Abstract This study examines the thermal and statistical characteristics of the Dirac oscillatorwithin the contexts of Amelino-Camelia and Magueijo-Smolin doubly special relativity (DSR)theories. The DSR expands on Einstein’s special relativity by incorporating an additionalinvariant scale, the Planck energy, resulting in altered energy-momentum equations that re-flect possible quantum-gravitational phenomena at extremely high energies. The researchdevelops the altered Dirac equations for both DSR models and analytically calculates theassociated energy spectra. The spectra are utilized to estimate the partition function andessential thermodynamic parameters, including specific heat, employing the Euler-Maclaurinformula for efficient approximation of the partition function. The investigation concentrateson the positive-energy sector, substantiated by the precise Foldy-Wouthuysen transforma-tion, which delineates positive and negative energy levels. The results illustrate how Planck-scale deformation factors significantly modify the energy levels and thermodynamic prop-erties of the Dirac oscillator inside each DSR framework, providing insights into potentialobservable fingerprints of quantum gravity effects in relativistic quantum systems.
Boumali et al. (Thu,) studied this question.