# On the Non-Relativistic Groundstate Energy of Positronium in Relativistic Schroedinger Theory

## TL;DR

This paper investigates the non-relativistic groundstate energy of positronium within the relativistic Schrödinger theory (RST), using a variational approach to compare RST predictions with conventional results and assess its practical validity.

## Contribution

It introduces a variational method to approximate positronium energies in RST and compares these with conventional theory, revealing notable deviations.

## Key findings

- RST groundstate energy is about 10% lower than conventional predictions.
- Approximate RST spectrum aligns with conventional results up to quantum number n≈100.
- Uncertainty remains whether deviations are due to RST or approximation methods.

## Abstract

The \emph{Relativistic Schr\"odinger Theory} (RST) has been set up as an alternative form of particle theory. This theory obeys the fundamental symmetries which are required to hold for any meaningful theory: gauge and Lorentz covariance (RST can be formulated even over a pseudo-Riemannian space-time). But the question is now whether obeying those fundamental symmetries is sufficient for the practical success of a theory, i.e. whether the predictions are in agreement with the experimental findings. In this context, the non-relativistic energy spectrum of positronium has been considered in some precedent papers. Here, the problem is that exact solutions of the RST eigenvalue system cannot be obtained and one has to resort to approximate solutions. For this purpose, a variational method is applied in the present paper which yields the RST groundstate energy smaller than the former results and than its conventional counterpart by some 10\%. Such deviations are also observed when one compares the approximate RST spectrum (up to quantum numbers $n\approx 100$) to the corresponding predictions of the conventional theory. It seems presently not possible to decide whether those deviations are due to RST itself or are merely due to the applied approximation technique. Thus the practical usefulness of RST must remain unclarified for the moment.