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, namely p²/2m is replaced by (p² c ² + m² c⁴)^1/2 - mc². The electrons are allowed to have q spin states (q = 2 in nature). For one electron and one nucleus instability occurs if zα > 2/π, where z is the nuclear charge and α is the fine structure constant. We prove that stability occurs in the many-body case if zα 2/π and α < 1/(47q). For small z, a better bound on α is also given. In the other direction we show that there is a critical α_c (no greater than 128/15π) such that if α > α_c then instability always occurs for all positive z (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.