A Knot Model Sugugested by the Standard Electroweak Theory

Abstract

We attempt to go beyond the standard electroweak theory by replacing SU(2) with its q-deformation: SU_q(2). This step introduces new degrees of freedom that we interpret as indicative of non-locality and as a possible basis for a solitonic model of the elementary particles. The solitons are conjectured to be knotted flux tubes labelled by the irreducible representations of SU_q(2), an alglebra which is not only closely related to the standard theory but also plays an underlying role in the description of knots. Each of the four families of elementary fermions is conjectured to be represented by one of the four possible trefoils. The three individual fermions belonging to any family are then assumed to occupy the three lowest states in the excitation spectrum of the local trefoil for that family. One finds a not unreasonable variation of q among the lepton and quark families. The model in its present form predicts a fourth generation of fermions as well as a neutrino mass spectrum. The model may be refined depending on whether or not the fourth generation is found.