 04397 Christian HAINZL, Mathieu LEWIN, Eric SERE
 Selfconsistent solution for the polarized vacuum
in a nophoton QED model
(58K, AMSTex)
Nov 26, 04

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Abstract. We study the BogoliubovDiracFock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 37913814, 1989) which is a meanfield theory deduced from nophoton 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 selfconsistent equation.
In a recent paper (ArXiv: {\tt mathph/0403005}), we proved the convergence of the iterative fixedpoint scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cutoff $\Lambda$ 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 cutoff $\Lambda$ 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 kills the external potential. Therefore the limit case of an infinite cutoff 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.
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