The stability and instability of relativistic matter

We consider the quantum mechanical many-body problem of electrons and fixed nuclei interacting via Coulomb forces, but with a relativistic form for the kinetic energy, namelyp²/2m is replaced by (p²c²+m²c⁴)^1/2–mc². The electrons are allowed to haveq spin states (q=2 in nature). For one electron and one nucleus instability occurs ifz>2/, wherez is the nuclear charge and is the fine structure constant. We prove that stability occurs in the many-body case ifz2/ and <1/(47q). For smallz, a better bound on is also given. In the other direction we show that there is a critical _c (no greater than 128/15) such that if >_c then instability always occurs forall positivez (not necessarily integral) when the number of nuclei is large enough. Several other results of a technical nature are also given such as localization estimates and bounds for the relativistic kinetic energy.Work partially supported by U.S. National Science Foundation grant PHY-85-15288-A02The author thanks the Institute for Advanced Study for its hospitality and the U.S. National Science Foundation for support under grant DMS-8601978