Renormalization and asymptotic expansion of Dirac's polarized vacuum

TL;DR

This paper rigorously analyzes charge renormalization in the reduced Bogoliubov-Dirac-Fock model, demonstrating an asymptotic expansion of the physical density considering vacuum polarization effects, with implications for atomic and molecular physics.

Contribution

It provides a rigorous mathematical framework for charge renormalization and asymptotic expansion in the rBDF model, accounting for vacuum polarization effects.

Findings

Asymptotic expansion of physical density to all orders in the coupling constant

Renormalization parameter behavior linked to ultraviolet cut-off

Validation of the model's consistency with physical vacuum polarization

Abstract

We perform rigorously the charge renormalization of the so-called reduced Bogoliubov-Dirac-Fock (rBDF) model. This nonlinear theory, based on the Dirac operator, describes atoms and molecules while taking into account vacuum polarization effects. We consider the total physical density including both the external density of a nucleus and the self-consistent polarization of the Dirac sea, but no `real' electron. We show that it admits an asymptotic expansion to any order in powers of the physical coupling constant $\alphaph$, provided that the ultraviolet cut-off behaves as $\Lambda\sim e^{3\pi(1-Z_3)/2\alphaph}\gg1$. The renormalization parameter $0