On the noncommutative and nonassociative geometry of octonionic space time, modified dispersion relations and grand unification

Abstract

The octonionic geometry (gravity) developed long ago by Oliveira and Marques, J. Math. Phys. 26, 3131 (1985) is extended to noncommutative and nonassociative space time coordinates associated with octonionic-valued coordinates and momenta. The octonionic metric Gμν already encompasses the ordinary space time metric gμν, in addition to the Maxwell U(1) and SU(2) Yang-Mills fields such that it implements the Kaluza-Klein Grand unification program without introducing extra space time 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 Fμν through the area-momentum Σμν in the internal isospin space yields corrections O(1∕MPlanck2) to the energy-momentum dispersion relations without violating Lorentz invariance as it occurs with Hopf algebraic deformations of the Poincare algebra. The known octonionic realizations of the Clifford Cl(8), Cl(4) algebras should permit the construction of octonionic string actions that should have a correspondence with ordinary string actions for strings moving in a curved Clifford-space target background associated with a Cl(3, 1) algebra.