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

Abstract

Our SLq(2) extension of the standard model is constructed by replacing the elementary field operators, [Formula: see text], of the standard model by [Formula: see text] where [Formula: see text] is an element of the [Formula: see text]-dimensional representation of the SLq(2) algebra, which is also the knot algebra. The allowed quantum states [Formula: see text] are restricted by the topological conditions [Formula: see text] postulated between the states of the quantum knot [Formula: see text] and the corresponding classical knot [Formula: see text] where the [Formula: see text] are (the number of crossings, the writhe, the rotation) of the 2d projection of the corresponding oriented classical knot. Here, [Formula: see text] is an odd number that is required by the difference in parity between [Formula: see text] and [Formula: see text]. There is also the empirical restriction on the allowed states [Formula: see text] that holds at the [Formula: see text] level, connecting quantum trefoils [Formula: see text] with leptons and quarks [Formula: see text]. The so-constructed knotted leptons and quarks turn out to be composed of three [Formula: see text] particles which unexpectedly agree with the preon models of Harrari and Shupe. The [Formula: see text] particles, being electroweak neutral, are dark and plausibly greatly outnumber the quarks and leptons. The SLq(2) or [Formula: see text] measure of charge has a direct physical interpretation since [Formula: see text] is the total number of preonic charges while [Formula: see text] and [Formula: see text] 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 [Formula: see text] and clockwise turns [Formula: see text] that any energy–momentum current makes in going once around the knot. In this way, the handedness of the knot reduces charge to a geometric concept similar to the way that curvature of space–time encodes mass and energy. According to this model, the leptons and quarks are [Formula: see text] particles, the preons are [Formula: see text] particles, and the [Formula: see text] particles are candidates for dark matter. It is possible to understand [Formula: see text] as a simple deformation parameter or as the ratio [Formula: see text] of the electroweak to the gluon coupling constants.