Many-Body Stability Implies a Bound on the Fine-Structure Constant

The Dirac equation for hydrogenic atoms has a well known instability when $z\alpha >1\mathrm{}$. A similar instability occurs for the "relativistic Schrödinger equation" with $\frac{p^2}{2m}$ replaced by ${(p^2{c}^2+m^2{c}^4)}^{\frac{1}{2}}-m{c}^2$ at $z\alpha =\frac{2}{\pi }$. These instabilities concern only the product $z\alpha$, but when the many-electron-many-nucleus problem is examined (in the relativistic Schrödinger theory) we find that a bound on $\alpha$ alone (independent of $z$) is then required for stability. If $\alpha <\frac{1}{94}$ we find that stability occurs all the way up to the critical value $z\alpha =\frac{2}{\pi }$, whereas if $\alpha >\frac{128}{15\pi }$ then the system is unstable for all values of $z$. Some implications of these findings are also discussed.