamics

Abstract

Using the Pauli-Villars regularization and arguments from convex analysis, we construct solutions to the classical time-independent Maxwell equations in Dirac's vacuum, in the presence of small external electromagnetic sources. The vacuum is not an empty space, but rather a quantum fluctuating medium which behaves as a nonlinear polarizable material. Its behavior is described by a Dirac equation involving infinitely many particles. The quantum corrections to the usual Maxwell equations are nonlinear and nonlocal. Even if photons are described by a purely classical electromagnetic field, the resulting vacuum polarization coincides to first order with that of full Quantum Electrodynamics.