Existence of a stable polarized vacuum in the Bogoliubov-Dirac-Fock approximation

Abstract

According to Dirac's ideas, the vacuum consists of infinitely many virtual electrons which completely fill up the negative part of the spectrum of the free Dirac operator $D^0$. In the presence of an external field, these virtual particles react and the vacuum becomes polarized. In this paper, following Chaix and Iracane ({\it J. Phys. B}, 22, 3791--3814, 1989), we consider the Bogoliubov-Dirac-Fock model, which is derived from no-photon QED. The corresponding BDF-energy takes the polarization of the vacuum into account and is bounded from below. A BDF-stable vacuum is defined to be a minimizer of this energy. If it exists, such a minimizer is solution of a self-consistent equation. We show the existence of a unique minimizer of the BDF-energy in the presence of an external electrostatic field, by means of a fixed-point approach. This minimizer is interpreted as the polarized vacuum.

arXiv commentfinal version, to appear in Commun. Math. Phys

References in corpus (7)

• Generalized Hartree-Fock Theory and the Hubbard Model
• Renormalization of the Regularized Relativistic Electron-Positron Field
• Non-Perturbative Mass and Charge Renormalization in Relativistic No-Photon Quantum Electrodynamics
• Electron-positron pair production in the external electromagnetic field of colliding relativistic heavy ions
• On the Hartree-Fock equations of the electron/positron field
• On the Vacuum Polarization Density Caused by an External Field
• Some connections between Dirac-Fock and Electron-Positron Hartree-Fock

Cited by in corpus (32)

• A new approach to the modelling of local defects in crystals: the reduced Hartree-Fock case
• The Hartree equation for infinitely many particles. I. Well-posedness theory
• The Hartree equation for infinitely many particles. II. Dispersion and scattering in 2D
• A Minimization Method for Relativistic Electrons in a Mean-Field Approximation of Quantum Electrodynamics
• Minimizers for the Hartree-Fock-Bogoliubov Theory of Neutron Stars and White Dwarfs
• Non-perturbative embedding of local defects in crystalline materials
• Ground State and Charge Renormalization in a Nonlinear Model of Relativistic Atoms
• Derivation of Pekar's Polarons from a Microscopic Model of Quantum Crystals
• The Dielectric Permittivity of Crystals in the reduced Hartree-Fock approximation
• A positive density analogue of the Lieb-Thirring inequality
• A Nonlinear Model for Relativistic Electrons at Positive Temperature
• A mathematical formulation of the random phase approximation for crystals
• Time Evolution of the External Field Problem in QED
• Construction of the Pauli-Villars-regulated Dirac vacuum in electromagnetic fields
• Renormalization and asymptotic expansion of Dirac's polarized vacuum
• A Formulation of Quantum Field Theory Realizing a Sea of Interacting Dirac Particles
• Low Density Limit of BCS Theory and Bose-Einstein Condensation of Fermion Pairs
• Ground state properties of graphene in Hartree-Fock theory
• The reduced Hartree-Fock model for short-range quantum crystals with defects
• The nonlinear Schrödinger equation for orthonormal functions: I. Existence of ground states
• Anyonic Topological Order in Twisted Equivariant Differential (TED) K-Theory
• The Scott correction in Dirac-Fock theory
• The Causal Perturbation Expansion Revisited: Rescaling the Interacting Dirac Sea
• A new definition of the Dirac-Fock ground state
• Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics
• Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in a Weak Homogeneous Magnetic Field
• Microscopic Derivation of Ginzburg-Landau Theory and the BCS Critical Temperature Shift in General External Fields
• Existence of Ground State of an Electron in the BDF Approximation
• Effective quantum electrodynamics: One-dimensional model of the relativistic hydrogen-like atom
• Relativistic electrons coupled with Newtonian nuclear dynamics
• Semi-classical Dirac vacuum polarisation in a scalar field
• Mean--field stability for the junction of quasi 1D systems with Coulomb interactions