  We have studied here the geometrodynamics of relativistic electron vortex
beams from the perspective of the geometric phase associated with the scalar
electron encircling the vortex line. It is pointed out that the electron vortex
beam carrying orbital angular momentum is a natural consequence of the skyrmion
model of a fermion. This follows from the quantization procedure of a fermion
in the framework of Nelson's stochastic mechanics when a direction vector
(vortex line) is introduced to depict the spin degrees of freedom. In this
formalism 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 electron
encircling the vortex line involves quantized Dirac monopole we have paraxial
(non-paraxial) beam when the vortex line is parallel (orthogonal) to the
wavefront propagation direction. Non-paraxial beams incorporate spin-orbit
interaction. When the vortex line is tilted with respect to the propagation
direction the Berry phase involves non-quantized monopole. The temporal
variation of the direction of the tilted vortices is studied here taking into
account the renormalization group flow of the monopole charge and it is
predicted that this gives rise to spin Hall effect.
