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.

在这项工作中，我们试图阐明自旋在量子相对论二维世界中的作用。为此，我们针对具有对称于垂直于运动平面的均匀磁场相互作用的费米子的3 + 1和2 + 1 Dirac方程，求解了具有圆对称的二维运动的Dirac方程。我们发现，正如其他作者已经提到的那样，可以在2 + 1哈密顿量中模拟磁场方向上的自旋投影s，作为保留哈密顿量中多个项之间反换向关系的参数，尽管它可以不是二维世界中的一个量子数。我们还发现，如果前者乘以s，则在圆形对称下，纯矢量势和纯张量势之间存在等价关系。这对于极坐标ρ具有任何依赖性。对于均匀磁场的特殊情况，这意味着该问题等效于二维Dirac振荡器。