Weinberg-Salam Theory in Non-Commutative Geometry
Abstract
Ordinary differential calculus on smooth manifold is generalized so as to construct gauge theory coupled to fermions on discrete space M4 × Z2 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 mH = sqrt{2}mW. The first model leaves the Weinberg angle arbitrary as usual, while the second one predicts sin2θW = 1/4 in the tree level. This prediction is the same as that of Connes but we obtain it from correct hypercharge assignment of 2 × 2 matrix-valued Higgs field and from vanishing photon mass, thereby dispensing with Connes' 0-trace condition or the equivalent.
- Publication:
-
Progress of Theoretical Physics
- Pub Date:
- May 1994
- DOI:
- 10.1143/ptp/91.5.959
- Bibcode:
- 1994PThPh..91..959M