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The mean‐field approximation in quantum electrodynamics: The no‐photon case
Author(s) -
Hainzl Christian,
Lewin Mathieu,
Solovej Jan Philip
Publication year - 2007
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20145
Subject(s) - fock space , hamiltonian (control theory) , quantum mechanics , physics , vacuum energy , photon , quantum field theory , quantum electrodynamics , mathematical physics , mathematics , mathematical optimization
We study the mean‐field approximation of quantum electrodynamics (QED) by means of a thermodynamic limit. The QED Hamiltonian is written in Coulomb gauge and does not contain any normal ordering or choice of bare electron/positron subspaces. Neglecting photons, we properly define this Hamiltonian in a finite box [− L /2; L /2) 3 , with periodic boundary conditions and an ultraviolet cutoff λ. We then study the limit of the ground state (i.e., the vacuum) energy and of the minimizers as L goes to infinity, in the Hartree‐Fock approximation. In the case with no external field, we prove that the energy per volume converges and obtain in the limit a translation‐invariant projector describing the free Hartree‐Fock vacuum. We also define the energy per unit volume of translation‐invariant states and prove that the free vacuum is the unique minimizer of this energy. In the presence of an external field, we prove that the difference between the minimum energy and the energy of the free vacuum converges as L goes to infinity. We obtain in the limit the so‐called Bogoliubov‐Dirac‐Fock functional. The Hartree‐Fock (polarized) vacuum is a Hilbert‐Schmidt perturbation of the free vacuum and it minimizes the Bogoliubov‐Dirac‐Fock energy. © 2006 Wiley Periodicals, Inc.