Sogami's Generalized Covariant Derivative and Non-Commutative Differential Geometry

Abstract

It is shown that there exists a close connection between Sogami's method in the standard model and our previous formulation of non-commutative differential geometry. This connection emerges from an identification of an extra one-form basis in the latter with an ordinary one-form valued in Dirac algebra. Application of the method to the standard model with (or without) singlet va's gives a new intepretation of SU(5) relations among gauge coupling constants in the model without SU(5) symmetry. In a recent paper, 1 > Sogami proposed a new method of deriving the bosonic sector of the standard model lagrangian from the fermionic one in an ingenious way. This is based on the observation that the generalized covariant derivative, which acts on a set of chiral fermion fields (total fermion field) interacting with both gauge and Higgs fields, determines generalized field strengths by taking a commutator, which then fixes the bosonic lagrangian. This procedure is applied 2 > to both gauge and non-gauge theories by performing the usual chiral decomposition of the Dirac spinor so as to make the generalized covariant derivative a 2 X 2 matrix in chiral space. This matrix structure resembles that of non-commutative geometry (NCG) 3 > as for­