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The SLq(2) extension of the standard model

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International Journal of Modern Physics A
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Robert J. Finkelstein
Robert J. Finkelstein
Robert J. Finkelstein
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Abstract

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.
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June 9, 2015 10:11 IJMPA S0217751X15300379 page 1
International Journal of Modern Physics A
Vol. 30, No. 16 (2015) 1530037 (50 pages)
c
World Scientific Publishing Company
DOI: 10.1142/S0217751X15300379
The SLq(2) extension of the standard model
Robert J. Finkelstein
Department of Physics and Astronomy,
University of California,
Los Angeles, CA 90095-1547, USA
finkel@physics.ucla.edu
Received 29 April 2015
Accepted 2 May 2015
Published 5 June 2015
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.
Keywords: Quantum group; electroweak; knot models; preon models.
PACS numbers: 02.20.Uu, 02.10.Kn, 12.60.Fr
Contents
1. Introduction ............................ 2
2. The Characterization of Oriented Knots ................ 3
3. The Kauffman Algorithm for Associating a Polynomial with a Knot3. . . 3
4. The Knot Algebra4–6 ........................ 4
5. Higher-Dimensional Representations of SLq(2) and SUq(2) ........ 5
6. The Gauge Group of the SLq(2) and SUq(2) Algebras .......... 8
7. Representation of an Oriented Knot .................. 9
8. The Quantum Knot ......................... 10
1530037-1
... The scheme presented in this article is based on a previous preon model of SM particles [1]. I define the updated, supersymmetric model as follows 1. the elementary fields are members of a supermultiplet, 2. the matter field is a light spin 1/2, charge 1/3 preon, 3. the gauge interaction is electromagnetism, 4. the quantum group SLq(2) is used to classify topologically scalar particles, preons, quarks and leptons [2,3]. ...
... Early work on knots in physics goes back in time to 19th and 18th century [7,8]. More recently Finkelstein has proposed a model based on the quantum group SLq(2) [2,3]. The idea stems from the fact that Lie groups are degenerate forms of quantum groups [9]. ...
... Therefore it is of interest to study a physical theory by replacing its Lie group by the corresponding quantum group. Finkelstein introduced the global group SLq(2) as an extension to the SM electroweak gauge group obtaining the group structure SU (2) × U (1) × SLq (2). ...
Supersymmetric Preons and the Standard Model
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The experimental fact that standard model superpartners have not been observed compels one to consider an alternative implementation for supersymmetry. The basic supermultiplet proposed here consists of a photon and a charged spin 1/2 preon field, and their superpartners. These fields are shown to yield the standard model fermions, Higgs fields and gauge symmetries. Supersymmetry is defined for unbound preons only. Quantum group SLq(2) representations are introduced to classify topologically scalars, preons, quarks and leptons.
View
... Beginning in 2005, in total ignorance of these phenomenological papers, I began to study the possibility of extending the standard model of elementary particles, in admitting topological degrees of freedom for the field particles by replacing the field operators Ψ(x) by Ψ(x) →Ψ j m,m (x)D j m,m (q) (1.1) where D j m,m (q) is an irreducible representation of the knot algebra SLq (2), whilẽ Ψ j m,m (x) satisfies the Lagrangian of the Standard Model after its modification by the form factors generated by the adjoined D j m,m (q) factors. [6][7][8][9][10][11] Unexpectedly, this topological model which we shall call the "knot model" agrees with the phenomenological models of Harari, Shupe, and Raitio. In this extension of the standard model, the states (j, m, m ) q of the SLq(2) algebra are postulated to be restricted by the topological conditions (j, m, m ) q = 1 2 (N, w, r + o) (1.2) where (j, m, m ) q labels a state of the quantum knot and (N, w, r) labels the 2d projection of a corresponding oriented classical knot. ...
... has been worked out in some detail as a SLq(2) extension of the standard lepton-quark model at the electroweak level. [9][10][11] It is successful when formulated as a preon theory at the electroweak level. Being a new model, however, it presents some unanswered questions and in particular it does not predict whether the preons are bound or are in fact observable. ...
... This view of the elementary particles as nonsingular lumps of field or as solitons has also been described as a unitary field theory. 10 The physical models suggested by Fig. 1 may be further studied in the context of gravitational and "preon binding" with the aid of generalized preon Lagrangians similar to that given in Ref. 11. The Hamiltonians of these three body systems can be parametrized by degrees of freedom characterizing both the preons (j = 1 2 ) and the binding field (j = 1). ...
Are dyons the preons of the knot model?
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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.
View
... Such quantum groups are deformations on Hopf algebras and depend on a deformation parameter q with the value q = 1 returning the undeformed universal enveloping algebra. First formalized by Jimbo [7] and Drinfeld [8] as a class of Hopf algebras, quantum groups have found many applications in theoretical physics, see [9][10][11][12][13][14][15] and references therein. ...
