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

**Authors:** Carlos Castro

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_{μν} already encompasses the ordinary spacetime metric g_{μν}, 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 G_{2}
group which is the automorphism group of the octonion algebra. It is shown
that the flux of the SU(2) Yang-Mills field strength F_{μν} through the area momentum
Σ^{μν} 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. 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.

**Comments:** 23 pages, This article appeared in the J. Math. Phys, 48, no. 7 (2007) 073517.

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### Submission history

[v1] 24 Aug 2009

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