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Young Seminar
Organizers: Douglas Lundholm (IHP) & Julien Sabin (Cergy)

 

Jérémy Sok (Paris-Dauphine): "Existence of ground states in the Bogoliubov-Dirac-Fock model"
Tuesday 07 May 2013, 15:30 - 16:30
Room 314

 

Abstract: The  Bogoliubov-Dirac-Fock model is a no-photon mean-field approximation of QED introduced by Chaix-Iracane [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 one-body density matrix, a self-adjoint operator in L2(R3,C4). The purpose is then to minimize the energy over these states leading to variational problems defined in subsets of bounded operators of L2(R3,C4). We will show some theorems dealing with the existence of minimizers. For instance this is guaranteed when there hold so-called binding inequalities. Also, we will show that in the non-relativistic 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:

[1] V. Bach, J-M. Helffer, H. Siedentop, On the Stability of the Relativistic Electron-Positron Field., Comm. Math. Phys, 201 445-60, 1999

[2] P. Chaix, D. Iracane, From quantum electrodynamics to mean-field theory: I. The Bogoliubov-Dirac-Fock formalism J. Phys. B: At. Mol. Opt. Phys. 22 3791-3814, 1989.

[3] C. Hainzl, M. Lewin, É. Séré, Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation, Comm. Math. Phys, 257, 2005.

[4] C. Hainzl, M. Lewin, É. Séré. Self-consistent solution for the polarized vacuum in a no-photon QED model, J. Phys. A: Math and Gen. 38, no 20, 4483-4499, 2005.

[5] C. Hainzl, M. Lewin, J.P. Solovej, The Mean-Field Approximation in Quantum Electrodynamics. The no-photon case., Comm. Pure Appl. Math. 60 (4) : 546-596, 2007

[6] P. Gravejat, M. Lewin, É. Séré, Ground state and charge renormalization in a non-linear model of relativistic atoms, Comm. Math. Phys, 286, 2009.

[7] C. Hainzl, M. Lewin, É. Séré, Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics., Arch. Rational Mech. Anal, 192 no. 3, 453-499, 2009

[8] J. Sok, Existence of Ground State of an Electron in the BDF Approximation., preprint, 2012.

 

 

 

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