Quasi-Classical Ground States. II. Standard Model of Non-Relativistic QED

Breteaux, Sébastien; Faupin, Jérémy; Payet, Jimmy

doi:10.5802/aif.3667

Quasi-Classical Ground States. II. Standard Model of Non-Relativistic QED
[États fondamentaux quasi classique. II. Le modèle standard de l’électrodynamique quantique non-relativiste]

Breteaux, Sébastien ¹ ; Faupin, Jérémy ¹ ; Payet, Jimmy ¹

¹ Université de Lorraine CNRS IECL F-57000 Metz (France)

Annales de l'Institut Fourier, Online first, 44 p.

Résumé
Abstract

On considère un électron non relativiste placé dans un potentiel extérieur et couplé au champ électromagnétique quantifié dans le modèle standard de l’électrodynamique quantique non relativiste. On s’intéresse à la fonctionnelle obtenue en calculant l’énergie du système total en des états produits de la forme $u \otimes Ψ_{f}$ , où $u$ est un état normalisé pour l’électron et $Ψ_{f}$ est un état cohérent dans l’espace de Fock pour le champ de photons. La minimisation de cette fonctionnelle fait apparaître, après une renormalisation triviale, l’énergie d’un système de Maxwell–Pauli. On prouve l’existence d’un état fondamental sous des conditions générales portant sur le potentiel extérieur et sur la fonction de couplage. En particulier, il n’est pas nécessaire d’imposer une troncature ultraviolette ni une troncature infrarouge. Nos résultats établissent la convergence dans la limite ultraviolette de l’énergie fondamentale des systèmes de Maxwell–Pauli, ainsi que le développement asymptotique au second ordre de cette énergie par rapport à la constante de couplage.

We consider a non-relativistic electron bound by an external potential and coupled to the quantized electromagnetic field in the standard model of non-relativistic QED. We compute the energy functional of product states of the form $u \otimes Ψ_{f}$ , where $u$ is a normalized state for the electron and $Ψ_{f}$ is a coherent state in Fock space for the photon field. The minimization of this functional yields a Maxwell–Pauli system up to a trivial renormalization. We prove the existence of a ground state under general conditions on the external potential and the coupling. In particular, neither an ultraviolet cutoff nor an infrared cutoff needs to be imposed. Our results provide the convergence in the ultraviolet limit and the second-order asymptotic expansion in the coupling constant of the ground state energy of Maxwell–Pauli systems.

Reçu le : 2022-10-27
Révisé le : 2023-04-19
Accepté le : 2023-08-04
Première publication : 2024-07-03

DOI : 10.5802/aif.3667

Classification : 00X99
Keywords: Ground states, quasi-classical limit, non-relativistic quantum electrodynamics, calculus of variations, Pauli-Fierz model, Maxwell-Pauli energy functional, Ultraviolet limit.
Mot clés : États fondamentaux, limite quasi-classique, électrodynamique quantique non relativiste, calcul variationnel, modèle de Pauli–Fierz, fonctionnelle d’énergie de Maxwell–Pauli, Limite ultraviolette.

Affiliations des auteurs :

Breteaux, Sébastien ¹ ; Faupin, Jérémy ¹ ; Payet, Jimmy ¹

¹ Université de Lorraine CNRS IECL F-57000 Metz (France)

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     title = {Quasi-Classical {Ground} {States.} {II.} {Standard} {Model} of {Non-Relativistic} {QED}},
     journal = {Annales de l'Institut Fourier},
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Breteaux, Sébastien; Faupin, Jérémy; Payet, Jimmy. Quasi-Classical Ground States. II. Standard Model of Non-Relativistic QED. Annales de l'Institut Fourier, Online first, 44 p.

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