dorsal/arxiv
View SchemaSelf-consistent solution for the polarized vacuum in a no-photon QED model
| Authors | Christian Hainzl, Mathieu Lewin, Eric Sere |
|---|---|
| Categories | |
| ArXiv ID | physics/0404047 |
| URL | https://arxiv.org/abs/physics/0404047 |
| DOI | 10.1088/0305-4470/38/20/014 |
Abstract
We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 3791--3814, 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 math-ph/0403005, 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 kills 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.
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"abstract": "We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane\n({\\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced\nfrom no-photon QED. The associated functional is bounded from below. In the\npresence of an external field, a minimizer, if it exists, is interpreted as the\npolarized vacuum and it solves a self-consistent equation.\n In a recent paper math-ph/0403005, we proved the convergence of the iterative\nfixed-point scheme naturally associated with this equation to a global\nminimizer of the BDF functional, under some restrictive conditions on the\nexternal potential, the ultraviolet cut-off $\\Lambda$ and the bare fine\nstructure constant $\\alpha$. In the present work, we improve this result by\nshowing the existence of the minimizer by a variational method, for any cut-off\n$\\Lambda$ and without any constraint on the external field.\n We also study the behaviour of the minimizer as $\\Lambda$ goes to infinity\nand show that the theory is \"nullified\" in that limit, as predicted first by\nLandau: the vacuum totally kills the external potential. Therefore the limit\ncase of an infinite cut-off makes no sense both from a physical and\nmathematical point of view.\n Finally, we perform a charge and density renormalization scheme applying\nsimultaneously to all orders of the fine structure constant $\\alpha$, on a\nsimplified model where the exchange term is neglected.",
"arxiv_id": "physics/0404047",
"authors": [
"Christian Hainzl",
"Mathieu Lewin",
"Eric Sere"
],
"categories": [
"physics.atom-ph",
"math-ph",
"math.MP"
],
"doi": "10.1088/0305-4470/38/20/014",
"title": "Self-consistent solution for the polarized vacuum in a no-photon QED model",
"url": "https://arxiv.org/abs/physics/0404047"
},
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