
Young Seminar
Organizers: Douglas Lundholm (IHP) & Julien Sabin (Cergy)
Jérémy Sok (ParisDauphine): "Existence of ground states in the BogoliubovDiracFock model"
Tuesday 07 May 2013, 15:30  16:30
Room 314
Abstract: The BogoliubovDiracFock model is a nophoton meanfield approximation of QED introduced by ChaixIracane [2]. It allows to describe relativistic electrons interacting with the Dirac sea in the presence of some external electrostatic potential. Starting from the hamiltonian of the QED in the Fock space we focus on special states entirely described by their onebody density matrix, a selfadjoint operator in L^{2}(R^{3},C^{4}). The purpose is then to minimize the energy over these states leading to variational problems defined in subsets of bounded operators of L^{2}(R^{3},C^{4}). We will show some theorems dealing with the existence of minimizers. For instance this is guaranteed when there hold socalled binding inequalities. Also, we will show that in the nonrelativistic limits (as the speed of light tends to infinity) the problem of one electron in the vacuum without any external potential has a minimizer. This is interpreted as follows: the vacuum reacts to the presence of the electron creating around him a cloud of positive charge.
Bibliography:
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[2] P. Chaix, D. Iracane, From quantum electrodynamics to meanfield theory: I. The BogoliubovDiracFock formalism J. Phys. B: At. Mol. Opt. Phys. 22 37913814, 1989.
[3] C. Hainzl, M. Lewin, É. Séré, Existence of a stable polarized vacuum in the BogoliubovDiracFock approximation, Comm. Math. Phys, 257, 2005.
[4] C. Hainzl, M. Lewin, É. Séré. Selfconsistent solution for the polarized vacuum in a nophoton QED model, J. Phys. A: Math and Gen. 38, no 20, 44834499, 2005.
[5] C. Hainzl, M. Lewin, J.P. Solovej, The MeanField Approximation in Quantum Electrodynamics. The nophoton case., Comm. Pure Appl. Math. 60 (4) : 546596, 2007
[6] P. Gravejat, M. Lewin, É. Séré, Ground state and charge renormalization in a nonlinear model of relativistic atoms, Comm. Math. Phys, 286, 2009.
[7] C. Hainzl, M. Lewin, É. Séré, Existence of Atoms and Molecules in the MeanField Approximation of NoPhoton Quantum Electrodynamics., Arch. Rational Mech. Anal, 192 no. 3, 453499, 2009
[8] J. Sok, Existence of Ground State of an Electron in the BDF Approximation., preprint, 2012.