Properties of Ground States in the BogoliubovDiracFock Model
Sok, Jérémy (2013), Properties of Ground States in the BogoliubovDiracFock Model. https://basepub.dauphine.fr/handle/123456789/12243
Type
Document de travail / Working paperExternal document link
http://hal.archivesouvertes.fr/hal00909168Date
2013Publisher
Université ParisDauphine
Published in
Paris
Pages
43
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Sok, JérémyAbstract (EN)
We study the BogoliubovDiracFock model that allows to consider relativistic electrons interacting with the vacuum in the presence of an external electrostatic field. It can be seen as a HartreeFock approximation of QED, where photons are neglected. A state is described by its onebody density matrix: an infinite rank, selfadjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator. We are interested in the properties of minimizers of the BDFenergy in the presence of an external field with charge density $\nu\ge 0$ in the regime $\alpha$, $\alpha \log(\Lambda)$ and $\alpha \nu$ (in some norms) small where $\alpha$ is the coupling constant and $\Lambda$ the ultraviolet cutoff. We prove that the density of such minimizer is integrable and compute the effective charge of the system. We also ensure the existence of minimizers under charge constraint $M\in\mathbf{N}^*$ provided that there holds $M1< \int \nu$ close to the nonrelativistic limit $\alpha\to 0$ with $\alpha\log(\Lambda)$ fixed to a small value. This contrasts with the assumptions of [Arch. Ration. Mech. Anal, 192(3):453499(2009)] where $\Lambda$ is fixed. As a consequence, the nonrelativistic model we obtain in the limit keeps track of the charge renormalisation: it is different from the HartreeFock model obtained.Subjects / Keywords
ground states; charge renormalisation; BogoliubovDircFock model; Dirac operatorRelated items
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