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Self-consistent solution for the polarized vacuum in a no-photon QED model. (English) Zbl 1073.81677

Summary: We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane [J. Phys. B, At. Mol. Opt. Phys. 22, 3791-814 (1989)] 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 \(\Lambda\) and the bare fine structure constant \(\alpha\). In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off \(\Lambda\) and without any constraint on the external field. We also study the behaviour of the minimizer as \(\Lambda\) 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 \(\alpha\) on a simplified model where the exchange term is neglected.

MSC:

81V10 Electromagnetic interaction; quantum electrodynamics
81T10 Model quantum field theories
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