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We study the replacement of SU(3) by SU q(3) in standard gauge theories. At the level of a global theory there is a physically sensible SU q(3) formalism with measurable differences from the SU (3) theory.

When the symmetry of a physical theory describing a finite system is deformed by replacing its Lie group by the corresponding quantum group, the operators and state function will lie in a new algebra describing new degrees of freedom. If the symmetry of a field theory is deformed in this way, the enlarged state space will again describe additional degrees of freedom, and the energy levels will acquire fine structure. The massive particles will have a stringlike spectrum lifting the degeneracy of the point-particle theory, and the resulting theory will have a non-local description. Theories of this kind naturally contain two sectors with one sector lying close to the standard theory while the second sector describes particles that should be more difficult to observe., Comment: 7 pages, TeX file

Our SLq(2) extension of the standard model is constructed by replacing the elementary field operators, $\Psi (x)$, of the standard model by $\hat{\Psi}^{j}_{mm'}(x) D^{j}_{mm'}$ where $D^{j}_{mm'}$ is an element of the $2j + 1$ dimensional representation of the SLq(2) algebra, which is also the knot algebra. The allowed quantum states $(j,m,m')$ 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 $(j,m,m')$ 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 $(\frac{3}{2},m,m')$ 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 $(j,m,m')$ measure of charge has a direct physical interpretation since $2j$ is the total number of preonic charges while $2m$ and $2m'$ 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., Comment: 21 pages, 3 figures. arXiv admin note: text overlap with arXiv:1401.1833

An affine field theory may be constructed by gauging an affine algebra. The momentum integrals of the affine N = 4 Yang–Mills theory are ultraviolet finite but diverge because the sum over states is infinite. If the affine symmetry is broken by postulating a nonvanishing vacuum expectation value for that component of the scalar field lying in the L0 direction, then the theory acquires a linear mass spectrum. This broken theory is ultraviolet finite too, but the mass spectrum is unbounded. If it is also postulated that the mass spectrum has an upper bound (say, the Planck mass), then the resulting theory appears to be altogether finite. The influence of the exotic states has been estimated and, according to the proposed scenario, is negligible below energies at which gravitational interactions become important. The final effective theory has the symmetry of a compact Lie algebra augmented by the operator L0.

The idea that the elementary particles might have the symmetry of knots has had a long history. In any modern formulation of this idea, however, the knot must be quantized. The present review is a summary of a small set of papers that began as an attempt to correlate the properties of quantized knots with empirical properties of the elementary particles. As the ideas behind these papers have developed over a number of years, the model has evolved, and this review is intended to present the model in its current form. The original picture of an elementary fermion as a solitonic knot of field, described by the trefoil representation of SUq(2), has expanded into its present form in which a knotted field is complementary to a composite structure composed of three preons that in turn are described by the fundamental representation of SLq(2). Higher representations of SLq(2) are interpreted as describing composite particles composed of three or more preons bound by a knotted field. This preon model unexpectedly agrees in important detail with the Harari–Shupe model. There is an associated Lagrangian dynamics capable in principle of describing the interactions and masses of the particles generated by the model.

There are suggestive experimental indications that the leptons, neutrinos, and quarks are composite and that their structure is described by the quantum group SLq(2). Since the hypothetical preons must be very heavy relative to the masses of the leptons, neutrinos, and quarks, there must be a very strong binding field to permit these composite particles to form. Unfortunately there are no experiments direct enough to establish the order of magnitude needed to make the SLq(2) Lagrangian dynamics quantitative. It is possible, however, to parametrize the preon masses and interactions that would be necessary to stabilize the three particle composite representing the leptons, neutrinos, and quarks. In this note we examine possible parametrizations of the masses and the interactions of these hypothetical structures. We also note an alternative view of SLq(2) preons., arXiv admin note: text overlap with arXiv:1301.6440

It is shown that there is no obstacle to the affine extension of supersymmetric Yang-Mills theories. This result holds equally for loop and Kac-Moody algebras. It also holds for 4, 6, and 10 dimensions. By dimensional reduction of the 10-dimensional theory one obtains a supersymmetric action describing interacting Kac-Moody fields in four dimensions.

The affine extension of supergravity is investigated. We discuss only the loop algebra since the affine extension of the super-Poincaré algebra appears inconsistent. The construction of the affine supergravity theory has been carried out by the group manifold method and leads to an action describing infinite towers of gravitons and gravitinos that interact subject to the symmetries of the loop algebra. The equations of motion satisfy the usual consistency check.