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The analysis of twisted (vortex) paraxial photons and electrons is fulfilledin the framework of relativistic quantum mechanics. The use of theFoldy-Wouthuysen representation radically simplifies a description ofrelativistic electrons and clarifies fundamental properties of twistedparticles. It is demonstrated that the twisted photon defined by theLaguerre-Gauss wave function is luminal. Its apparent subluminality takes placebecause the photon energy is contributed by the hidden transversal motion whilethe average transversal momentum vanishes. Nevertheless, the Laguerre-Gaussphoton can be self-consistently modeled by a subluminal centroid with a nonzerokinematic (Lorentz-invariant) mass. An applicability of this model is aconsequence of independence of centroid parameters from the longitudinalcoordinate. All twisted and untwisted Laguerre-Gauss modes are subliminal. TheLaguerre-Gauss electron has similar properties. The new effect of aquantization of the velocity and mass of the Laguerre-Gauss photon and electronis predicted. This effect is observable for the photon. The twisted photon andelectron modeled by the centroids are considered in the general noninertialframe. Their inertial masses coinciding with kinematic ones are determined. Theorbital magnetic moment of the Laguerre-Gauss electron does not depend on theradial quantum number.

Relativistic quantum-mechanical description of twisted paraxial electron and photon beams