无光子QED模型中极化真空的自洽解
Self-consistent solution for the polarized vacuum in a no-photon QED model
摘要：
本文研究了由Chaix和Iracane（{ \it J. Phys. B.}，22，3791- 3814，1989），这是从无光子QED推导出的平均场论。相关的泛函是从下有界的。在外场存在的情况下，极小元，如果它存在的话，被解释为极化真空，它解决了一个自洽方程。在最近的一篇论文math-ph/0403005中，我们证明了与该方程自然相关的迭代不动点格式在一定的外势、紫外截止值$Λ$和裸精细结构常数$α$的限制条件下收敛到BDF泛函的全局极小值.在目前的工作中，我们改进了这一结果，通过变分方法显示存在的极小值，任何截止$Λ$和没有任何约束的外部字段。我们还研究了行为的最小$Λ$走向无穷大，并表明，该理论是“无效”的限制，第一预测的朗道：真空完全杀死外部潜力。因此，从物理和数学的角度来看，无限截止的极限情况是没有意义的。最后，我们执行的电荷和密度重整化计划，同时适用于所有订单的精细结构常数$α$，在一个简化的模型，其中交换项被忽略。
英文摘要：
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 $Λ$ 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 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 $α$, on a simplified model where the exchange term is neglected.
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