Stark Energy Shifts due to Quantum Gravity in RGUP Algebra

TL;DR

This work develops a relativistic generalization of the uncertainty principle (RGUP) in Minkowski spacetime and applies it to the Stark effect in hydrogen. Using the Stetsko–Tkachuk approximation, the authors derive a RGUP-modified Hamiltonian that includes momentum- and field-related corrections, and they obtain energy shifts for non-degenerate ($n=1$) and degenerate ($n=2$) states through perturbation theory. The non-degenerate ground-state shift is a linear-in-$β$ term, while the degenerate manifold exhibits β-dependent modifications to Stark splitting, with an upper bound on $β$ estimated as $β < 10^{42}$. The results recover the standard Stark effect in the limits $β\to0$ and $c\to\infty$, providing a relativistic minimal-length framework for quantum-gravity phenomenology and a pathway to constrain RGUP via precision spectroscopy.

Abstract

In this paper, we investigate the Stark effect in the hydrogen atom under an external electric field, incorporating relativistic generalized uncertainty principle (RGUP) corrections within Minkowskian spacetime and calculate the upper bound on $β$ the RGUP parameter. Employing RGUP algebra and the Stetsko-Tkachuk approximation, we derive modifications to the energy spectrum for degenerate and non-degenerate states. The perturbed Hamiltonian, modified by RGUP, enfold quantum gravitational effects. Our results reveal quantum gravitational corrections to the Stark energy spectrum in the relativistic regime, with energy shifts for non-degenerate ($n=1$) and degenerate ($n \neq 1$) cases showing additional terms proportional to $β$. These findings reduce to standard Stark effect results and non-relativistic GUP frameworks in the limits $β\rightarrow 0$ and $c \rightarrow \infty $, establishing our model as a generalized framework for analyzing minimal length effects in relativistic quantum systems.