The Dirac equation as the origin of symmetry breaking

Peter Rowlands
Department of Physics
University of Liverpool
Oliver Lodge Laboratory
Oxford Street
Liverpool, L69 7ZE, UK
e-mail prowlands@liverpool.ac.uk

A multivariate 4-vector representation for space-time combined with a quaternion representation for mass and the electric, strong and weak charges (Vigier I) compactifies into a nilpotent form of the Dirac equation (Vigier II), incorporating the entire physical information available about a fermion state. The act of compactification serves to break the symmetry between the strong, electric and weak interactions, by linking their respective charges with vector, scalar and pseudoscalar operators in the nilpotent state vector. The SU(3) ´ SU(2)L ´ U(1) symmetry which results is obtained by direct solution of the nilpotent Dirac equation for all possible spherically-symmetric distance-dependent potentials (which are interpreted here as those which conserve type of charge – Vigier I). It is found that there are just three such solutions, and that these correspond in character to those expected for strong, weak and electric interactions. All the solutions require a scalar phase term which is associated here with the coupling constant. The weak solution, however, requires a complex potential, which is equivalent to the complex coupling constant required to generate CP violation. The scalar phases may additionally be combined in a strong-electroweak solution, and would be the only parts of the interactions remaining at Grand Unification as previously envisaged (Vigier III). That unification also requires a gravitational term (here interpreted as a spin 1 inertial repulsion), which may be incorporated into the combined solution as an additional component of the scalar phase.