On the Schr\"odinger spectrum of a hydrogen atom with electrostatic Bopp-Land\'e-Thomas-Podolsky interaction between electron and proton

TL;DR

This paper investigates the energy spectrum of a hydrogen atom under a modified electrostatic interaction from BLTP theory, providing bounds, numerical results, and implications for fundamental physics and measurements.

Contribution

It offers rigorous bounds and numerical analysis of hydrogen energy levels with BLTP interaction, revealing potential observable effects and constraints on the Bopp length scale.

Findings

BLTP predicts a non-relativistic correction to Lyman-alpha splitting.

Experimental data constrains the Bopp length to less than approximately 10^{-18} meters.

Proton size effects do not alter the main conclusions.

Abstract

The Schr\"odinger spectrum of a hydrogen atom, modelled as a two-body system consisting of a point electron and a point proton, interacting with a modification of Coulomb's law proposed, in the 1940s, by Bopp, Land\'e--Thomas, and Podolsky (BLTP). The BLTP theory hypothesizes the existence of an electromagnetic length scale of nature --- the Bopp length $\varkappa ^{-1}$ ---, to the effect that the electrostatic pair interaction deviates significantly from Coulomb's law only for distances much shorter than $\varkappa^{-1}$. Rigorous lower and upper bounds are constructed for the Schr\"odinger energy levels of the hydrogen atom, $E_{\ell,n}(\varkappa)$, for all $\ell\in\{0,1,2,...\}$ and $n >\ell$. The energy levels $E_{0,1}(\varkappa)$, $E_{0,2}(\varkappa)$, and $E_{1,2}(\varkappa)$ are also computed numerically and plotted versus $\varkappa^{-1}$. It is found that the BLTP theory predicts a non-relativistic correction to the splitting of the Lyman-$\alpha$ line in addition to its well-known relativistic fine-structure splitting. Under the assumption, that this splitting doesn't go away in a relativistic calculation, it is argued that present-day precision measurements of the Lyman-$\alpha$ line suggest that $\varkappa^{-1}$ must be smaller than $\approx 10^{-18}\,\mathrm{m}$. Finite proton size effects are found not to modify this conclusion. As a consequence, the electrostatic field energy of an elementary point charge, although finite in BLTP electrodynamics, is much larger than the empirical rest energy of an electron. If, as assumed in all `renormalized theories' of the electron, the empirical rest mass of a physical electron is the sum of its bare rest mass plus its electrostatic field energy ($/c^2$), then in BLTP electrodynamics the electron has to be assigned a negative bare rest mass.