Self-consistent solution for the polarized vacuum in a no-photon QED model
Séré, Eric; Hainzl, Christian; Lewin, Mathieu (2005), Self-consistent solution for the polarized vacuum in a no-photon QED model, Journal of Physics A: Mathematical and General, 38, 20, p. 4483-4499. http://dx.doi.org/10.1088/0305-4470/38/20/014
TypeArticle accepté pour publication ou publié
Journal nameJournal of Physics A: Mathematical and General
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Abstract (EN)We study the Bogoliubov–Dirac–Fock model introduced by Chaix and Iracane (1989 J. Phys. B: At. Mol. Opt. Phys. 22 3791–814) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off Λ and the bare fine structure constant α. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off Λ and without any constraint on the external field. We also study the behaviour of the minimizer as Λ goes to infinity and show that the theory is 'nullified' in that limit, as predicted first by Landau: the vacuum totally cancels the external potential. Therefore, the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant α, on a simplified model where the exchange term is neglected.
Subjects / KeywordsMathematical Physics
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