Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms

Abstract

37 pages, 1 figureInternational audienceWe study the reduced Bogoliubov-Dirac-Fock (BDF) energy which allows to describe relativistic electrons interacting with the Dirac sea, in an external electrostatic potential. The model can be seen as a mean-field approximation of Quantum Electrodynamics (QED) where photons and the so-called exchange term are neglected. A state of the system is described by its one-body density matrix, an infinite rank self-adjoint operator which is a compact perturbation of the negative spectral projector of the free Dirac operator (the Dirac sea). We study the minimization of the reduced BDF energy under a charge constraint. We prove the existence of minimizers for a large range of values of the charge, and any positive value of the coupling constant $\alpha$. Our result covers neutral and positively charged molecules, provided that the positive charge is not large enough to create electron-positron pairs. We also prove that the density of any minimizer is an $L^1$ function and compute the effective charge of the system, recovering the usual renormalization of charge: the physical coupling constant is related to $\alpha$ by the formula $\alpha_{\rm phys}\simeq \alpha(1+2\alpha/(3\pi)\log\Lambda)^{-1}$, where $\Lambda$ is the ultraviolet cut-off. We eventually prove an estimate on the highest number of electrons which can be bound by a nucleus of charge $Z$. In the nonrelativistic limit, we obtain that this number is $\leq 2Z$, recovering a result of Lieb. This work is based on a series of papers by Hainzl, Lewin, Sere and Solovej on the mean-field approximation of no-photon QED

Construction of the Pauli–Villars-Regulated Dirac Vacuum in Electromagnetic Fields

“…In this paper we have considered a second-quantized Dirac field which only interacts with a classical electromagnetic field. There is another way to arrive at exactly the same Lagrangian action (2.28), which is closer to our previous works [22,23,25,24,18,19]. We start with Coulombgauge QED with quantized transverse photons.…”

A Nonlinear Model for Relativistic Electrons at Positive Temperature

“…The end of the proof of Proposition 1 is then the same as in [9,Proposition 17]. First, we notice that as C(r) is bounded and has a compact support, b 1 and b 2 are in L ∞ (R 3 ).…”

“…The proof proceeds along the same lines as in [9,Appendix]. In the following, we shall denote by P + 0 and P − 0 the projection onto the positive and negative spectral subspace of D 0 , respectively.…”

“…This was indeed the source of complications concerning the definition of the trace (and hence of the charge) of Hartree-Fock states [10] when T = 0. This is related to the issue of renormalization [9,11,14]. Although we do not minimize in the trace-class in the case T = 0 but rather in the Hilbert-Schmidt class because the free energy is only coercive for the Hilbert-Schmidt norm, it turns out that the minimizer is trace-class nevertheless.…”

“…2.2. In particular we prove that any minimizer γ satisfies the following nonlinear equation (9) where the (self-consistent) Dirac operator reads…”