Abstract
We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. 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. The first set predicts the same singularity in the circulation as in the case of nonrelativistic vortex beams. 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. These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation.
Highlights
We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation
Nonrelativistic electron vortex beams carrying orbital angular momentum (OAM) have recently been studied and are well-understood using the paraxial approximation of the Schrödinger equation[1,2,3,4,5,6]
The wavefunction of a nonrelativistic electron vortex includes a phase singularity factor, eilφ, where φ is the azimuthal angle around the axis of the vortex, and the electron vortex beam can carry orbital angular momentum of l in which l is an integer known as the topological charge[1]
Summary
We have studied the phase singularity of relativistic vortex beams for two sets of relativistic operators using circulation. 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 These contradistinctive predictions suggest that a relativistic electron beam experiment with spin-polarized electrons could for the first time answer a long-standing fundamental question, i.e., what are the proper relativistic observables, raised from the beginning of relativistic quantum mechanics following the discovery of the Dirac equation. Bialynicki-Birula et al proved the assertion that any acceptable solutions of the Dirac equation cannot be eigenstates of the (usual) Dirac OAM, and showed that the vortex lines continuously smeared out into all space for their exponential solutions, which become the standard vortex wavefunction in the nonrelativistic limit[12] This raised the question of whether a relativistic vortex can be generated from high-energy electron beams. The new spin operator was shown to be the generator of the SU(2) little group of the Poincaré group and admits a ( ) SN
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