Clifford Algebras, Spinors and $Cl(8,8)$ Unification

Abstract

It is shown how the vector space $V_{8,8}$ arises from the Clifford algebra $Cl(1,3)$ of spacetime. The latter algebra describes fundamental objects such as strings and branes in terms of their $r$-volume degrees of freedom, $x^{{\mu }_1{\mu }_2...{\mu }_r}$ $\equiv x^M$, $r=0,1,2,3$, that generalizethe concept of center of mass. Taking into account that there are sixteen $x^M$, $M=1,2,3,...,16$, and in general $16\times 15/2=120$ rotations of the form $x^{\prime M}={R^M}_N{x}^N$, we can consider $x^M$ as components of a vector $X=x^M{q}_M$, where $q_M$ are generators of the Clifford algebra $Cl(8,8)$. The vector space $V_{8,8}$ has enough room for the unification of the fundamental particles and forces of the standard model. The rotations in $V_{8,8}\otimes \mathbb{C}$ contain the grand unification group $SO(10)$ as a subgroup, and also the Lorentz group $SO(1,3)$. It is shown how the Coleman-Mandula no go theorem can be avoided. Spinors in $V_{8,8}\otimes \mathbb{C}$ are constructed in terms of the wedge products of the basis vectors rewritten in the Witt basis. They satisfy the massless Dirac equation in $M_{8,8}$ with the internal part of the Dirac operator giving the non vanishing masses in four dimensions.

arXiv comment20 pages, 2 figures

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