Construction of the Pauli-Villars-regulated Dirac vacuum in   electromagnetic fields
  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. 
Philippe Gravejat (CMLS-EcolePolytechnique), Christian Hainzl, Mathieu   Lewin (AGM), Eric S\'er\'e (CEREMADE)
math-ph math.MP