Preon

In particle physics, preons are postulated "point-like" particles, conceived to be subcomponents of quarks and leptons. The word was coined by Jogesh Pati and Abdus Salam in 1974. Interest in preon models peaked in the 1980s but has slowed as some proposed models were ruled out by collider experiments and no model was able to predict a new experimental result. Additionally, after the first superstring revolution, many particle physicists became convinced that string theory offers the most logical research program to pursue, in understanding the nature of elementary particles. It was thought string theory was more promising than preon theory, and many particle physicsts committed their research efforts in this direction. More recently confidence in string theory has waned in some quarters, leading to renewed interest in alternative approaches, including preon models.

Background: The urge to simplify the Standard Model

Before the Standard Model (SM) was developed in the 1970s (the key elements of the standard model known as quarks were proposed by Gell-Mann and Zweig in 1964), physicists observed hundreds of different kinds of particles in particle accelerators. These were organized into relationships on their physical properties in a largely ad-hoc system of hierarchies, not entirely unlike the way taxonomy grouped animals based on their physical features. Not surprisingly, the huge number of particles was referred to as the "particle zoo".

The Standard Model, which is now the prevailing model of particle physics, dramatically simplified this picture by showing that most of the observed particles were mesons, which are combinations of two quarks, or baryons which are combinations of three quarks, plus a handful of other particles. The particles being seen in the ever-more-powerful accelerators were, according to the theory, typically nothing more than combinations of these quarks.

Within the Standard Model, there are several different types of particles. One of these, the quarks, has six different kinds, of which there are three varieties in each (dubbed "colours", red, green, and blue, giving rise to QCD: quantum chromodynamics). Additionally, there are six different types of what are known as leptons. Of these six leptons, there are three charged particles: the electron, muon, and tauon. The neutrinos comprise the other three leptons, and for each neutrino there is a corresponding member from the other set of three leptons. In the Standard Model, there are also the photons, W⁺, W⁻, and Z particles, gluons, and a few open spaces left for the graviton and Higgs boson, which have not yet been discovered. Almost all of these particles come in "left handed" and "right handed" versions.

The Standard Model also has a number of problems which have not been entirely solved. In particular, no successful theory of gravitation based on a particle theory has yet been proposed. Although the Model assumes the existence of a graviton, all attempts to produce a consistent theory based on them has failed. Additionally, mass remains a mystery in the Standard Model. Although the mass of each successive particle follows certain patterns, predictions of the rest mass of most particles can not be made precisely. The Higgs boson is assumed to "solve" this problem, but to date the Higgs mechanism remains unproven.

The Model also has problems predicting the large scale structure of the universe. For instance, the Model generally predicts equal amounts of matter and anti-matter in the universe, something that is observably not the case. A number of attempts have been made to "fix" this through a variety of mechanisms, but to date none have won widespread support. Likewise, basic adaptations of the Model suggest the presence of proton decay, another mechanism that has not yet been observed, and appears unlikely to exist.

Preon theory is motivated by a desire to replicate the achievements of the periodic table, and the later Standard Model which tamed the "particle zoo", by finding more fundamental answers to the huge number of arbitrary constants present in the Standard Model.

Theoretical particle physics considerations to research preon theory

Preon research is motivated by the desire to explain already existing facts (postdiction), which include:

