q-Electroweak, q-Gravity, and Knotted Solitons

Abstract

If the Lie group of a non-Abelian theory is replaced by the corresponding q-group, one is led to replace the Lie algebra by two dual algebras. The first of these lies close to the Lie algebra that it is replacing while the second introduces new degrees of freedom. We interpret the theory based on the first algebra as a modification of standard field theory while we propose that the new degrees of freedom introduced by the second algebra describe solitonic rather than point particle sources. We have earlier found that the modified q-electroweak theory differs very little from the standard theory. Here we find a similar result for q-gravity. Both of the modified theories are incomplete, however, and must be completed by the solitonic sector. We propsoe that the solitonic sector of both q-electroweak and q-gravity have the symmetry of knots associated with SU_q(2). Since the Lorentz group is here deformed, there is no longer the standard classification of particles described by mass and spin. There is instead a classification of irreducible structures determined by SU_q(2).