The Symmetry Properties of a Non-Linear Relativistic Wave Equation: Lorentz Covariance, Gauge Invariance and Poincare Transformation

TL;DR

This paper demonstrates the Lorentz covariance and gauge invariance of a non-linear relativistic wave equation, explores its Poincare symmetry breaking, and introduces a quaternion-based operator for the mass invariant.

Contribution

It establishes the relativistic properties of a non-linear wave equation and introduces a quaternionic operator for the Poincare invariant mass.

Findings

The wave equation is Lorentz covariant.

The equation is gauge invariant.

Poincare symmetry is broken via time translation.

Abstract

The Lorentz covariance of a non-linear, time-dependent relativistic wave equation is demonstrated; the equation has recently been shown to have highly interesting and significant empirical consequences. It is established here that an operator already exists which ensures the relativistic properties of the equation. Furthermore, we show that the time-dependent equation is gauge invariant. The equation however, breaks Poincare symmetry via time translation in a way consistent with its physical interpretation. It is also shown herein that the Casimir invariant PmuPmu of the Poincare group, which corresponds to the square of the rest mass M-squared can be expressed in terms of quaternions such that M is described by an operator Q which has a constant norm and a phase phi which varies in hypercomplex space.