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This workbook contains many outdoor activities which were developed in New York's Central Park to help children explore and understand their city parks. Involvement in the activities is intended to increase appreciation and awareness of the role of parks in the urban environment. The publication can serve as an example of what others can do with similar facilities. (SB)

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

We consider the possibility that the preons defined by the SLq(2) extension of the Standard Model may be identified with Schwinger dyons. The SLq(2) extension is here presented as a model that may exist in either a currently observable electric phase or in a magnetic phase that is predicted but currently unobservable., Comment: arXiv admin note: text overlap with arXiv:1511.07919 and substantial text overlap with arXiv:1809.03324

We examine a generic field theory in which the field particle has two couplings. It is of particular interest when these are the electroweak, e, and the hypothetical magnetoweak, g. The new field operators are obtained by replacing the field operators $\Psi (x)$ of the standard model or of similar models by $\tilde{\Psi} (x) D^j_q (m,m')$ where $ D^j_q (m,m')$ is an element of the $2j+1$ dimensional representation of the SLq(2) algebra, which is also the knot algebra. The field is assumed to exist in two phases distinguished by two values of $q$: $q_e = \frac{e}{g}$ and $q_g = \frac{g}{e}$ which label the electroweak and magnetoweak phases respectively. We assume that the observed leptons and quarks are composed of e-preons and are in agreement with the observed charge spectrum of leptons and quarks. It is now proposed that there is also a g-phase where g-leptons and g-quarks are composed of g-preons. It is assumed that the g-charge is very large compared to the e-charge and the mass of the g charged particle is even larger since the mass of all of these particles is partially determined by the eigenvalues of $\bar{D}^j_q (m,m') D_q^j (m,m')$, a polynomial in $q$, that multiplies the Higgs mass term and where \begin{equation*} \frac{q_g}{q_e} = \left ( \frac{\hbar c}{e^2} \right)^2 \approx (137)^2. \end{equation*} These values of $q$ indicate that particles in the g-phase are much more massive, they should be harder to produce or to observe., Comment: arXiv admin note: substantial text overlap with arXiv:1511.07919

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

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., Comment: arXiv admin note: text overlap with arXiv:1301.6440

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., Comment: 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., Comment: 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., Comment: 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., Comment: 10 pages, 4 Tables

It is shown that the four trefoil solitons that are described by the irreducible representations D^{3/2}_{mm'} of the quantum algebra SL_q(2) (and that may be identified with the four families of elementary fermions (e,\mu,\tau;\nu_e\nu_\mu\nu_\tau;d,s,b;u,c,t) may be built out of three preons, chosen from two charged preons with charges (1/3,-1/3) and two neutral preons. These preons are Lorentz spinors and are described by the D^{1/2}_{mm'} representation of SL_q(2). There are also four bosonic preons described by the D^1_{mm'} and D^0_{00} representations of SL_q(2). The knotted standard theory may be replicated at the preon level and the conjectured particles are in principle indirectly observable., Comment: LaTex document; 12 pages; 4 tables

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

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., Comment: 16 pages, LaTex file

We explore a knot model of the elementary particles that is compatible with electroweak physics. The knots are quantized and their kinematic states are labelled by $D^j_{mm'}$, irreducible representations of $SU_q(2)$, where j = N/2, m = w/2, m' = (r+1)/2 and (N,w,r) designate respectively the number of crossings, the writhe, and the rotation of the knot. The knot quantum numbers (N,w,r) are related to the standard isotopic spin quantum numbers $(t,t_3,t_0)$ by $(t=N/6,t_3=-w/6,t_0=-(r+1)/6)$, where $t_0$ is the hypercharge. In this model the elementary fermions are low lying states of the quantum trefoil (N=3) and the gauge bosons are ditrefoils (N=6). The fermionic knots interact by the emission and absorption of bosonic knots. In this framework we have explored a slightly modified standard electroweak Lagrangian with a slightly modified gauge group which agrees closely but not entirely with standard electroweak theory., Comment: 29 pages; LaTex file

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

The q-electroweak theory suggests a description of elementary particles as solitons labelled by the irreducible representations of SU_q(2). Since knots may also be labelled by the irreducible representations of SU_q(2), we study a model of elementary particles based on a one-to-one correspondence between the four families of Fermions (leptons, neutrinos, (-1/3) quarks, (2/3) quarks) and the four simplest knots (trefoils). In this model the three particles of each family are identified with the ground and first two excited states of their common trefoil. Guided by the standard electroweak theory we calculate conditions restricting the masses of the fermions and the interactions between them. In its present form the model predicts a fourth generation of fermions as well as a neutrino spectrum. The same model with q almost equal to 1 is compatible with the Kobayashi-Maskawa matrix. Depending on the test of these predictions, the model may be refined., Comment: 40 pages, 2 figures, latex format

We attempt to go beyond the standard electroweak theory by replacing SU(2) with its q-deformation: SU_q(2). This step introduces new degrees of freedom that we interpret as indicative of non-locality and as a possible basis for a solitonic model of the elementary particles. The solitons are conjectured to be knotted flux tubes labelled by the irreducible representations of SU_q(2), an alglebra which is not only closely related to the standard theory but also plays an underlying role in the description of knots. Each of the four families of elementary fermions is conjectured to be represented by one of the four possible trefoils. The three individual fermions belonging to any family are then assumed to occupy the three lowest states in the excitation spectrum of the local trefoil for that family. One finds a not unreasonable variation of q among the lepton and quark families. The model in its present form predicts a fourth generation of fermions as well as a neutrino mass spectrum. The model may be refined depending on whether or not the fourth generation is found., Comment: 10 pages, Latex file; changed content

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