Non-Relativistic Positronium Spectrum in Relativistic Schroedinger Theory

TL;DR

This paper investigates the positronium energy spectrum using Relativistic Schrödinger Theory, establishing an approximation scheme that aligns with standard quantum results at lowest order and reveals small deviations at higher orders.

Contribution

It introduces an approximation method within RST for positronium energy levels and compares its predictions to conventional quantum theory, highlighting the effects of spherical symmetry approximation.

Findings

Ground state energy matches standard theory at lowest approximation order.

Higher-order approximations show small deviations (~0.9 eV) from conventional results.

Excited states require higher-order approximations beyond those studied.

Abstract

The lowest energy levels of positronium are studied in the non-relativistic approximation within the framework of Relativistic Schr\"odinger Theory (RST). Since it is very difficult to find the exact solutions of the RST field equations (even in the non-relativistic limit), an approximation scheme is set up on the basis of the hydrogen-like wave functions (i.e. polynomial times exponential). For any approximation order $\NN (\NN=0,1,2,3,...)$ there arises a spectrum of approximate RST solutions with the associated energies, quite similarly to the conventional treatment of positronium in the standard quantum theory (Appendix). For the lowest approximation order $(\NN=0)$ the RST prediction for the \emph{groundstate} energy exactly agrees with the conventional prediction of the standard theory. However for the higher approximation orders $(\NN=1,2,3)$, the corresponding RST prediction differs from the conventional result by (roughly) $0,9 [eV]$ which confirms the previous estimate of the error being due to the use of the spherically symmetric approximation. The excited states require the application of higher-order approximations $(\NN>>3)$ and are therefore not adequately described by the present orders $(\NN\le 3)$.