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

Abstract

Abstract. We consider a generalized Dirac-Fock type evolution equation deduced from no-photon Quantum Electrodynamics, which describes the selfconsistent time-evolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by Chaix-Iracane (J. Phys. B., 22, 3791-3814, 1989), and recently established by Hainzl-Lewin-Séré, we prove the existence of global-in-time solutions of the considered evolution equation.

The Hartree Equation for Infinitely Many Particles I. Well-Posedness Theory

“…To our knowledge, the results contained in this article are the first of this kind for the Hartree model with infinitely many particles. A similar problem has been considered before at zero temperature for Dirac particles in [28] and for crystals in [11], but there the mean-field operator H (t) has a gap in its spectrum, which dramatically simplifies the study.…”

“…To this end, we need two estimates which are stated in the next lemmas. (27) and (28). Then we have [ρ Q * w, γ f ] ∈ S p,s for every Q ∈ S p,s and…”

“…Finally, the unique solution Q(t) depends continuously on Q 0 ∈ S p,s , and on w and f : if Q (n) 0 → Q 0 strongly in S p,s , w n → w and f n → f for the norms corresponding to (26), (27) and (28), then the unique solution Q (n) (t) has maximal times of existence T ± n such that lim inf n→∞ T ± n T ± and Q (n) …”

“…Let q p, s r > d(q − 1)/q, and Q 0 ∈ S p,s ⊂ S q,r . We assume that w, f satisfy (26), (27) and (28). Then, the unique solution Q to (20) has a priori distinct maximal times of existence T ± p,s and T ± q,r , corresponding to the two spaces S p,s and S q,r .…”

The Hartree equation for infinite quantum systems

“…In particular, since γ ref is not trace-class, the operator γ 0 is not trace-class either and we are naturally lead to study (1.1) for non-trace-class initial data. The dynamics of infinite quantum systems in interaction has already been studied by Hainzl, Lewin, and Sparber [16] in a relativistic setting, and by Cancès and Stoltz [7] for crystals. Both these works show the global well-posedness of their respective equations in the energy space, and in particular leave open the question of the large time behaviour of the solutions.…”