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Three-Dimensional Modified Dirac Oscillator in Standard and Generalized Doubly Special Relativity

Abdelmalek Boumali, Nosratollah Jafari

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 ($k\to\infty$ or $l_p\to0$) is recovered smoothly and the deformation signal increases % with excitation through the oscillator scale and spin--orbit splitting.

Three-Dimensional Modified Dirac Oscillator in Standard and Generalized Doubly Special Relativity

Abstract

% Doubly Special Relativity (DSR) introduces, besides the invariant speed of light , 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- 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 ) and in a generalized DSR framework based on a % first-order expansion in the Planck length . 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 and the relativistic energy, producing branch-dependent shifts for both particle and antiparticle % solutions. The undeformed limit ( or ) is recovered smoothly and the deformation signal increases % with excitation through the oscillator scale and spin--orbit splitting.
Paper Structure (23 sections, 43 equations, 2 figures)

This paper contains 23 sections, 43 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Three-dimensional Dirac oscillator (undeformed problem)
  3. Dirac equation and oscillator coupling
  4. Stationary reduction and large/small components
  5. Decoupling and effective oscillator operator
  6. Spherical symmetry, quantum numbers, and spectrum
  7. Proof of the undeformed spectral conditions
  8. DSR-deformed Dirac oscillator
  9. Standard DSR realizations
  10. Amelino--Camelia type
  11. Magueijo--Smolin type
  12. Large-$k$ expansion and nonrelativistic limit
  13. AC vs MS: deformation pattern and spin--orbit structure.
  14. Degeneracy of the energy spectrum: comparison with the Moshinsky solution.
  15. Generalized DSR: first-order Planck-length expansion
  16. ...and 8 more sections

Figures (2)

  • Figure 1: Representative Dirac-oscillator energy spectrum $E_{N j}$ as a function of the principal oscillator number $N$ for fixed $j$ (or as a function of $j$ at fixed $N$), showing particle and antiparticle branches. The undeformed families \ref{['eq:E0_familyA']}--\ref{['eq:E0_familyB']} are compared with standard DSR realizations \ref{['eq:AC_spectrum_DO']}, \ref{['eq:MS_spectrum_DO']}, and with generalized $l_p$ expansions \ref{['eq:E_firstorder_DO']}--\ref{['eq:genMS_spectrum_DO']}.
  • Figure 2: Relative spectral shift $\delta_{N j}\equiv(E_{N j}-E_{N j}^{(0)})/E_{N j}^{(0)}$ for particle and antiparticle branches, illustrating the deformation signal and its dependence on $(N,j)$ through $\Lambda_{N j}^{(\pm)}$.