A Possible SLq(2) Substructure of the Standard Model

TL;DR

This paper explores a quantum group extension of the standard model incorporating $SLq(2)$ symmetry, proposing a knot-based substructure for elementary particles, and suggests a correspondence between quantum knots and fermion families.

Contribution

It introduces a novel $SLq(2)$ quantum group framework linking knot theory to particle substructure, extending the standard model with quantum knots and preons.

Findings

Empirical correspondence between quantum trefoils and fermion families.

Elementary fermions are associated with the $j=3/2$ representation of $SLq(2)$.

Preons are identified as elements of the fundamental $j=1/2$ representation.

Abstract

We examine a quantum group extension of the standard model with the symmetry $SU(3) \times SU(2) \times U(1)\times $ global $SLq(2)$. The quantum fields of this extended model lie in the state space of the $SLq(2)$ algebra. The normal modes or field quanta carry the factors $D^j_{mm^\prime} (q|abcd)$, which are irreducible representations of $SLq(2)$ (which is also the knot algebra). We describe these field quanta as quantum knots and set $(j,m,m^\prime)= 1/2 (N,w, \pm r+1)$ where the $(N,w,r)$ are restricted to be (the number of crossings, the writhe, the rotation) respectively, of a classical knot. There is an empirical one-to-one correspondence between the four quantum trefoils and the four families of elementary fermions, a correspondence that may be expressed as $(j,m,m^\prime)=3(t,-t_3, -t_0)$, where the four quantum trefoils are labelled by $(j,m,m^\prime)$ and where the four families are labelled in the standard model by the isotopic and hypercharge indices $(t,t_3,-t_0)$. We propose extending this correlation to all representations by attaching $D_{-3t-3t_0}^{3t} (q| abcd) $ to the field operator of every particle labelled by $(t,t_3, t_0)$ in the standard model. Then the elementary fermions $(t=1/2)$ belong to the $j=3/2$ representation of $SLq(2)$. The elements of the fundamental representation $j=1/2$ will be called preons and $D_{-3t,-3t_o}^{3t}$ may be interpreted as describing the creation operator of a composite particle composed of elementary preons. $D_{m m^\prime}^j$ also may be interpreted to describe a quantum knot when expressed as $D_{\frac w2 \frac{\pm r+1}2} ^{N/2}$ These complementary descriptions may be understood as describing a composite particle of $N$ preons bound by a knotted boson field with $N$ crossings.