C. Castro: Octonionic Spacetime and Grand Unification


On the Noncommutative and Nonassociative Geometry of Octonionic Spacetime, Modified Dispersion Relations and Grand Unification

by Carlos Castro
Center for Theoretical Studies of Physical Systems
Clark Atlanta University, Atlanta, GA. 30314, castro@ctsps.cau.edu
30 April, 2007

Submitted to: JMP

Abstract

The Octonionic Geometry (Gravity) developed long ago by Oliveira and Marques is extended to Noncommutative and Nonassociative Spacetime coordinates associated with octonionic-valued coordinates and momenta. The octonionic metric G_{mu nu} already encompasses the ordinary spacetime metric g_{mu nu}, in addition to the Maxwell U(1) and SU(2) Yang-Mills fields such that implements the Kaluza-Klein Grand Unification program without introducing extra spacetime dimensions. The color group SU(3) is a subgroup of the exceptional G2 group which is the automorphism group of the octonion algebra. It is shown that the flux of the SU(2) Yang-Mills field strength \vec{F}_{mu nu} through the areamomentum \vec{\Sigma}_{mu nu} in the internal isospin space yields corrections O(1/M^2_Planck) to the energy-momentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. Known Octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic strings where the 1+1-octonionic-dim worldsheet of an octonionic string has a correspondence with an 8+8 real-dimensional spacetime of split signature corresponding to the Clifford space associated with a Cl(3, 1) algebra.

Keywords: Nonassociative Geometry, Clifford algebras, Quaternions, Octonionic Gravity, Unification, Strings.

PDF Online Version: here

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