Thermal properties of the Dirac oscillator in Amelino-Camelia and Magueijo-Smolin DSR theories

Abstract This study investigates the thermal and statistical properties of the Dirac oscillator within the framework of two prominent formulations of doubly special relativity (DSR): the Amelino-Camelia and Magueijo-Smolin models. DSR extends Einstein’s special relativity by introducing an additional invariant scale—the Planck energy—leading to modified energy-momentum relations that encode potential quantum-gravitational effects at ultra-high energies. In this context, we derive the modified Dirac equations for both DSR scenarios and analytically determine the corresponding energy spectra. These spectra are subsequently used to compute the partition function and key thermodynamic quantities, including specific heat, by employing the Euler–Maclaurin formula to facilitate an efficient approximation of the partition function. The analysis is restricted to the positive-energy sector, enabled by the exact Foldy–Wouthuysen transformation, which effectively decouples positive and negative energy states. The findings reveal that Planck-scale deformation parameters induce significant modifications in the energy spectrum and thermodynamic behavior of the Dirac oscillator in each DSR framework, thereby offering valuable insights into possible observable imprints of quantum gravitational phenomena in relativistic quantum systems.