From quantum electrodynamics to mean-field theory. I. The Bogoliubov-Dirac-Fock formalism

Abstract

A relativistic mean-field theory for interacting Dirac particles in an external field is derived from quantum-field theory using a minimisation principle, and discussed in the context of atomic physics. In this approach, electrons and positrons are treated on the same footing, and neither final 'reinterpretation' nor 'positive energy projection' are needed. We obtain mean-field equations of Dirac-Fock type containing a vacuum polarisation term that does not exist in the standard Dirac-Fock equations. However, the standard Dirac-Fock equations, as well as the equations resulting from earlier attempts to build a mean-field theory from quantum electrodynamics, are recovered as non-variational approximations. The minimisation principle also leads to a new way of introducing finite-basis relativistic calculations.

Existence of Global-In-Time Solutions to a Generalized Dirac-Fock Type Evolution Equation

“…As we shall see, another conserved quantity for Q(t) is the Bogoliubov-Dirac-Fock energy [3,1,21,22]. This energy is defined, for any Q = P − P 0 ∈ H Λ (P being a orthogonal projector), by…”

“…The interpretation of (1.1), in terms of the BDF model in no-photon QED, is roughly speaking as follows: in [3], Chaix and Iracane consider the free relativistic Fock space in which they define a class of states furnished by Bogoliubov transformations of the free vacuum. Each of these states can then be equivalently represented by its density matrix which is an orthogonal projector P , such that Q = P − P 0 is Hilbert-Schmidt.…”

“…Each of these states can then be equivalently represented by its density matrix which is an orthogonal projector P , such that Q = P − P 0 is Hilbert-Schmidt. In [3], the energy of such a BDF state Ω P is computed and proved to be only depending on Q = P − P 0 . Details can also be found in the appendix of [21].…”

“…In comparison to other relativistic theories, which usually do not take into account the behavior of the vacuum, the BDF model of [3] admits the extremely important property that its associated energy is a bounded-below functional, cf. [1,4,21].…”

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

“…For A ≡ 0, the electrostatic stability of the free Dirac vacuum was pointed out first by Chaix, Iracane and Lions [4,5] and proved later in full generality in [1,22,23]. It is possible to include the exchange term and even establish the global stability of the free Dirac vacuum [22,23,24].…”