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The Lorentz covariance of a non-linear, time-dependent relativistic waveequation is demonstrated; the equation has recently been shown to have highlyinteresting and significant empirical consequences. It is established here thatan operator already exists which ensures the relativistic properties of theequation. Furthermore, we show that the time-dependent equation is gaugeinvariant. The equation however, breaks Poincare symmetry via time translationin a way consistent with its physical interpretation. It is also shown hereinthat the Casimir invariant PmuPmu of the Poincare group, which corresponds tothe square of the rest mass M-squared can be expressed in terms of quaternionssuch that M is described by an operator Q which has a constant norm and a phasephi which varies in hypercomplex space.

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