NONLINEAR KLEIN-GORDON EQUATION IN CAUCHY-NAVIER ELASTIC SOLID
We show that the quaternionic field theory can be rigorously derived from the classical
balance equations in an isotropic ideal crystal where the momentum transport and the field energy are described by the Cauchy-Navier equation. The theory is presented in the form of the non-linear wave and Poisson equations with quaternion valued wave functions. The derived quaternionic form of the Cauchy-Navier equation couples the compression and torsion of the displacement. The wave equation has the form of the nonlinear Klein-Gordon equation and describes a spatially localized wave function that is equivalent to the particle. The derived wave equation avoids the problems of negative energy and probability. We show the self-consistent classical interpretation of wave phenomena and gravity.
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