Generalized Dirac equation and its symmetries

Existence of a minimal observable distance on the order of Planck length is an immediate consequence of the Generalized Uncertainty Principle (GUP). As a result, relativistic quantum mechanics should be generalized to incorporate such a novel feature. The resulting theory may be considered as an effective relativistic quantum mechanics for the Planck scale regime. In this paper, we find generalized Dirac equation in GUP framework and solve its eigenvalue problem for a free particle. Lagrangian of generalized Dirac field is obtained and its symmetry properties are discussed. Finally, CPT symmetries will be re-examined in the framework of generalized Dirac theory. We show that, up to first order in GUP parameter, Noether’s theorem and CPT symmetries are not broken in GUP framework.