• To reduce the large number of particles, many that differ only in charge, to a smaller number of more fundamental particles. For example, the electron and positron are identical except for charge, and preon research is motivated by explaining that electrons and positrons are composed of similar preons with the relevant difference accounting for charge. The hope is to reproduce the reductionist strategy that has worked for the periodic table of elements.
• The second and third generation fermions are supposedly fundamental, yet they have have higher masses than those of the first generation, and the quarks are unstable and decay into their first generation counterparts. Historically, the instability and radiactivity of some chemical elements were explained in terms of isotopes. By analogy this suggests a more fundamental structure for at least some fermions. [[1]]
• To unify particle physics with gravity, for example, Bilson-Thompson model with loop quantum gravity.
• To give prediction for parameters that are otherwise unexplained by the Standard Model, such as particle masses and charges and color, and reduce the number of experimental input parameters required by the standard model.
• To provide reasons for the very large differences in energy-masses observed in supposedly fundamental particles, from the electron neutrino to the top quark.
• The top quark has a mass comparable to a gold nuclei. Where as the gold nuclei is stable, and is composed of a more elementary structure of 79 protons and electrons, and on average, 118 neutrons, the top quark is currently described as being a particle of pure mass, with no internal structure, and one that spontaneously decays. Preon models try to model the top quark mass in a manner similar to gold nucleus mass.
• To provide a theoretical framework of unstable particle half-life.
• To explain the number of generations of fermions.
• To provide alternative explanations for the electro-weak symmetry breaking without invoking a Higgs field, which in turn possibly needs a supersymmetry to correct the theoretical problems involved with the Higgs field. Supersymmetry itself has theoretical problems.
• To explain the features of particle physics without the need for higher dimensions, supersymmetry, higgs field, or string theory.
• To account for neutrino oscillation and mass.
• The desire to make new nontrivial predictions, for example, to provide possible cold dark matter candidates, or to predict that the LHC will not observe a Higgs boson or superpartners.
• The desire to reproduce only observed particles, and to prevent prediction within its framework for non-observed particles (which is a theoretical problem with supersymmetry).
• The experimental falsification of certain grand unified theories of particle physics as the result of not observing proton decay may suggest that the grand unification scenario, which string theory is predicated on, and supersymmetry, may be false, and different solutions and thinking will be required for the progress of particle physics.

Were string theory successful in its original objectives, preon theory research would not be necessary. String theory was supposed to account for the above issues in terms of string dynamics. The different particles of the standard model were accounted for as different frequencies (tension) of a planck-scale string, particle dynamics were explained in terms of the worldsheet diagrams, (the string theory equivalent of Feynman diagrams) and the three generations of fermions were explained in terms of strings "wrapping around" specific configuration of higher-dimensional moduli. The continuing failure of string theory to achieve the above objectives as a theory of particle physics Relevant literature include: Peter Woit Not Even Wrong, or Lee Smolin's The Trouble with Physics, or Daniel Friedan's "String theory is a complete scientific failure". Andrew Oh-Willeke states "as string theory develops more doubters, I think [preon theory] will be an obvious direction for non-string theory investigators and theorists."

History: Pre-quark theories

A number of physicists have attempted to develop a theory of "pre-quarks" (from which the name preon derives) in an effort to justify theoretically the many parts of the Standard Model that are known only through experimental data.

Other names which have been used for these proposed fundamental particles (or particles intermediate between the most fundamental particles and those observed in the Standard Model) include: prequarks, subquarks, maons, alphons, quinks, rishons, tweedles, helons, haplons, and Y-particles. Preon is the leading name in the physics community.

Efforts to develop a substructure date at least as far back as 1974 with a paper by Pati and Salam in Physics Review. Other attempts include a 1977 paper by Terazawa, Chikashige and Akama, similar, but independent 1979 papers by Ne'eman, Harari[2] and Shupe, a 1981 paper by Frizsch and Mandelbaum, a 1992 paper by D'Souza and Kalman, and a 1997 paper by Larson [3]. None has gained wide acceptance in the physics world.

Each of the preon models identifies a set of fundamental particles far smaller in number than those of the Standard Model, explains rules governing how those fundamental particles operate, and shows how those proposed particles and rules can explain the Standard Model, often with predicted small discrepancies from the existing model, proposed new particles, and certain phenomena in the standard model that remain unexplained. The Harari Rishon Model illustrates some of the typical efforts in the field.

Many of the Preon models theorize that the apparent imbalance of matter and anti-matter in the universe is in fact illusory, with large quantities of preon level anti-matter confined within more complex structures.

Many preon models either do not account for the Higgs boson, or rule it out, and propose composite preons as what breaks the electro-weak symmetry, rather than a scalar Higgs fields. For example, Fredriksson preon theory does not need the Higgs boson, and explains the electro-weak breaking as the rearrangement of preons, rather a Higgs-mediated field. In fact, Fredriksson preon model predicts that the Higgs boson does not exist. In the above cited paper, Fredricksson acknowledges the mass paradox represents a problem in his accounting for neutrino mass, however, he proposes a specific arrangement of preons in his model, which he calls the X-quark, which his theory suggests could be a stable good cold, dark matter candidate.

