On Electrons and Nuclei in a Magnetic Field

1 0 Introduction In this paper, we prove a lower bound for the energy of N electrons and M nuclei in a magnetic eld. This bound, together with work of J. Frr ohlich and G.-M. Graf, proves the H-stability of non-relativistic quantum electrodynamics (QED) with an ultraviolet cutoo. H-stability for a system with many particles says that the energy per (charged) particle remains bounded below, as the number of particles tends to innnity. This is a very natural requirement for physical systems. In this introduction, we begin by stating carefully the main inequality to be proven in this paper. We then give a quick sketch of the construction of non-relativistic QED with an ultraviolet cutoo. Next, we explain the relevance of our main estimate to QED, via the work of Frr ohlich and Graf. The introduction concludes with a quick sketch of the main ideas in our proof. The results proven here are announced in FFG2] and applied to QED in FFG1]. See also Bugliaro-Frr ohlich-Graf BFG].