  The analysis of twisted (vortex) paraxial photons and electrons is fulfilled
in the framework of relativistic quantum mechanics. The use of the
Foldy-Wouthuysen representation radically simplifies a description of
relativistic electrons and clarifies fundamental properties of twisted
particles. It is demonstrated that the twisted photon defined by the
Laguerre-Gauss wave function is luminal. Its apparent subluminality takes place
because the photon energy is contributed by the hidden transversal motion while
the average transversal momentum vanishes. Nevertheless, the Laguerre-Gauss
photon can be self-consistently modeled by a subluminal centroid with a nonzero
kinematic (Lorentz-invariant) mass. An applicability of this model is a
consequence of independence of centroid parameters from the longitudinal
coordinate. All twisted and untwisted Laguerre-Gauss modes are subliminal. The
Laguerre-Gauss electron has similar properties. The new effect of a
quantization of the velocity and mass of the Laguerre-Gauss photon and electron
is predicted. This effect is observable for the photon. The twisted photon and
electron modeled by the centroids are considered in the general noninertial
frame. Their inertial masses coinciding with kinematic ones are determined. The
orbital magnetic moment of the Laguerre-Gauss electron does not depend on the
radial quantum number.
