Towards a derivation of Classical ElectroDynamics of charges and fields from QED
[Vers une dérivation de l’électromagnétisme des charges et des champs à partir de l’électrodynamique quantique]
Ammari, Zied1 ; Falconi, Marco2 ; Hiroshima, Fumio3
1 Univ Rennes, [UR1], CNRS, IRMAR – UMR 6625, F-35000 Rennes (France)
2 Politecnico di Milano, D-Mat, Piazza Leonardo da Vinci, 20133 Milano (Italy)
3 Faculty of Mathematics, Kyushu University, Motooka 744, Nishiku, Fukuoka 819-0385 (Japan)
Annales de l'Institut Fourier, Online first, 87 p.

The purpose of this article is twofold:

  • (1) On one hand, we rigorously derive the Newton–Maxwell equation in the Coulomb gauge from first principles of quantum electrodynamics in agreement with the formal Bohr correspondence principle of quantum mechanics.
  • (2) On the other hand, we establish the global well-posedness of the Newton–Maxwell system on energy-spaces under weak assumptions on the charge distribution.

Both results improve the state of the art, and are obtained by incorporating semiclassical and measure theoretical techniques. One of the novelties is the use of quantum propagation properties in order to build global solutions of the Newton–Maxwell equation.

L’objectif de cet article est double :

  • (1) D’une part, la dérivation rigoureuse et cohérente avec le principe de correspondance de Bohr de l’équation de Newton–Maxwell dans la jauge de Coulomb à partir de l’électrodynamique quantique.
  • (2) D’autre part, la preuve du caractère bien posé du système de Newton–Maxwell sur des espaces d’énergies adaptés avec des hypothèses faibles sur la distribution des charges.

Les deux résultats améliorent l’état de l’art et sont obtenus en incorporant des techniques semi-classiques et de la théorie des mesures. L’une des nouveautés est l’utilisation des propriétés de propagation quantique afin de construire des solutions globales de l’équation de Newton–Maxwell.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3720
Classification : 81V10, 35Q61, 81Q20, 81S08, 28C20
Keywords: Pauli–Fierz, Newton–Maxwell system, semiclassical analysis, Wigner measures, Liouville equation, generalized flow
Mots-clés : Modèle de Pauli–Fierz, système d’équations de Newton–Maxwell, analyse semi-classique, mesures de Wigner, équation de Liouville, flot généralisé

Ammari, Zied 1 ; Falconi, Marco 2 ; Hiroshima, Fumio 3

1 Univ Rennes, [UR1], CNRS, IRMAR – UMR 6625, F-35000 Rennes (France)
2 Politecnico di Milano, D-Mat, Piazza Leonardo da Vinci, 20133 Milano (Italy)
3 Faculty of Mathematics, Kyushu University, Motooka 744, Nishiku, Fukuoka 819-0385 (Japan) 
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Ammari, Zied; Falconi, Marco; Hiroshima, Fumio. Towards a derivation of Classical ElectroDynamics of charges and fields from QED. Annales de l'Institut Fourier, Online first, 87 p.

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