On the Deformation Parameter in SLq(2) Models of the Elementary Particles

Abstract

When the fundamental invariant of $SLq(2)$ is expressed as $\epsilon_q = (\matrix{0 & \alpha_2 \cr -\alpha_1 & 0})$, then the deformation parameter, $q$, defining the knot algebra is $q = \frac{\alpha_1}{\alpha_2}$. We consider models in which the elementary particles carry more than one kind of charge with running coupling constants, $\alpha_1$ and $\alpha_2$, having different energy dependence and belonging to different gauge groups. Let these coupling constants be normalized to agree with experiment at hadronic energies and written as $\alpha_1 = \frac{e}{\sqrt{\hbar c}}$ and $\alpha_2 = \frac{g}{\sqrt{\hbar c}}$. Then $q = \frac{e}{g}$. If $e$ is an electroweak coupling and $g$ is a gluon coupling, $q$ will increase with energy. In previous discussions of $SLq(2)$ it has been assumed that $\epsilon_q^{2} = -1$. If this condition is maintained, then $eg = \hbar c$. If the elementary particle is like a Schwinger dyon and therefore the source of magnetic as well as electric charge, $eg = \hbar c$ is the Dirac condition for magnetic charge.