Thermal Properties of the Dirac Oscillator in Amelino-Camelia andMagueijo-Smolin DSR Theories

This study examines the thermal and statistical characteristics of the Dirac oscillator within the contexts of Amelino-Camelia and Magueijo-Smolin doubly special relativity (DSR) theories. The DSR expands on Einstein’s special relativity by incorporating an additional invariant scale, the Planck energy, resulting in altered energy-momentum equations that re- flect possible quantum-gravitational phenomena at extremely high energies. The research develops the altered Dirac equations for both DSR models and analytically calculates the associated energy spectra. The spectra are utilized to estimate the partition function and essential thermodynamic parameters, including specific heat, employing the Euler-Maclaurin formula for efficient approximation of the partition function. The investigation concentrates on 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 potential observable fingerprints of quantum gravity effects in relativistic quantum systems.