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 ( E 2 / k 2 ) . 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 k c ( 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 k eff (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 k eff → ∞. The analytically exact solutions presented here offer precise directives, including parameter regimes, the significance of k c ( n ), and NR approximations, for empirical inquiries aimed at probing the form of DSR modifications, despite the inaccessibility of genuine Planck-scale effects.