### NONLINEAR KLEIN-GORDON EQUATION IN CAUCHY-NAVIER ELASTIC SOLID

M. Danielewski, L. Sapa

#### Abstract

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.

#### Keywords

quantum wave; quaternion algebra; Klein-Gordon equation; gravity.

#### References

Landau L. D., Lifshitz E. M. (1986). Theory of Elasticity, 3rd ed. Oxford:Butterworth-Heinemann Elsevier Ltd. ISBN 0-7506-2633-X.

Maxwell J. C. (1865). A Dynamical Theory of the Electromagnetic Field. Phil. Trans. R. Soc. Lond., 155, 459-512, pp. 488 and 493; doi: 10.1098/rstl.1865.0008.

Kleinert H. (1987). Gravity as a Theory of Defects in a Crystal with only Second Gradient of Elasticity, Ann. Phys., 44, 117.

Kleinert H., Zaanen J. (2004). Nematic world crystal model of gravity explaining absence of torsion in spacetime, Phys. Lett. A, 324 ,361-365.

Danielewski M. (2007). The Planck-Kleinert Crystal, Z. Naturforsch, 62a, 564-568.

Klein O. (1926) The Atomicity of Electricity as a Quantum Theory Law, Nature, 118, 516, doi:10.1038/118516a0.

Dirac P. A. M. (1978). Mathematical Foundations of Quantum Theory, Ed. Marlow A. R. New York: Academic.

Hamilton W. R. (1844). On Quaternions, or on a New System of Imaginaries in Algebra. Edinburgh and Dublin Phil. Magazine and J. of Sci. (3rd Series) 25, 10-13.

National Institute of Standards and Technology. (2010). http://physics.nist.gov

Lakes R. S. (1998). Elastic freedom in cellular solids and composite materials in Mathematics of Multiscale Materials. Springer, NY, Berlin, 99, 129-153.

Gürlebeck K. and Sprößig W. (1989). Quaternionic Analysis and Elliptic Boundary Value Problems. Berlin: Akademie-Verlag.

Weinberg S. (1995). The Quantum Theory of Fields. Cambridge: University Press, 1. 7-8.

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