Search

We reformulate the quantization of the gravitational field and its sources, including the electric and magnetic fields as they appear in the knot algebra., 22 pages, 3 figures, 2 tables. arXiv admin note: substantial text overlap with arXiv:1809.03324, arXiv:1809.05394, arXiv:1511.07919

A short autobiography written for a centennial party., 8 pages

We describe a Lagrangian defining the preon sector of the knot model. The preons are the elements of the fundamental representation of SLq(2), and unexpectedly agree with the preons conjectured by Harari and by Shupe. The leptons, neutrinos, up and down quarks, described as $j=3/2$ representations, and the electroweak vectors, described as $j=3$ representations, of SLq(2) also have the preon composition required by the schemes of Harari and of Shupe. The coupling constants and masses required by the preon Lagrangian introduce form factors that are simple functions of the knot model parameters. As previously shown the knot model has similarly provided a possible parametrization of the masses and electroweak coupling constants (Kobayashi-Maskawa matrix) of the standard model. There is an alternative formulation of the kinematics permitting possible additional isotopic partners of the quarks and the neutrinos., Comment: 28 Pages. Paper updated by the author on May 23, 2013

We examine a quantum group extension of the standard model. The field operators of the extended theory are obtained by replacing the field operators $\psi$ of the standard model by ${\psi}$D$^j_{mm'}$, where D$^{j}_{mm'}$ are elements of a representation of the quantum algebra SLq(2), which is also the knot algebra. The D$^j_{mm'}$ lie in this algebra and carry the new degrees of freedom of the field quanta. The D$^j_{mm'}$ are restricted jointly by empirical constraints and by a postulated correspondence with classical knots. The elementary fermions are described by elements of the trefoil ($j=\frac{3}{2}$) representation and the weak vector bosons by elements of the ditrefoil ($j=3$) representation. The adjoint ($j=1$) and fundamental ($j=\frac{1}{2}$) representations define hypothetical bosonic and fermionic preons. All particles described by higher representations may be regarded as composed of the fermionic preons. This preon model unexpectedly agrees in important detail with the Harari-Shupe model. The new Lagrangian, which is invariant under gauge transformations of the SLq(2) algebra, fixes the relative masses of the elementary fermions within the same family. It also introduces form factors that modify the electroweak couplings and provide a parametrization of the Cabbibo-Kobayashi-Maskawa matrix. It is additionally postulated that the preons carry gluon charge and that the fermions, which are three preon systems, are in agreement with the color assignments of the standard model., Comment: 40 pages. Paper updated by the author on May 23, 2013

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., Comment: 12 Pages, LaTeX2e

The idea that the elementary particles might have the symmetry of knots has had a long history. In any current 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 the 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 current form in which a knotted field is complementary to a composite structure composed of three or more preons that in turn are described by the fundamental representation of SLq(2). These complementary descriptions may be interpreted as describing single composite particles composed of three or more preons bound by a knotted field., 59 pages

We discuss flavor states of the knot model and their relation to the KM and the PMNS matrices. These states are eigenstates of absorption-emission operators and are analogous to the coherent states of the Maxwell field. The underlying model has been proposed as a possible substructure of the standard model. We include a knot parameterization of the CKM matrix., This revision combines hep-th_1011.0764_v2 with submission, "Note on the Knot Parameterization of the CKM Matrix," as suggested by manager; Latex file; 23 pages

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., 22 pages, LaTex file

Previous studies have suggested complementary models of the elementary particles as (a) quantum knots and (b) preonic nuclei that are field and particle descriptions, respectively, of the same particles. This earlier work, carried out in the context of standard electroweak (SU(2) x U(1)) physics, is here extended to the strong interactions by the introduction of color (SU(3)) charges., 10 pages, 4 Tables

The knot model is extended by assuming that the trefoils are realized as either chiral fermions or as scalar bosons. There are then four scalar trefoils with electric charges (0, -1,2/3,-1/3) that may be classified in the same way as the chiral fermions: as two isotopic doublets where the two doublets have different hypercharge and the two members of the doublets have different t_3. Only the neutral scalar plays the role of the standard Higgs in fixing the mass ratios of the vector bosons, wile the charged scalars, in addition to having the usual electromagnetic interactions of scalar particles, fix the mass spectrum of the fermions. The extended model would suggest a search for the charged scalars., 16 pages, LaTex file

The conjecture that the elementary fermions are knotted flux tubes permit the construction of a phenomenology that is not accessible from the standard electroweak theory. In order to carry these ideas further we have attempted to formulate the elements of a field theory in which local SU(2) x U(1), the symmetry group of standard electroweak theory, is combined with global SU_q(2), the symmetry group of knotted solitons., Comment: 21 pages, LaTex document

We extend our earlier study of the electroweak interactions of quantum knots to their gravitational and strong interactions. The knots are defined by appropriate quantum groups and are intended to describe all knotted field structures that conserve mass and spin, charge and hypercharge, as well as color charge and color hypercharge. As sources of the gravitational fields the knots are described as representations of the quantum group $SL_q(2)$ and as sources of the electroweak and strong fields they are described by $SU_q(2)$. When the point sources of the standard theory are replaced by the quantum knots, the interaction terms of the new Lagrangian density acquire knot form factors and the standard local gauge invariance is supplemented by an additional global $U(1)\times U(1)$ invariance of the $SU_q(2)$ algebra., Comment: LaTex file; 23 pages

By utilizing the gauge invariance of the SU_q(2) algebra we sharpen the basis of the q-knot phenomenology., Comment: 12 pages, LaTex file

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. In contrast to the SU_q(2) case, where it is possible to construct a q-electroweak theory, there is no local (Yang-Mills) formalism for SU_q(3)., Comment: 18 pages, 2 figures, TeX file