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We have studied here the geometrodynamics of relativistic electron vortexbeams from the perspective of the geometric phase associated with the scalarelectron encircling the vortex line. It is pointed out that the electron vortexbeam carrying orbital angular momentum is a natural consequence of the skyrmionmodel of a fermion. This follows from the quantization procedure of a fermionin the framework of Nelson's stochastic mechanics when a direction vector(vortex line) is introduced to depict the spin degrees of freedom. In thisformalism a fermion is depicted as a scalar particle encircling a vortex line.It is here shown that when the Berry phase acquired by the scalar electronencircling the vortex line involves quantized Dirac monopole we have paraxial(non-paraxial) beam when the vortex line is parallel (orthogonal) to thewavefront propagation direction. Non-paraxial beams incorporate spin-orbitinteraction. When the vortex line is tilted with respect to the propagationdirection the Berry phase involves non-quantized monopole. The temporalvariation of the direction of the tilted vortices is studied here taking intoaccount the renormalization group flow of the monopole charge and it ispredicted that this gives rise to spin Hall effect.

The geometric phase and the geometrodynamics of relativistic electron vortex beams