Spin in a planar relativistic fermion problem

• Published in: Physics letters. A 2021-07, Vol.404, p.127412, Article 127412
• Main Authors: de Castro, A.S., Alberto, P.
• Format: Article
• Language: English
• Subjects: 2+1 Dirac equation; Circular symmetry; Spin
• ISSN: 0375-9601; 1873-2429
• DOI: 10.1016/j.physleta.2021.127412

•The problem of how to describe spin of a relativistic fermion moving in a plane under circular symmetry is addressed.•The formalism for this planar motion is fully developed in both 3+1 and 2+1 dimensional worlds under general Lorentz potentials.•A general equivalence between a pure vector potential and a pure tensor potential if the former is multiplied by a spin parameter s is derived.•The formalism in 2+1 and 3+1 dimensions is applied to a relativistic fermion subject to a magnetic field perpendicular to the plane of motion.•A correspondence between the solutions in both cases is discussed, highlighting the different physical interpretations of the s parameter. 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.