Quaternionic Dirac free particle

TL;DR

This paper solves the quaternionic Dirac equation in real Hilbert space, revealing a richer set of free particle solutions for both massive and massless cases compared to the complex Dirac equation, which is crucial for developing quaternionic quantum field theory.

Contribution

It provides the first explicit solutions to the quaternionic Dirac equation, expanding the understanding of quaternionic quantum mechanics and aiding in the construction of quaternionic quantum field theory.

Findings

Massive particle solutions: 8 elements

Massless particle solutions: 4 elements

New solutions are essential for quaternionic QFT

Abstract

We solve the quaternionic Dirac equation ($\mathbbm H$DE) in the real Hilbert space, and we ascertain that their free particle solutions set comprises eight elements in the case of a massive particle, and a four elements solution set in the case of a massless particle, a richer situation when compared to the four elements solutions set of the usual complex Dirac equation ($\mathbbm C$DE). These free particle solutions were unknown in the previous solutions of anti-hermitian quaternionic quantum mechanics, and constitute an essential element in order to build a quaternionic quantum field theory ($\mathbbm H$QFT).