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Physical theories in hypercomplex geometric description
Ефремов А.П. (Yefremov A.P.) Physical theories in hypercomplex geometric description // International Journal of Geometric Methods in Modern Physics. 2014. 11(06): 1450062. doi: 10.1142/S0219887814500625

Категории: Исследование, Авторский указатель

Physical theories in hypercomplex geometric description
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Аннотация

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 (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) 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

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