Yershov's preon model

The vast bulk of recent theoretical research into the particle zoo has been string theory. It was thought string theory has completely supplanted preon research, and that one dimensional supersymmetric strings can reproduce all the particles of the standard model, and their superpartners, the MSSM, their properties, color, charge, parity, chirality, and energy-masses, obviating any need for preon research. To date, string theory has been unable to reproduce the standard model. A search through Spires and Arxiv, show that approximately over 30, 000 papers in string theory or supersymmetry since 1982, with several hundred new papers being published every month. In comparison, in 2006, since 2003, there have been about a dozen papers in preon theory listed as such in arxiv. String theories continuing failure to reproduce the particle spectrum of the standard model has given some life for preon theories, and there have been recent papers on preon theory. As of 2006, Yershov, Fredriksson, and Bilson-Thompson have published papers in Preon theory within the past 5 years: a 2003 paper by Fredriksson [4], and a 2005 paper by Bilson-Thompson [5].

When the term "preon" was coined, it was primarily to explain the two families of spin 1/2 fermions: leptons and quarks. More recent preon models also account for spin-1 bosons, and are still called "preons". The term "preon" is the term of choice for Bilson-Thompson, Yershov, and Fredrickson, although they expand the meaning of the term, in addition to accounting for spin 1/2 fermions of leptons and quarks, which the term was used in its early history, the latter theories also accounts for spin-1 bosons.

Yershov's model is patterned after the idea naked singularities in general relativity, and closely resembles geon from John Archibald Wheeler research program into Geometrodynamics. Electron structure in Yershov's theory was further elaborated on in 2006 [6]. 2003 papers by Yershov [7] [8] are notable for being some of the only papers in the field to use the Preon model as a basis for providing specific numerical values from first principles for the masses of the particles described in the Standard Model. Yershov's model does not predict the mass of the Higgs Boson, and does not need the Higgs boson, and predicts it will not be found. Yershov's model deals with the mass paradox by prosposing a huge binding energy for his preons, which acts as a source of mass-energy, as through mass defect. To get around the mass paradox, Yershov's model proposes a new force that is 10^5 stronger than the strong nuclear force, that binds his preons together.

The basic building block of the composite particle model is a primitive particle (preon) regarded as a source of a spherically-symmetric pairwise field with the colour (tripolar) symmetry. Except for its tripolarity, the field is analogous to the Lennard-Jones fields used in molecular physics for modelling long-range attractive and short-range repulsive forces. A trade-off between these forces leads to equilibrium particle configurations, the simplest of which are colour dipoles and tripoles composed of, respectively, two and three preons.

The dipoles are deficient in one colour, whereas the tripole fields are colourless at infinity and colour-polarised nearby.This allows axial coupling of tripoles into strings. Due to the Z3-symmetry of the tripoles, they are consecutively 120-degree rotated within a string of like-charged tripoles. Given the cyclic property of the group, a three-component string is likely to close in a loop, which would minimise its potential energy.

The tripoles in such a system retain their rotational and translational degrees of freedom (around and along their common ring-closed axis), while maintaining their relative 120-degree orientation with respect to each other. The colour-currents formed due to the motions of the looped tripoles are helices with either clockwise or anticlockwise winding. Helicity is found to be one of the very important properties of these structures.

A string of tripole-antitripole pairs has a similar symmetry, which allows its closure in a neutral loop containing six such pairs (twelve tripoles). By their properties, the three- and twelve-tripole loops can be identified with the electron and its neutrino, respectively. It can be seen that different combinations of these loops (involving tripoles as well) lead to a variety of structures identifiable with the observed variety of elementary particles.

It is noted, that there is no place for arbitrariness in the variety of these structures because they are hierarchically self-similar and each particular (relatively stable) structure obeys kinematic constraints of a topological nature corresponding to the minimum of its potential energy. The number of constituents (preons) in each structure is well-defined and, therefore, - surprisingly enough - it turns out that the model does not require free or experimental parameters to explain the origin and the properties of the fundamental fermions.

The masses of these structures are assumed to arise as the energies of the preon motions inside of the structures, combined with the preon binding energies (mass defects), which are known as two standard mass-generating mechanisms for composite systems. In a first approximation, the masses of the charged structures are derived from the number of their constituent preons by setting for simplicity the energy of each preon to unity.

