On the SLq(2) extension of the standard model and the measure of charge

Abstract

Our SLq(2) extension of the standard model is constructed by replacing the elementary field operators, $\Psi (x)$, of the standard model by ${\hat{\Psi }}_{m{m}^{\prime }}^j(x)D_{m{m}^{\prime }}^j$ where $D_{m{m}^{\prime }}^j$ is an element of the $2j+1$ dimensional representation of the SLq(2) algebra, which is also the knot algebra. The allowed quantum states $(j,m,m^{\prime })$ are restricted by the topological conditions \begin{equation*} (j,m,m') = \frac{1}{2}(N,w,r+o) \end{equation*} postulated between the states of the quantum knot $(j,m,m^{\prime })$ and the corresponding classical knot $(N,w,r+o)$ where the $(N,w,r)$ are (the number of crossings, the writhe, the rotation) of the 2d projection of the corresponding oriented classical knot. Here $o$ is an odd number that is required by the difference in parity between $w$ and $r$. There is also the empirical restriction on the allowed states \begin{equation*} (j,m,m')=3(t,-t_3,-t_0)_L \end{equation*} that holds at the $j=\frac{3}{2}$ level, connecting quantum trefoils $(\frac{3}{2},m,m^{\prime })$ with leptons and quarks $(\frac{1}{2},-t_3,-t_0{)}_L$. The so constructed knotted leptons and quarks turn out to be composed of three $j=\frac{1}{2}$ particles which unexpectedly agree with the preon models of Harrari and Shupe. The $j=0$ particles, being electroweak neutral, are dark and plausibly greatly outnumber the quarks and leptons. The SLq(2) or $(j,m,m^{\prime })$ measure of charge has a direct physical interpretation since $2j$ is the total number of preonic charges while $2m$ and $2m^{\prime }$ are the numbers of writhe and rotation sources of preonic charge. The total SLq(2) charge of a particle, measured by writhe and rotation and composed of preons, sums the signs of the counterclockwise turns $(+1)$ and clockwise turns $(-1)$ that any energy-momentum current makes in going once around the knot... Keywords: Quantum group; electroweak; knot models; preon models; dark matter.

21 pages, 3 figures. arXiv admin note: text overlap with arXiv:1401.1833

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