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M. Danielewski
L. Sapa


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.

Article Details

Materials Physics
Author Biographies

M. Danielewski, Faculty of Materials Science and Ceramics, AGH University of Science and Technology


L. Sapa, Faculty of Applied Mathematics, AGH University of Science and Technology

Assistant professor


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).

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.