arXiv:2603.15632v1 Announce Type: new
Abstract: % Doubly Special Relativity (DSR) introduces, besides the invariant speed of light $c$, an observer-independent high-energy % scale that deforms relativistic kinematics and can be implemented through modified dispersion relations or effective % wave equations with energy-dependent spatial operators. In this work we develop a three-dimensional, exactly solvable % benchmark for such deformations in the spin-$tfrac12$ sector: the Dirac oscillator. Following the original % construction of Moshinsky and Szczepaniak, the oscillator is introduced through a linear non-minimal momentum coupling, % which preserves Hermiticity and yields, after decoupling the Dirac equation into large and small components, a % three-dimensional isotropic harmonic-oscillator operator supplemented by a strong spin–orbit term. % We then incorporate Planck-scale deformations in two standard DSR realizations (Amelino–Camelia and % Magueijo–Smolin, characterized by an invariant energy scale $k$) and in a generalized DSR framework based on a % first-order expansion in the Planck length $l_p$. In all cases the bound-state eigenfunctions retain the % oscillator-spinor structure dictated by spherical symmetry, while DSR deforms the algebraic relation between quantum % numbers $(N,j,ell)$ and the relativistic energy, producing branch-dependent shifts for both particle and antiparticle % solutions. The undeformed limit ($ktoinfty$ or $l_pto0$) is recovered smoothly and the deformation signal increases % with excitation through the oscillator scale and spin–orbit splitting.