The fields of opposite polarities (or colours) cancel each other if two unlike-charged preons get close to each other or overlap, which would also nullify the mass of the system. In this case the energy would be stored in the form of the mass defect (also dampening the oscillatory motions of the components). In fact, in the neutral systems, such as the electron neutrino, the preons' centres do not coincide exactly, therefore the complete cancellation of fields (and masses) in such systems does not occur, and the neutrinos are supposed to have a non-vanishing mass. The mass of a neutral loop (neutrino) is recovered when an electric field source (e.g., a tripole) is enclosed by the loop.

For more complicated structures (like the second and third generation fermions regarded in this model as clusters of simpler structures belonging to the first generation) the computation of masses is unlikely to be so simple because of possible multiple resonances in the clusters. However, an empirical (non-physical) summation rule was proposed to show that, at least in principle, these masses are computable and that the variety of these structures does, indeed, resemble the observed variety of particles.

Criticisms specific to Yershov's model

Yershov's model has been described in terms of classical general relativity, rather than quantum mechanics. The standard model is framed entirely in the language of Quantum Field Theory. It is unclear whether Yershov's model can take quantum field theory into account, and successfully address T'Hooft's anomaly matching constraints. A theory that can post-dict particle mass and half-lives would be a major breakthrough, and Yeshov's model purports to do this. However, some physicists are unconviced with Yershov's mass formula for elementary particles in his research papers on preons on the grounds that it is highly arbitrary, both in the derivation of his formula, and the manner in which he assigns values to his preons.[[9]]. Proposing unobserved "preon binding forces" to account for the preon mass paradox may result in predictions in conflict with observation. The non-quantum nature of this preon model has attracted little, if any, interest, among particle physicists and there appears to be no citations to his research in the literature.

Loop quantum gravity and Bilson-Thompson Preon theory

In the 2005 paper mentioned above, Sundance Bilson-Thompson proposed a model which adapted the Harari Rishon Model to preon-like objects which were extended ribbons, rather than point-like particles. This provided a possible explanation for why ordering of the subcomponents matters (giving rise to colour charge) whereas in the older point-particle preon model (Rishon Model), this feature must be treated as an ad hoc assumption. Bilson-Thompson refers to his extended ribbons as "helons", and his model as the Helon Model. This model leads to an interpretation of electric charge as a topological quantity (twists carried on the individual ribbons). In a subsequent paper [10] from 2006 Bilson-Thompson, Fotini Markopolou, and Lee Smolin suggested that in any of a class of quantum gravity theories similar to loop quantum gravity (LQG) in which spacetime comes in discrete chunks, excitations of spacetime itself may play the role of preons, and give rise to the standard model of particle physics as an emergent property of the quantum gravity theory.

Consequently, Bilson-Thompson et al proposed that loop quantum gravity could reproduce the standard model. In this scenario the four forces would already be unified. The first generation of fermions (leptons and quarks) with correct charge and parity properties have been modelled using preons constituted of braids of spacetime as the building blocks. Bilson-Thompson's original paper suggested that the higher-generation fermions could be represented by more complicated braidings, although explicit constructions of these structures were not given. The electric charge, colour, and parity properties of such fermions would arise in the same way as for the first generation. Utilization of quantum computing concepts made it possible to demonstrate that the particles are able to survive quantum fluctuations.

The ribbon-like structures of Bilson-Thompson's model have been described as "pieces of spacetime ribbon-tape", in that they may be made of the same structure that makes up spacetime itself. [11]. While Bilson-Thompson's papers do offer an explanation on how to get fermions and spin-1 bosons, he does not show a braiding that would account for the Higgs boson [12].

In a 2006 paper [13], L. Freidel, J. Kowalski--Glikman, A. Starodubtsev suggests that elementary particles are Wilson lines of gravitational field, which implies that the properties of elementary particles, such as mass, energy, and spin, can be described by LQG's Wilson loops, and particle dynamics can be modeled on breaks in these Wilson loops, adding theoretical support to Bilson-Thompson's preon proposals.

The spin foam formalism (closely related to loop quantum gravity) allows for the derivation of certain other particles of the standard model, the spin-1 bosons, such as photons and gluons, [[14]] and gravitons [[15]], [[16]] from loop quantum gravity's fundamental principles, and independent of Bilson-Thompson's braiding scheme for fermions. However, as of 2006, there is not a derivation of the helon model from spin foam formalism, as described by braiding. The helon model does not offer a braiding that would account for a Higgs, but does not rule out the possibility of a Higgs boson as a composite object. Bilson-Thompson himself observes that since the particles with higher masses have (generally) more complicated internal "structure" - including twists which are identified as electric charges - it is possible that this internal structure gives rise to inertial mass through some unspecified mechanism (The massless photon is untwisted in Bilson-Thompson's preon scheme). As of 2006, it remains to be seen whether the derivation of the photon from the spin foam formalism in [[17]] can be matched with Bilson-Thompson's braiding of three untwisted ribbons [18], or perhaps, there are multiple ways to derive photons from the spin foam formalism.

