Exceptional gauge groups and quantum theory

Journal Article · Thu Feb 01 00:00:00 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), Vol. 20:2, Issue 2
• Country of Publication: United States
• Language: English

Related Subjects

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