Singularity of relativistic vortex beam and proper relativistic observables

We have studied the phase singularity of the relativistic vortex beams for the two sets of relativistic operators. One includes the new spin and orbital angular momentum (OAM) operators, which is derived from the parity-extended Poincar\'e group, and the other is composed of the (usual) Dirac spin and OAM operators. The first set predicts the same singular circulation as the nonrelativistic vortex beams. On the other hand, the second set anticipates that the singularity of the circulation is spin orientation-dependent and can be disappeared especially for relativistic paraxial electron beam with spin parallel to the propagating direction. These contradistinctive predictions suggest the relativistic electron beam experiment with spin-polarized electrons for the first time to answer the long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics since the discovery of the Dirac equation.