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

Abstract

When the fundamental invariant of $SLq(2)$ is expressed as $ε_q = (\matrix{0 & α_2 \cr -α_1 & 0})$, 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 ${\alpha }_1=\frac{e}{\sqrt{\hslash c}}$ and ${\alpha }_2=\frac{g}{\sqrt{\hslash c}}$. 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 $eg=\hslash c$. If the elementary particle is like a Schwinger dyon and therefore the source of magnetic as well as electric charge, $eg=\hslash c$ is the Dirac condition for magnetic charge.

12 Pages, LaTeX2e