When the term "preon" was first coined, it was used to describe pointlike subparticles that describe spin-1/2 fermions that include leptons and quarks. Such sub-quark pointlike particles would suffer from the mass paradox described below. It is observed that Bilson-Thompson's ribbon structures are not actually "classical" pointlike preons, as defined in the introduction to this article. Bilson-Thompson uses the term "preon" in the sense of more fundamental "subparticles" of quarks, leptons, and gauge bosons, and to maintain continuity in terminology with the larger physics community. His braiding also accounts for spin-1 bosons.

One way to understand Bilson-Thompson's "preon" theory, in contrast to older particle preon theories, is that elementary particles such as electrons are described in terms of a wave function, and the state-sum of coherent phase spin foam is also described in terms of a wave function. It is hoped that it might be possible to derive a wave function from the spin foam formalism that matches the wave function used to describe elementary particles such as photons and electrons. Coupling matter to loop quantum gravity remains an active area of research [[19]]

Bilson-Thompson has recently updated his paper dated October 27, 2006, [[20]] and notes that his model, while not preon in the strict sense of the term, nevertheless is apreon-inspired model, and is open to the possibility that other more fundamental theories, such as M-Theory, may account for his topological diagrams, as well as the Higgs boson and gravity. The theoretical objections that apply to classic preon models do not necessarily apply to his preon inspired model, as it is not the particles themselves, but the relations between his preons (braiding) that give rise to the properties of particles. In this newer version of his paper, he has added a new section, section IV, called "unresolved issues" and acknowledges that open issues include mass, spin, cabbibo mixing, and grounding in a more fundamental theory. He states that grounding preons in M-theory is a possibility, as well as loop quantum gravity.

Theoretical objections to preon theories: Mass paradox, chirality, and 't Hooft anomaly-matching constraints

Heisenberg's uncertainty principle states that ΔxΔp ≥ ħ/2 and thus anything confined to a box smaller than Δx would have a momentum of uncertainty proportionally greater. Some candidate preon models propose particles smaller than the elementary particles they make up, therefore, the momentum of uncertainty Δp should be greater than the particles themselves.

One preon model started as an internal paper at the Collider Detector at Fermilab (CDF) around 1994. The paper was written after the occurrence of an unexpected and inexplicable excess of jets with energies above 200 GeV were detected in the 1992—1993 running period.

Scattering experiments have shown that quarks and leptons are "pointlike" down to distance scales of less than 10⁻¹⁸ m (or 1/1000 of a proton diameter). The momentum uncertainty of a preon (of whatever mass) confined to a box of this size is about 200 GeV, 50,000 times larger than the rest mass of an up-quark and 400,000 times larger than the rest mass of an electron.

Thus, the preon model represents a mass paradox: How could quarks or electrons be made of smaller particles that would have many orders of magnitude greater mass-energies arising from their enormous momenta? Yershov's model, referenced above, proposes that when both particles and anti-particles of the proposed Y-particles in the theory are present, such as in the model's proposed neutrino composition, the mass of the constituent parts "cancels out", but can appear again when the structure of the Y-particles is changed. Yershov's model also proposes that particle mass arises as a mass defect binding energy among his preons, which helps account for the mass paradox.

Sundance preon model may avoid this by denying that preons are pointlike particles confined in a box less than 10⁻¹⁸ m, and instead positing that preons are extended 2-dimensional ribbon-like structures, not necessarily smaller than the elementary particles they compose, not necessarily confined in a small box as point particles preon models propose, and not necessarily "particle-like", but more like glitches and topological folds of spacetime that exist in three-fold bound states that interact as though they were point particles when braided in groups of three as a bound state with other particle properties such as mass and pointlike interaction arising as an emergent property so that their momentum uncertainty would be on the same order as the elementary particles themselves.

