The Dirac–Klein–Gordon system in the strong coupling limit
[Limite de couplage fort du système de Dirac–Klein–Gordon]
Lampart, Jonas1 ; Le Treust, Loïc2 ; Rota Nodari, Simona3 ; Sabin, Julien4
1 CNRS & Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université de Bourgogne-Franche-Comté, 9, avenue Alain Savary, 21078 Dijon Cedex, France
2 Institut de Mathématiques de Marseille, UMR CNRS 7373, Aix-Marseille Université, Rue Frédéric Joliot-Curie, 13453 MARSEILLE Cedex 13, France
3 Laboratoire Jean Alexandre Dieudonné, UMR CNRS 7351, Université Côte d’Azur, 28, avenue Valrose, 06108 Nice Cedex 2, France
4 Centre de mathématiques Laurent Schwartz, UMR CNRS 7640, École polytechnique, 91128 Palaiseau Cedex, France
Annales Henri Lebesgue, Tome 6 (2023), pp. 541-573

We study the Dirac equation coupled to scalar and vector Klein–Gordon fields in the limit of strong coupling and large masses of the fields. We prove convergence of the solutions to those of a cubic non-linear Dirac equation, given that the initial spinors coincide. This shows that in this parameter regime, which is relevant to the relativistic mean-field theory of nuclei, the retarded interaction is well approximated by an instantaneous, local self-interaction. We generalize this result to a many-body Dirac–Fock equation on the space of Hilbert–Schmidt operators.

Nous étudions une équation de Dirac couplée à des champs de Klein–Gordon scalaires et vectoriels dans la limite de couplages forts et de grandes masses. Nous démontrons la convergence de ses solutions vers celles d’une équation de Dirac non-linéaire, si les spineurs initiaux coïncident. Cela montre que dans ce régime de paramètres, pertinent en théorie de champ moyen relativiste des noyaux, l’interaction retardée est bien approchée par une auto-interaction locale et instantanée. Nous généralisons ensuite ce résultat à une équation de Dirac–Fock à plusieurs corps dans l’espace des opérateurs Hilbert–Schmidt.

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Accepté le :
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DOI : 10.5802/ahl.171
Classification : 35Q40, 81Q05, 81V35, 47J35
Keywords: Relativistic mean-field, nuclear physics, nonlinear analysis, asymptotic analysis, highly oscillatory equations, nonlinear Dirac equation, Klein-Gordon equation

Lampart, Jonas 1 ; Le Treust, Loïc 2 ; Rota Nodari, Simona 3 ; Sabin, Julien 4

1 CNRS & Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR CNRS 6303, Université de Bourgogne-Franche-Comté, 9, avenue Alain Savary, 21078 Dijon Cedex, France
2 Institut de Mathématiques de Marseille, UMR CNRS 7373, Aix-Marseille Université, Rue Frédéric Joliot-Curie, 13453 MARSEILLE Cedex 13, France
3 Laboratoire Jean Alexandre Dieudonné, UMR CNRS 7351, Université Côte d’Azur, 28, avenue Valrose, 06108 Nice Cedex 2, France
4 Centre de mathématiques Laurent Schwartz, UMR CNRS 7640, École polytechnique, 91128 Palaiseau Cedex, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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     title = {The {Dirac{\textendash}Klein{\textendash}Gordon} system in the strong coupling limit},
     journal = {Annales Henri Lebesgue},
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Lampart, Jonas; Le Treust, Loïc; Rota Nodari, Simona; Sabin, Julien. The Dirac–Klein–Gordon system in the strong coupling limit. Annales Henri Lebesgue, Tome 6 (2023), pp. 541-573. doi: 10.5802/ahl.171

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