A.P.Yefremov, "Physical Theories in Hypercomplex Geometric Description",
Int. Journal of Geometric Methods in Modern Physics,
Vol.11 (2014) 1450062 (33 pages);
DOI: 10.1142/S0219887814500625
Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras
are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrodinger-Pauli and Hamilton-Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ?) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy-Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang-Mills field theory.
Keywords: Hypercomplex numbers; quaternions; fundamental surface; spinors; quantum and classical mechanics; theory of relativity; electrodynamics; Yang-Mills field