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The Deformed Dirac Oscillator in Linear-Fractional Doubly Special Relativity

N. Jafari, A. Boumali

Abstract

We study the $(1+1)$-dimensional Dirac oscillator within a class of doubly special relativity (DSR) models generated by linear-fractional (projective) transformations on momentum space that preserve both the invariant speed of light and a high-energy observer-independent scale $\Ep$. Starting from the associated deformed Casimir invariants, we construct the coordinate-space Dirac equations for three inequivalent choices of the deformation vector (time-like, space-like, and light-like). For the time-like and light-like realizations the deformation induces momentum-dependent effective mass operators, which makes the coordinate-space formulation sensitive to operator ordering. To retain locality and obtain solvable second-order equations we adopt a reverted-product ordering prescription. Closed-form relativistic energy spectra and eigenfunctions are obtained in all three geometries, and the standard Dirac-oscillator results are recovered smoothly in the limit $\Ep\to\infty$. Finally, we derive the nonrelativistic expansion of the positive-energy branch and show that the deformation geometry controls the leading rest-energy shift and the renormalization of the oscillator level spacing.

The Deformed Dirac Oscillator in Linear-Fractional Doubly Special Relativity

Abstract

We study the -dimensional Dirac oscillator within a class of doubly special relativity (DSR) models generated by linear-fractional (projective) transformations on momentum space that preserve both the invariant speed of light and a high-energy observer-independent scale . Starting from the associated deformed Casimir invariants, we construct the coordinate-space Dirac equations for three inequivalent choices of the deformation vector (time-like, space-like, and light-like). For the time-like and light-like realizations the deformation induces momentum-dependent effective mass operators, which makes the coordinate-space formulation sensitive to operator ordering. To retain locality and obtain solvable second-order equations we adopt a reverted-product ordering prescription. Closed-form relativistic energy spectra and eigenfunctions are obtained in all three geometries, and the standard Dirac-oscillator results are recovered smoothly in the limit . Finally, we derive the nonrelativistic expansion of the positive-energy branch and show that the deformation geometry controls the leading rest-energy shift and the renormalization of the oscillator level spacing.
Paper Structure (15 sections, 48 equations)

This paper contains 15 sections, 48 equations.

Table of Contents

  1. Introduction
  2. Objective and outline.
  3. Linear-fractional DSR Dirac oscillator
  4. Projective momentum map and modified mass shell
  5. Geometric deformations and linearized Dirac equations
  6. Dirac-oscillator coupling and ordering
  7. Exact solutions and nonrelativistic limits (time-like, space-like, light-like)
  8. Eigenfunctions.
  9. Nonrelativistic limit (positive-energy branch).
  10. Eigenfunctions.
  11. Nonrelativistic limit (positive-energy branch).
  12. Eigenfunctions.
  13. Nonrelativistic limit (positive-energy branch).
  14. Geometric comparison (within this solvable model).
  15. Conclusion