# Twisted Localized Solutions of the Dirac Equation: Hopfion-like States of Relativistic Electrons

## TL;DR

This paper introduces exact, localized solutions to the Dirac equation with angular momentum, exhibiting topological properties akin to hopfions, thus advancing the understanding of electron states with realistic localization.

## Contribution

The paper presents the first analytic localized solutions of the Dirac equation with angular momentum, featuring topological properties similar to hopfions.

## Key findings

- Solutions are localized along the propagation direction.
- Solutions possess intricate topological structures.
- Eigenstates of total angular momentum are constructed.

## Abstract

All known solutions of the Dirac equation describing states of electrons endowed with angular momentum are very far from our notion of the electron as a spinning charged bullet because they are not localized in the direction of propagation. We present here analytic exact solutions, eigenstates of the total angular momentum component $M_z$, that come very close to this notion. These new solutions of the Dirac equation have also intricate topological properties similar to the hopfion solutions of the Maxwell equations.