New solutions of the Dirac, Maxwell and Weyl equations from the fractional Fourier transform

TL;DR

This paper introduces a unified method to derive solutions to Dirac, Maxwell, and Weyl equations using fractional Fourier transforms and Gaussian generating functions, leading to explicit wave functions including hopfion solutions.

Contribution

It presents a novel unified approach employing fractional Fourier transforms and Gaussian generating functions to solve relativistic wave equations.

Findings

Derived new wave solutions for Dirac, Maxwell, and Weyl equations.

Obtained explicit Maxwell and Dirac hopfion solutions.

Unified framework simplifies solution construction for relativistic equations.

Abstract

New solutions of relativistic wave equations are obtained in a unified manner from generating functions of spinorial variables. The choice of generating functions as Gaussians leads to representations in the form of generalized fractional Fourier transforms. Wave functions satisfying the Dirac, Maxwell, and Weyl equations are constructed by simple differentiations with respect to spinorial arguments. In the simplest case, one obtains Maxwell and Dirac hopfion solutions.