WEINBERG-SALAM THEORY IN NONCOMMUTATIVE GEOMETRY

Ordinary differential calculus on smooth manifold is generalized so as to construct gauge theory coupled to fermions on discrete space M4xZ2 which is an underlying space-time in the non-commutative geometry for the standard model. We can reproduce not only the bosonic sector but also the fermionic sector of the Weinberg-Salam theory without recourse to the Dirac operator at the outset. Treatment of the fermionic sector is based on the generalized spinor one-forms from which the Dirac lagrangian is derived through taking the inner product. Two model constructions are presented using our formalism, both giving the classical mass relation m(H) = square-root 2m(W). The first model leaves the Weinberg angle arbitrary as usual, while the second one predicts sin2thetaW = 1/4 in the tree level. This prediction is the same as that of Connes but we obtain it from correct hypercharge assignment of 2x2 matrix-valued Higgs field and from vanishing photon mass, thereby dispensing with Connes' 0-trace condition or the equivalent.