Quaternionic factorization of the Schroedinger operator and its applications to some first order systems of mathematical physics

TL;DR

This paper demonstrates that various first order systems in mathematical physics, including Dirac and Maxwell equations, can be unified into a single quaternionic framework, simplifying their analysis to solving a Schrödinger equation with biquaternionic potential.

Contribution

It introduces a quaternionic factorization approach that reduces multiple physical systems to a single Schrödinger equation with biquaternionic potential, offering a unified analytical method.

Findings

All considered systems reduce to a single quaternionic equation.

The biquaternionic potential can often be diagonalized into scalar potentials.

This approach simplifies the analysis of complex physical systems.

Abstract

We consider the following first order systems of mathematical physics. 1.The Dirac equation with scalar potential. 2.The Dirac equation with electric potential. 3.The Dirac equation with pseudoscalar potential. 4.The system describing non-linear force free magnetic fields or Beltrami fields with nonconstant proportionality factor. 5.The Maxwell equations for slowly changing media. 6.The static Maxwell system. We show that all this variety of first order systems reduces to a single quaternionic equation the analysis of which in its turn reduces to the solution of a Schroedinger equation with biquaternionic potential. In some important situations the biquaternionic potential can be diagonalized and converted into scalar potentials.