... Lie-type deformations (those that deform a Lie algebra) have proven very useful in generalizing spacetime symmetries [30][31][32][33]. q-Deformation on the other hand seem to have particular applications in generalized descriptions of internal and gauge symmetries [10,34,35]. Considered together therefore, it seems that Lie-type and q-deformations offer a consistent framework within which to develop physics in the 21st century [36][37][38][39]. ...
Charge specific baryon mass relations with deformed SU_q(3) flavor symmetry
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  • May 2016
The quantum group SUq(3)=Uq(su(3))SU_q(3)=U_q(su(3)) is taken as a baryon flavor symmetry. Accounting for electromagnetic contributions to baryons masses to zeroth order, new charge specific q-deformed octet and decuplet baryon mass formulas are obtained. These new mass relations have errors of only 0.02\% and 0.08\% respectively; a factor of 20 reduction compared to the standard Gell-Mann-Okubo mass formulas. A new relation between the octet and decuplet baryon masses that is accurate to 1.2\% is derived. An explicit formula for the Cabibbo angle, taken to be π14\frac{\pi}{14}, in terms of the deformation parameter q and spin parity JPJ^P of the baryons is obtained.
View
... The basic idea behind the asymmetry is the following. Superons in one region 4 The superons have two dimensional anyon statistics and form composite states by Chern-Simons interaction [21]. of the universe can form quarks and leptons with charges like in uud and e − , or (1.1) first line. But in other regions of the universe, nearby or far away, the same superons may combine differently forming a u ud and e + , or pe + pair, i.e. an atom of antihydrogen H . ...
View
... This new expression for q would directly relate Gamow cosmology to Bohr quantum mechanics, and would be required to be experimentally realized. 7 ...
A Possible Relation Between the Magnetic Pole and the Gravitational Field
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  • May 2020
We reformulate the quantization of the gravitational field and its sources, including the electric and magnetic fields as they appear in the knot algebra.
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... The original version [1] of this scenario was redefined in terms of fields with charge, spin, and light mass in [2]. As shown by Finkelstein [3,4], this kind of preon model (his as well as ours) can be extended to possess topological symmetry property of the quantum group SLq(2) which provides consistent representations for quarks, leptons and preons. Both scenarios agree with the standard model group structure. ...
A Model for Matter-Antimatter Asymmetry
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Full-text available
  • Jan 2020
A previous preon scenario based on unbroken global supersymmetry is developed further to provide a natural physical reason for the observed matter-antimatter asymmetry. A tentative mechanism for asymmetric genesis of matter in early cosmology is proposed. With global supersymmetry made local the scenario can be extended to supergravity. PACS 12.60.Rc
View
... Jehle [55,56], for one, proposed a classification based on quantized flux that has much in common with the present approach, but it is weak on connection with field equations. Finkelstein [57,58] has developed a detailed match of knot topology with structure of the Standard Model. Knot topology can be taken as supplementing the present approach, which is based on differential geometry using STA. ...
Zitterbewegung structure in electrons and photons
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The Dirac equation is reinterpreted as a constitutive equation for singularities in the electromagnetic vacuum, with the electron as a point singularity on a lightlike toroidal vortex. The diameter of the vortex is a Compton wavelength and its thickness is given by the electron's anomalous magnetic moment. The photon is modeled as an electron-positron pair trapped in a vortex with energy proportional to the photon frequency. The possibility that all elementary particles are composed of similar vortices is discussed.
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A Chern-Simons model for baryon asymmetry
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In search of a phenomenological model that would describe physics from Big Bang to the Standard Model (SM), we propose a model with the following properties (i) above an energy about Λcr>1016 GeV there are Wess-Zumino supersymmetric preons and Chern-Simons (CS) fields, (ii) at Λcr∼1016 GeV spontaneous gauge symmetry breaking takes place in the CS sector and the generated topological mass provides an attractive interaction to equal charge preons, (iii) well below 1016 GeV the model reduces to the standard model with essentially pointlike quarks and leptons, having a radius ∼1/Λcr≈10−31 m. The baryon asymmetry turns out to have a fortuitous ratio nB/nγ≪1.
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A Stringy Model of Pointlike Particles
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A previous supersymmetric preon scenario for visible matter particles is extended to the dark sector. In addition, the scenario is reformulated as a Double Field Theory (DFT) with four extra dimensions, to avoid a singular Big Bang in cosmology. T-duality and doubled local Lorentz symmetry of the model are genuine stringy properties. It is proposed that DFT preons may be an approximate pointlike projection of string theory.
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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.
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We examine a quantum group extension of the standard model. The field operators of the extended theory are obtained by replacing the field operators ψ\psi of the standard model by ψ{\psi}Dmm′j^j_{mm'}, where Dmm′j^{j}_{mm'} are elements of a representation of the quantum algebra SLq(2), which is also the knot algebra. The Dmm′j^j_{mm'} lie in this algebra and carry the new degrees of freedom of the field quanta. The Dmm′j^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=32j=\frac{3}{2}) representation and the weak vector bosons by elements of the ditrefoil (j=3) representation. The adjoint (j=1) and fundamental (j=12j=\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.
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