Exceptional gauge groups and quantum theory

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Abstract
It is shown that a Hilbert space over the real Clifford algebra C7 provides a mathematical framework, consistent with the structure of the usual quantum mechanical formalism, for models for the unification of weak, electromagnetic and strong interactions utilizing the exceptional Lie groups. In particular, in case no further structure is assumed beyond that of C7, the group of automorphisms leaving invariant a minimal subspace acts, in the ideal generated by that subspace, as G2, and the subgroup of this group leaving one generating element (e7) fixed acts, in this ideal, as the color gauge group SU(3). A generalized phase algebra A⊇C7 is defined by the requirement that quantum mechanical states can be consistently constructed for a theory in which the smallest linear manifolds are closed over the subalgebra C(1,e7) (isomorphic to the complex field) of C7. Eight solutions are found for the generalized phase algebra, corresponding (up to an overall sign), in effect, to the use of ±e7 as imaginary unit in each of four superselection sectors. Operators linear over these alternative forms of imanary unit provide distinct types of ’’lepton–quark’’ and ’’quark–quark’’ transitions. The subgroup in A which leaves expectation values of operators linear over A invariant is its unitary subgroup U(4), and is a realization (explicitly constructed) of the U(4) invariance of the complex scalar product. An embedding of the algebraic Hilbert space into the complex space defined over C(1,e7) is shown to lead to a decomposition into ’’lepton and ’’quark’’ superselection subspaces. The color SU(3) subgroup of G2 coincides with the SU(3) subgroup of the generalized phase U(4) which leaves the ’’lepton’’ space invariant. The problem of constructing tensor products is studied, and some remarks are made on observability and the role of nonassociativity.

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