On the other hand, the second set anticipates that the singularity of the circulation is spin-orientation-dependent and can disappear, especially for a relativistic paraxial electron beam with spin parallel to the propagating direction. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. One set includes new spin and orbital angular momentum (OAM) operators, which are derived from the parity-extended Poincaré group, and the other set consists of the (usual) Dirac spin and OAM operators. ![]() ![]() We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation.
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