Dirac Oscillator in DSR: A Comparative Study of Magueijo-Smolin and Amelino-Camelia Models

TL;DR

The paper investigates how Planck-scale deformations from Doubly Special Relativity modify the energy spectrum of the 1D Dirac oscillator, comparing the Magueijo–Smolin and Amelino-Camelia models. Using an $O(E^{2}/k^{2})$ approximation, it derives modified Dirac equations and analytically solves for the energy branches, highlighting model-dependent deviations. The MS realization yields non-uniform shifts at small $k$ that gradually vanish as $k$ grows, recovering the standard spectrum in the large-$k$ limit, while the AC realization shows stronger deviations at low $k$ and a singularity at $k_c(n)=\sqrt{m\omega n/2}$, indicating a breakdown of the low-energy approximation. Overall, the study demonstrates how Planck-scale physics can imprint detectable corrections on relativistic bound states and clarifies differences between DSR models, with implications for high-precision spectroscopic or astrophysical tests of quantum-gravity effects.

Abstract

This paper investigates the energy spectrum of the Dirac oscillator within the framework of Doubly Special Relativity (DSR), focusing on two prominent models: the Magueijo--Smolin (MS) and Amelino-Camelia models. We derive the modified Dirac equations in both MS and Amelino-Camelia DSR models under the approximation of $$O(E^{2}/k^{2})$$ for a single particle and examine the resulting energy spectra. The study reveals significant corrections to the standard relativistic Dirac oscillator spectrum due to the Planck-scale deformation parameter $$k$$, which introduces distinct deviations depending on the DSR model employed. For the MS model, we observe non-uniform shifts in both positive and negative energy branches at small $$k$$, with the spectrum gradually flattening toward the canonical result as $$k$$ increases. In the Amelino-Camelia model, the energy levels show larger deviations at lower values of $$k$$, and these anomalies diminish more slowly compared to the MS model. The results provide insights into the impact of quantum gravity effects on quantum systems, with potential applications in high-precision spectroscopic or astrophysical observations at energies near the Planck scale. Furthermore, the comparative analysis of these two DSR models highlights the robustness of Planck-scale predictions and guides future experimental efforts aimed at detecting quantum-gravity signatures.

Figures (2)

• Figure 1: Plot of Energy Levels for the 1D Dirac Oscillator in MS DSR for Various Values of $k$
• Figure 2: Plot of Energy Branches $E_{\pm}(n)$ for Different $k$ Values in Amelino-Camelia DSR