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

TL;DR

This paper explores the deformation parameter in SLq(2) models of elementary particles, linking it to running coupling constants and their energy dependence, with implications for electric and magnetic charge conditions.

Contribution

It introduces a model where the deformation parameter q relates to different gauge couplings, considering their energy dependence and magnetic charge implications.

Findings

q equals the ratio of two coupling constants, e and g.

The deformation parameter q varies with energy if e and g are different.

Maintains the condition eg = ħc for magnetic charge consistency.

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.