A Relativistic Quaternionic Wave Equation

TL;DR

This paper introduces a relativistically covariant quaternionic wave equation with conserved currents, gauge invariance, and initial steps towards quantum field theory, expanding the mathematical framework of relativistic quantum mechanics.

Contribution

It presents a novel quaternionic wave equation that maintains relativistic covariance and gauge invariance, with preliminary results on scattering and two-particle states.

Findings

Conserved 4-current and antisymmetric tensor derived from the wave equation

Waves propagate within the light-cone, consistent with relativity

Initial steps towards second quantization and scattering theory

Abstract

We study a one-component quaternionic wave equation which is relativistically covariant. Bi-linear forms include a conserved 4-current and an antisymmetric second rank tensor. Waves propagate within the light-cone and there is a conserved quantity which looks like helicity. The principle of superposition is retained in a slightly altered manner. External potentials can be introduced in a way that allows for gauge invariance. There are some results for scattering theory and for two-particle wavefunctions as well as the beginnings of second quantization. However, we are unable to find a suitable Lagrangian or an energy-momentum tensor.