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

## TL;DR

This paper develops a relativistic quantum mechanical framework for twisted photon and electron beams, revealing new effects like velocity and mass quantization, and clarifying their luminal or subliminal nature through centroid modeling.

## Contribution

It introduces a relativistic quantum description of twisted beams using the Foldy-Wouthuysen representation, and predicts observable quantization effects in their velocity and mass.

## Key findings

- Laguerre-Gauss photons are luminal but modeled as subluminal via centroid mass.
- Laguerre-Gauss electrons share similar properties with photons.
- Velocity and mass quantization effects are predicted and potentially observable.

## Abstract

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.