Positronium Groundstate in Relativistic Schroedinger Theory

The usefulness of the Relativistic Schr\"odinger Theory (RST) is studied in the field of atomic physics. As a concrete demonstration, the positronium groundstate is considered in great detail; especially the groundstate energy $E_0$ is worked out in the non-relativistic approximation and under neglection of the magnetic interactions between the positron and the electron. The corresponding RST prediction $(E_0\simeq -6,48 [eV])$ misses the analogous conventional Schr\"odinger result $(E_0\simeq -6,80 [eV])$ but is closer to the latter than the corresponding Hartree approximation $(-2,65 [eV])$. The missing binding energy of $6,80-6,48=0,32 [eV]$ can be attributed to the approximative use of an SO(3) symmetric interaction potential which in RST, however, is actually only SO(2) invariant against rotations around the z-axis. It is expected that, with the correct use of an anisotropic interaction potential due to the SO(2) symmetry, the RST predictions will come even closer to the conventional Schr\"odinger result, where however the mathematical structure of RST relies on exotic (i.e. double-valued) wave functions and on the corresponding unconventional interaction potentials (e.g. Struve-Neumann potential).