String theory posits one-dimensional strings on the order of the Planck scale as giving rise to all the particles of the Standard Model, which would appear to also have the mass paradox problem. String theorist Lubos Motl has offered explanations as to how string theory gets around the mass paradox [21], and consents to have his explanation shared here. The center-of-mass position X0 and the total momentum of a string, P0, behave just like for pointlike particles. They don't commute and follow the uncertainty principle. A certain X0 means uncertain P0 and vice versa, the product being above hbar/2.

Besides the zero modes (center-of-mass degrees of freedom), every string has infinitely many internal degrees of freedom. That's like an atom with many electrons, but you have infinitely many arranged along a string. The relative motion of pieces of string gives energy - expressed as the usual sum of the kinetic and potential contributions. And because they're relativistic strings, energy also means mass via E=mc².

The result is that the minimal size of the string - in the lowest-energy or lowest-mass states - is a compromise in which the kinetic terms from the internal degrees of freedom contribute the same as the potential terms, just like for X and P in the harmonic oscillator, attempting to minimize the energy while satisfying the uncertainty relation for the internal X,P degrees of freedom. This minimum occurs if the typical size of the string is comparable to a typical distance scale derivable from the tension of the string - the string length - that is conventionally believed to be close to the Planck scale 10⁻³⁵ meters (a bit longer than that).

The actual numerical coefficient of the string is actually logarithmically divergent but this fact doesn't affect any finite-energy experiments. The question has the same answer in string theory just like for ordinary particles because it is the zero modes that matter here. The internal degrees of freedom are only relevant for experimental uncertainty considerations if your probe the internal structure of a string, and indeed, you will always find out that the "radius" of it seems to be of order the string length.

Any candidate preon theory must address particle chirality and the 't Hooft anomaly-matching constraints, and ideally be more parsimonious in theoretical structure than the Standard Model itself. Often, preon models propose additional unobserved forces or dynamics to account for their proposed preons compose the particle zoo, which may make the theory even more complicated than the Standard Model, or have implications in conflict with observation. One specific example: should the LHC observe a Higgs boson, or superpartners, or both, the observation would be in conflict with the predictions of many preon models, which predict the Higgs boson does not exist, or are unable to derive a combination of preons which would give rise to a Higgs Boson.

String theory and preon theory

String theory proposes that a one dimensional string on the order of a Planck scale has a tension, and differences in tension give rise directly to all the particles of the standard model and their super partners, in interaction with the proper compactified 6 or 7 dimensional Yau-Calabi mainfold and SUSY breaking. To date, string theory has been no more successful than preon theory in achieving this goal. John Baez and Lubos Motl have discussed the possibility [22] that should preon theory prove successful, it may be possible to formulate a version of string theory that gives rise to a successful model of preons.

There have been recent research papers that have proposed preon models that are made of superstrings in Arxiv [[23]], [[24]] or supersymmetry [[25]]

Preons in popular culture

In the 1948 reprint/redit of his 1930 novel Skylark Three, E. E. Smith postulated a series of 'subelectrons of the first and second type' with the latter being material properties that corresponded to gravitation. While this may not have been an element of the original novel (the scientific basis of some of the other novels in the series was revised extensively due to the additional thirty years of scientific development), even the edited publication may be the first, or one of the first, mentions of the possibility that electrons are not elementary particles.

See additional discussion in the article on the Rishon Model.

See also

• Preon star
• Preon-degenerate matter
• Rishon

References

• Pati, J. C.; Salam, A. (1974); Lepton number as the fourth "color", Phys. Rev. D10, 275-289
• Dugne, J.-J.; Fredriksson, S.; and Hansson, J.; Preon Trinity - A Schematic Model of Leptons, Quarks and Heavy Vector Bosons, Europhys. Lett., 60, 188 (2002) [26]
• Bilson-Thompson, S. O.; A topological model of composite preons, submitted to Phys. Lett. B (2005) [27]
• Bilson-Thompson, S. O.; Markopoulou, F.; and Smolin, L.; Quantum Gravity and the Standard Model, eprint (2006) [28]
• Analysis of selected preon models
• Yershov, V. N.; Equilibrium configurations of tripolar charges, Few-Body Systems, 37 (2005) 79-106, [29]
• Yershov, V. N.; Fermions as topological objects, Progr. Phys., 1 (2006) 19-26, [30]
• Yershov, V. N.; Quantum properties of a cyclic structure based on tripolar fields, Physica D (in press),[31]