Many-body stability implies a bound on the fine-structure constant

The Dirac equation for hydrogenic atoms has a well known instability when z±>1. A similar instability occurs for the "relativistic Schrödinger equation" with p22m replaced by (p2c2+m2c4)12-mc2 at z±=2. These instabilities concern only the product z±, but when the many-electron-many-nucleus problem is examined (in the relativistic Schrödinger theory) we find that a bound on ± alone (independent of z) is then required for stability. If ±<194 we find that stability occurs all the way up to the critical value z±=2, whereas if ±>12815 then the system is unstable for all values of z. Some implications of these findings are also discussed.