Three-dimensional modified dirac oscillator in standard and generalized doubly special relativity

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- $$\tfrac{1}{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> 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 $$l_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> . 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 $$(N,j,\ell )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>N</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>,</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and the relativistic energy, producing branch-dependent shifts for both particle and antiparticle solutions. The undeformed limit ( $$k\rightarrow \infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>→</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> or $$l_p\rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>l</mml:mi> <mml:mi>p</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> ) is recovered smoothly and the deformation signal increases with excitation through the oscillator scale and spin–orbit splitting.