Connes' gauge theory and Weinberg-Salam model of leptons on noncommutative space-time

Connes' gauge theory is defined on noncommutative space-time. It is applied to formulate a noncommutative Weinberg-Salam (WS) model in the leptonic sector with nu (R). It is shown that the model has two Higgs doublets and a gauge boson sector after the Higgs mechanism contains the massive charged gauge fields, two massless and two massive neutral gauge fields. It is also shown that, at the tree level, the neutrino couples to one of two 'photons', the electron interacts with both 'photons', and there exists a nontrivial nu (R)-interaction on noncommutative space-time. To investigate the commutative limit of the model at the Lagrangian level, we generalize the charge conjugation transformation in QED to that in noncommutative QED. We show that there are two different generalizations, the C and C' transformations, which are based on two equivalent forms of the charge conjugation transformation in QED. It turns out that the two 'photons' are C' conjugates but become C conjugates (up to a sign) in the commutative limit. It is then proved that the bosonic action of the model has Z(2) symmetry, which reflects the C' invariance in the bosonic sector. Our noncommutative WS model is reduced to the commutative WS theory at the Lagrangian level. Thus, in the neutral gauge boson sector, there are only one massless photon and one Z(o) in the commutative limit. This limit is studied by means of a theta -expansion, which is obtained by expanding the action with respect to the noncommutativity parameter.