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We consider a matrix space based on the spin degree of freedom, describing both a Hilbert state space and its corresponding symmetry operators. Under the requirement that the Lorentz symmetry be kept, at given dimension, scalar symmetries and their representations are determined. Symmetries are flavor or gauge-like, with fixed chirality. After spin 0, 1/2 and 1 fields are obtained in this space, we construct associated interactive gauge-invariant renormalizable terms, showing their equivalence to a Lagrangian formulation, using as example the previously studied (5+1)-dimensional case, with many standard-model connections. At 7 + 1 dimensions, a pair of Higgs-like scalar Lagrangian is obtained naturally producing mass hierarchy within a fermion flavor doublet.

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Standard-model fields and their associated electroweak Lagrangian are equivalently expressed in a shared spin basis. The scalar-vector terms are written with scalar-operator components acting on quark-doublet elements, and shown to be parametrization-invariant. Such terms, and the t-and b-quark Yukawa terms are linked by the identification of the common mass-generating Higgs operating upon the other fields, after acquiring a vacuum expectation value v.Thus, the customary vector masses are related to the fermions', fixing the tquark mass m t with the relation m 2 t + m 2 b = v 2 /2 either for maximal hierarchy, or given the b-quark mass m b , implying m t 173.9 GeV, for v = 246 GeV. A sum rule is derived for all quark masses that generalizes this restriction. An interpretation follows that electroweak bosons and heavy quarks belong in a multiplet.

A spin-space extension is reviewed, which provides information on the standard model. Its defining feature is a common matrix space that describes symmetries and representations, and leads to limits on these, for given dimension. The model provides additional information on the standard model, whose interpretation requires an interactive formulation. Within this program, we compare the model's lepton-W generated interactive Lagrangian in (5+1)-dimensions, and that of the standard model. We derive the conditions for this matching, which apply to other Lagrangian terms. We also discuss the advantages of this extension, as compared to others.Comment: Talk presented at the 10th Latin American Symposium on High Energy Physics (SILAFAE 2014) 24-28 Nov 2014. Medellin, Colombia. 15 pages, 1 figure, 1 table, 15 reference