QUATERNIONS AND NONCOMMUTATIVE GEOMETRY

It is shown that the Dirac operator becomes 2 x 2 matrix acting on the space of chiral fermions in the previously-proposed quaternionic formulation of the Dirac theory. The interaction part of thus-obtained Dirac operator formally corresponds to the 2 x 2 matrix of the generalized one-form of Coquereaux et al. in the non-commutative geometric approach to the electro-weak unification. It is pointed out that the mathematical formulation of Coquereaux et al. allows to introduce an arbitrary parameter which can be reduced to unity unless it vanishes and whose presence spoils a unique construction of Yang-Mills-Higgs theory in their approach. For instance, one may choose the parameter to vanish, resulting in only pure Yang-Mills theory. It is also pointed out that the original model construction of Coquereaux et al. corresponds to employ a quaternion algebra which is isomorphic to but totally independent of that used in the Dirac theory. According to this model construction the Higgs boson mass is predicted to be the same as that of the gauge boson in a toy model with U(l) x U(l) down to U(l). Bosonic sector of the Weinberg-Salam theory is also constructed along the same line using quaternions with components being 2 x 2 matrices. The construction is unique without recourse to SU(2\1) and predicts m(H) = square-root 2m(w) and sin2theta(w) = 1/4 at the classical level.