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Advances in Mathematical Physics
Volume 2015, Article ID 196708, 14 pages
http://dx.doi.org/10.1155/2015/196708
Research Article

Geometrical Applications of Split Octonions

1Tbilisi Ivane Javakhishvili State University, 3 Chavchavadze Avenue, 0179 Tbilisi, Georgia
2Andronikashvili Institute of Physics, 6 Tamarashvili Street, 0177 Tbilisi, Georgia

Received 16 August 2015; Accepted 28 September 2015

Academic Editor: Yao-Zhong Zhang

Copyright © 2015 Merab Gogberashvili and Otari Sakhelashvili. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties from conventional (3 + 1)-theory (e.g., number of dimensions, existence of maximal velocities, Heisenberg uncertainty, and particle generations). This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonions, the noncompact form of the exceptional Lie group . This group generates specific rotations of (3 + 4)-vector parts of split octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations. In this picture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates. It is shown how the algebra’s chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin), as well as a parity violating effect on light emitted by a moving quantum system. Elementary particles are connected with the special elements of the algebra which nullify octonionic intervals. Then the zero-norm conditions lead to free particle Lagrangians, which allow virtual trajectories also and exhibit the appearance of spatial horizons governing by mass parameters.