Geometrical Applications of Split Octonions

Merab Gogberashvili and Otari Sakhelashvili

Advances in Mathematical Physics, 2015, vol. 2015, issue 1

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 G2. 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 G2 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.

Date: 2015
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https://doi.org/10.1155/2015/196708

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnlamp:v:2015:y:2015:i:1:n:196708

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