Exceptional gauge groups and quantum theory

Journal Article · Wed Jan 31 23:00:00 EST 1979 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.524076· OSTI ID:6416029

Horwitz, L P; Biedenharn, L C

It is shown that a Hilbert space over the real Clifford algebra C/sub 7/ 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 C/sub 7/, the group of automorphisms leaving invariant a minimal subspace acts, in the ideal generated by that subspace, as G/sub 2/, and the subgroup of this group leaving one generating element (e/sub 7/) fixed acts, in this ideal, as the color gauge group SU(3). A generalized phase algebra AcontainsC/sub 7/ 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,e/sub 7/) (isomorphic to the complex field) of C/sub 7/. Eight solutions are found for the generalized phase algebra, corresponding (up to an overall sign), in effect, to the use of +- e/sub 7/ 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,e/sub 7/) is shown to lead to a decomposition into ''lepton and ''quark'' superselection subspaces. The color SU(3) subgroup of G/sub 2/ 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.

• Research Organization: Tel Aviv University, Ramat Aviv, Israel
• OSTI ID: 6416029
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Journal Issue: 2 Vol. 20:2; ISSN JMAPA
• Country of Publication: United States
• Language: English

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71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
ALGEBRA
BANACH SPACE
CLIFFORD ALGEBRA
COLOR MODEL
COMPOSITE MODELS
HILBERT SPACE
LIE GROUPS
MATHEMATICAL MODELS
MATHEMATICAL OPERATORS
MATHEMATICAL SPACE
MATHEMATICS
MECHANICS
PARTICLE MODELS
QUANTUM MECHANICS
QUARK MODEL
SELECTION RULES
SPACE
SU GROUPS
SU-3 GROUPS
SUPERSELECTION RULES
SYMMETRY GROUPS
TENSORS
U GROUPS
U-4 GROUPS
UNIFIED GAUGE MODELS