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 , where is a normalized state for the electron and 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.
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 , où est un état normalisé pour l’électron et 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.
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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.
Breteaux, Sébastien 1; Faupin, Jérémy 1; Payet, Jimmy 1
@unpublished{AIF_0__0_0_A122_0, author = {Breteaux, S\'ebastien and Faupin, J\'er\'emy and Payet, Jimmy}, title = {Quasi-Classical {Ground} {States.} {II.} {Standard} {Model} of {Non-Relativistic} {QED}}, journal = {Annales de l'Institut Fourier}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, year = {2024}, doi = {10.5802/aif.3667}, language = {en}, note = {Online first}, }
TY - UNPB AU - Breteaux, Sébastien AU - Faupin, Jérémy AU - Payet, Jimmy TI - Quasi-Classical Ground States. II. Standard Model of Non-Relativistic QED JO - Annales de l'Institut Fourier PY - 2024 PB - Association des Annales de l’institut Fourier N1 - Online first DO - 10.5802/aif.3667 LA - en ID - AIF_0__0_0_A122_0 ER -
%0 Unpublished Work %A Breteaux, Sébastien %A Faupin, Jérémy %A Payet, Jimmy %T Quasi-Classical Ground States. II. Standard Model of Non-Relativistic QED %J Annales de l'Institut Fourier %D 2024 %I Association des Annales de l’institut Fourier %Z Online first %R 10.5802/aif.3667 %G en %F AIF_0__0_0_A122_0
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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