Space-Time Structure of Weak and Electromagnetic Interactions

It has often been suggested that observed symmetries among elementary particles may well reflect symmetries of some fundamental dynamical equation or system of equations. The suggestion has gained credence with the recent successes of gauge theories, the WeinbergSalam (W-S) model in particular. However, in current theories no definite connection has yet been established between the so-called internal symmetries of particle interactions and space-time symmetries of Lorentz invariant fields. This paper aims to show that the basis for a connection between internal symmetries and spacetime symmetries exists already in the Dirac equation for an electron. The key step in the argument is the recognition that the generator of electromagnetic gauge transformations in the Dirac equation has a definite geometric interpretation. We show, in fact, that it is the generator of local spatial rotations which leave the Dirac current and spin invariant. All this is hidden in the conventional representation of the Dirac equation, but we employ a representation that makes it explicit. Having established that the generator of electromagnetic gauge transformations has a geometric interpretation, we certainly expect the same for generators of weak gauge transformations if the unification of weak and electromagnetic interactions in a gauge theory is on the right track. Sure enough, we find that the group SU(2) × U(1) of the W-S model exists already as a symmetry group of spinor fields in space-time. We find that the W-S model can be accommodated by generalization of the conventional spinors with a definite geometrical and physical meaning. The electron and its neutrino can then be interpreted as orthogonal eigenstates of a single lepton in the same way that orthogonal spin states are interpreted as distinct states of a single particle. This unification of electron and neutrino by representing them as components of a single spinor field is geometrically analogous to the unification of electric and magnetic fields by representing them as components of a single-rank two-tensor. By adhering to the geometrical interpretation of the generators, we are led to a representation of the W-S model with new features. This provides some theoretical justification for the W-S model by establishing an intrinsic connection to the Dirac theory. This in-