Sogami's Generalized Covariant Derivative in Gauge and Non-Gauge Theories

Abstract

We describe Sogami's method of generating the bosonic sector of the standard model lagrangian from the generalized covariant derivative acting on chiral fermion fields in a simpler setting using well-known field theory models with either global or local symmetries. It is shown that Sogami's method is applicable not only to gauge theories but also to non-gauge theories. It is also shown that the present method offers a different interpretation of Sogami's high energy limit with SU(5) relations among gauge coupling constants in the standard model. Recently, SogamP) proposed an interesting method of generating the bosonic sector of the standard model lagrangian from the generalized covariant derivative operator acting on a set of chiral fermion fields (total fermion field). Its spirit is similar to Cannes' approach based on the non-commutative geometry, 2 ), 3 ), 4 ) but its mathematical nature is quite different, and much easier to understand, using more familiar tools. The purpose of this note is to show that Sogami's method is appli­ cable to both gauge and non-gauge theories in greater generality. We first explain it in terms of simpler field theory models with either global or local symmetries, and then re-study Sogami's reformulation of the standard model in a different setting described in this note. A gauge theory coupled to massive fermions of mass m is described by the lagrangian