Spin in a planar relativistic fermion problem

Abstract In this work we seek to clarify the role of spin in a quantum relativistic two-dimensional world. To this end, we solve the Dirac equation for two-dimensional motion with circular symmetry using both 3+1 and 2+1 Dirac equations for a fermion interacting with a uniform magnetic field perpendicular to the plane of motion. We find that, as remarked already by other authors, the spin projection s in the direction of the magnetic field can be emulated in the 2+1 Hamiltonian as a parameter preserving the anti-commutation relations between the several terms in the Hamiltonian, although it is not a quantum number in the two-dimensional world. We also find that there is an equivalence between a pure vector potential and a pure tensor potential under circular symmetry, if the former is multiplied by s. This holds for any dependence on the polar coordinate ρ. For the particular case of the uniform magnetic field, this means that this problem is equivalent to the two-dimensional Dirac oscillator.