Archive for Rational Mechanics and Analysis

, Volume 208, Issue 2, pp 603–665 | Cite as

Construction of the Pauli–Villars-Regulated Dirac Vacuum in Electromagnetic Fields

  • Philippe Gravejat
  • Christian Hainzl
  • Mathieu Lewin
  • Éric Séré


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.


Electromagnetic Field Quantum Electrodynamic Fourth Order Term Lagrangian Action Resolvent Expansion 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Philippe Gravejat
    • 1
  • Christian Hainzl
    • 2
  • Mathieu Lewin
    • 3
  • Éric Séré
    • 4
  1. 1.Centre de Mathématiques Laurent Schwartz (UMR 7640)Palaiseau CedexFrance
  2. 2.Mathematisches InstitutTübingenGermany
  3. 3.CNRS & Laboratoire de Mathématiques (UMR 8088)Université de Cergy-PontoiseCergy-PontoiseFrance
  4. 4.Ceremade (UMR 7534), Université Paris-DauphinePlace du Maréchal de Lattre de TassignyParis Cedex 16France

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