Eight-lepton and eight-quark unification based on Grassmann algebra (original) (raw)

1980, Lettere Al Nuovo Cimento

Anticommuting operators and numbers afford convenient tools to describe the system with bounded discrete spectrum (1). The purpose of this note is to propose a new unification of leptons and quarks, the numbers of which are severely restricted, respectively, by the observational limits to the primordial abundance of 4He (~-) and by the requirement of asymptotic freedom in the colour gauge theory of strong interaction (a), in terms of Grassmann algebra along the approach developed by CASALBUO~ and GATTO (4). Contrary to the standard unified gauge theories such as those based on the groups SU 5 and Olo (5), where the fundamental fermions are classified in families generation by generation, eight leptons and eight quarks are unified in one grand family without disparity of generations in our scheme. We postulate that all fundamental fermion fields arc represented by a single supcrfield T(x, ~Q) which depends on the internal space ~2 made out of Grassmann elements Wa, satisfying the anticommutation relation (1) o~ae)b-4-wbw~: O, as well as on ordinary space-time coordinates x~. The superfield is assumed to have neither colour nor flavour other than B-L = baryon number-lepton number =-1 and to transform like a Dirac spinor under the Lorentz transformation. Colour and charge are attributed to the internal scalar elements COa, and lepton fields and quark fields are identified with coefficient-fields arising from the expansion of the superfield with respect to Grassmann variables. The structure of the internal space ~2 is detcrmined

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