No 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.

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.