Equivalence between the Dirac oscillator and a spin-1/2 fermion embedded in a transverse homogeneous magnetic field: movement in a (2 + 1)-dimensional world

The planar Dirac oscillator embedded in a transverse uniform magnetic field has emerged in applications to optical models [1][3] and graphene [4][10]. In a recent work, it has been shown in a simple way that the planar Dirac oscillator and a spin-1/2 fermion embedded a transverse homogeneous magnetic field are equivalent problems [11]. This achievement, put into effect with 4×4 realizations of the Dirac matrices in a (3+1)-dimensional world, fills with suspicions of wrongdoing many works diffused in the literature. In the present work, even with a content easy to deal with by graduate students in Physics, the same achievement is reached by using 2 × 2 realizations of the Dirac matrices in a (2+1)-dimensional world. In the Minkowski spacetime, the behaviour of a spin1/2 fermion of mass m subject to the most generic set of interactions is governed by the Dirac equation (in natural units ~ = c = 1)