From quantum electrodynamics to mean-field theory. II. Variational stability of the vacuum of quantum electrodynamics in the mean-field approximation

Abstract

We use a minimisation principle to analyse the variational stability of the translationally invariant vacuum of quantum electrodynamics, with a Coulomb two-body interaction. We show how the magnitude of the coupling constant cy determines the existence of a stable variational ground state. This ground state does exist within the considered variational space, provided that cy is smaller than a critical value cyc. The ground state collapses if cy is larger than a,. Bounds on the critical value U , are given, and the physical value cy = 1/137 is shown to be undercritical.

Existence of Atoms and Molecules in the Mean-Field Approximation of No-Photon Quantum Electrodynamics

“…Notice that compared to [10,11,9,4,27,28], we have not only (15) of the BDF energy, following the results of [29].…”

“…The Chaix-Iracane model was first mathematically studied in the free case by Chaix, Iracane and Lions in [11] and then by Bach, Barbaroux, Helffer and Siedentop in [4]. The external field case was rigorously defined and studied by the authors of the present paper in [27,28].…”

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

“…[1,4,21]. This property in the following enables us to prove the existence of globalin-time solutions to equation (1.1).…”

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].…”

“…∀p ≥ 2, f (x)g(−i∇) Sp 5) where S p is the usual Schatten class [39]. Applying (2.5) with p = 6 together with the Sobolev inequality, we obtain…”