The mean‐field approximation in quantum electrodynamics: The no‐photon case

Abstract

We study the mean-field approximation of Quantum Electrodynamics, 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 define properly this Hamiltonian in a finite box [−L/2; L/2) 3 , with periodic boundary conditions and an ultraviolet cut-off 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 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.

Mathematical Modeling of Point Defects in Materials Science

“…Note that the renormalized model for defects in crystals we will obtain in the rHF case is formally very similar to the model introduced in [36,37,38] in the context of the no-photon approximation of quantum electrodynamics (QED) to describe atoms embedded in the QED vacuum.…”

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

“…, where the density ρ A and the current j A are defined as 25) and with Q A refering to the kernel of the operator…”

“…(62)- (64)]). Solutions have been rigorously constructed in the previous works [22,23,25], with a sharp ultraviolet cut-off, but in the purely electrostatic case A ext = A * = 0. In this special case it is possible to obtain the polarized vacuum as a global minimizer.…”

“…In this special case it is possible to obtain the polarized vacuum as a global minimizer. The method of [22,23,25] does not seem to be applicable with magnetic fields, however. To our knowledge, Theorem 2.4 is the first result dealing with optimized magnetic fields in interaction with Dirac's vacuum.…”

The Thermodynamic Limit of Quantum Coulomb Systems: A New Approach

“…It is interesting to note that for these other models, we do not need the cone property and we can weaken the assumptions on the regularity of the boundary by replacing t on the r.h.s. of (14) by any t p , 0 < p ≤ 1. Details may be found in our article [16].…”