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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 Ψ (x) of the Standard Model or of similar models by Ψ ̃ (x) D q j (m , m ′) where D q j (m , m ′) is an element of the (2 j + 1) -dimensional representation of the SLq(2) algebra, which is also the knot algebra. The new field is assumed to exist in two phases distinguished by two values of q : q e = e q and q g = g q which label the electroweak and magnetoweak phases, respectively. We assume that the observed leptons and quarks are mainly 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 mainly 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 D ̄ q j (m , m ′) D q j (m , m ′) , a polynomial in q , that multiplies the Higgs mass term and where q g q e = ℏ c e 2 2 ≈ (1 3 7) 2 . Since these values of q g / q e indicate that particles in the g -phase are much more massive, they should be harder to produce or to observe. Since the remote parts of the universe are at increasingly higher temperatures, magnetic poles are perhaps most likely to be found in deep probes of space as well as in high energy accelerators. The section entitled "Introduction" was added only after it was generally realized that the birth of the present universe was probably due to a nuclear explosion. [ABSTRACT FROM AUTHOR]

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

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. [ABSTRACT FROM AUTHOR]

There are suggestive experimental indications that the leptons, neutrinos, and quarks might be 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. [ABSTRACT FROM AUTHOR]

It is shown that the four quantum trefoil solitons that are described by the irreducible representations ${\mathcal D}^{3/2}_{mm^\prime}$ of the quantum algebra SLq(2) [and that may be identified with the four families of elementary fermions (e, μ, τ; νeνμντ;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 fermions and are described by the ${\mathcal D}^{1/2}_{mm^\prime}$ representation of SLq(2). There are also four bosonic preons described by the ${\mathcal D}^1_{mm^\prime}$ and ${\mathcal D}^0_{00}$ representations of SLq(2). The knotted standard theory may be replicated at the preon level and the conjectured particles are in principle indirectly observable. [ABSTRACT FROM AUTHOR]

We explore a knot model of the elementary particles that is compatible with electroweak physics. The knots are quantized and their kinematic states are labeled by $D^j_{mm^\prime}$, irreducible representations of SUq(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, t3, t0) by (t = N/6, t3 = -w/6, t0 = -(r+1)/6), where t0 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. [ABSTRACT FROM AUTHOR]

The q-electroweak theory suggests a description of elementary particles as solitons labeled by the irreducible representations of SUq(2). Since knots may also be labeled by the irreducible representations of SUq(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 ≅ 1 is compatible with the Kobayashi–Maskawa matrix. Depending on the test of these predictions, the model may be refined. [ABSTRACT FROM AUTHOR]

We attempt to go beyond the standard electroweak theory by replacing SU(2) with its q-deformation: SUq(2). This step introduces new degrees of freedom that we interpret as indicative of nonlocality and as a possible basis for a solitonic model of the elementary particles. The solitons are conjectured to be knotted flux tubes labeled by the irreducible representations of SUq(2), an algebra 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 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. [ABSTRACT FROM AUTHOR]

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