{
  "id": "0712.2911",
  "version": "v1",
  "published": "2007-12-18T09:53:27.000Z",
  "updated": "2007-12-18T09:53:27.000Z",
  "title": "Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms",
  "authors": [
    "Philippe Gravejat",
    "Mathieu Lewin",
    "Eric Sere"
  ],
  "comment": "37 pages, 1 figure",
  "doi": "10.1007/s00220-008-0660-9",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-18T09:53:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "charge renormalization",
      "nonlinear model",
      "relativistic atoms",
      "ground state",
      "dirac sea"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Springer",
      "journal": "Communications in Mathematical Physics",
      "year": 2009,
      "month": "Feb",
      "volume": 286,
      "number": 1,
      "pages": 179
    },
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009CMaPh.286..179G"
    }
  }
}