Dirac oscillator in DSR: A comparative study of Magueijo–Smolin and Amelino-Camelia models

This investigation examines the one-dimensional Dirac oscillator in the framework of Doubly Special Relativity (DSR), with a comparative analysis of the Magueijo–Smolin (MS) and Amelino-Camelia (AC) formulations up to order O(E2/k2). Exact closed-form energy spectra and non-relativistic (NR) expansions are derived for both particle and antiparticle sectors, revealing distinctive features inherent to each model: a gradual flattening of the energy-level dispersion in the MS realisation and an n-dependent critical deformation scale kc(n) in the AC realisation that constrains the accessible excitation states. Crucially, when k is set to the Planck energy, authentic DSR modifications remain undetectable in laboratory spectroscopic measurements: for energies spanning eV to GeV, relative deviations are bounded by ≲10−38 to 10−18. In light of this, two viable experimental strategies are proposed to optimise detectability: (i) analogue simulation employing a tunable effective deformation parameter keff (e.g., via trapped ions or photonic/microwave lattices) to replicate the anticipated functional dependence of level-spacing perturbations; and (ii) Dirac materials (such as graphene and moiré heterostructures), wherein engineered departures from linear dispersion mimic DSR-inspired kinematics. Across these platforms, the undeformed regime is seamlessly restored as keff → ∞. The analytically exact solutions presented here offer precise directives, including parameter regimes, the significance of kc(n), and NR approximations, for empirical inquiries aimed at probing the form of DSR modifications, despite the inaccessibility of genuine Planck-scale effects.