Designing new apartment buildings for strings and conformal field theories. First steps

The concepts of apartments and buildings were suggested by Tits for description of the Weyl-Coxeter reflection groups. We use these and many additional facts from the theory of reflection and pseudo-reflection groups along with results from the algebraic and symplectic geometry of toric varieties in order to obtain the tachyon-free Veneziano-like multiparticle scattering amplitudes and the partition function generating these amplitudes.Although the obtained amplitudes reproduce the tachyon-free spectra of both open and closed boisonic string, the generating (partition) function is not that of the traditional bosonic string. It is argued that it is directly related to the N=2 sypersymmetric quantum mechanical model proposed by Witten in 1982 in connection with his development of the Morse theory.Such partition function can be independently obtained with help of the results by Solomon (published in 1963) on invariants of finite (pseudo) reflection groups. Although the formalism developed in this work is also applicable to conformal field theories (CFT), it leaves all CFT results unchangedComment: 90 pages, 2 figures; this version was carefully proofread to eliminate numerous noticed typo

Role of general relativity and quantum mechanics in dynamics of Solar System

Let m(i) be the mass of i-th planet and M be the Solar mass. From astronomical data it is known that ratios r(i)=m(i)/(m(i)+M) are of order 10^(-3)-10^(-6) for all planets. The same is true for all satellites of heavy planets. These results suggest that Einstein's treatment of Mercury dynamics can be extended to almost any object in the Solar System. This fact does not explain the existing order in the Solar System. Indeed, all planets lie in the same (Sun's equatorial) plane and move in the same direction coinciding with that for the rotating Sun.The same is true for regular satellites of heavy planets and for planetary rings associated with these satellites.In addition to regular satellites, there are irregular satellites (and at least one irregular (Saturn) ring associated with such a satellite (Phoebe)) grouped in respective planes (other than equatorial) in which they all move in "wrong" directions on stable orbits. These are located strictly outside of those for regular satellites. This filling pattern is reminiscent to that in atomic mechanics. Based on the original Heisenberg's ideas, we develop quantum celestial mechanics explaining this filling pattern and that for rings of heavy planets. The formalism takes essentially into account that planets and satellites are moving on geodesics.Comment: 41 page

Heisenberg Honeycombs Solve Veneziano Puzzle

In this paper we reformulate some results obtained by Heisenberg into modern mathematical language of honeycombs. This language was developed in connection with complete solution of the Horn conjecture problem. Such a reformulation is done with the purpose of posing and solving the following problem. Is by analysing the (spectroscopic) experimental data it possible to restore the underlying microscopic physical model generating these data? Development of Heisenberg's ideas happens to be the most useful for this purpose. Solution is facilitated by our earlier developed string-theoretic formalism. In this paper only qualitative arguments are presented (with few exceptions). These arguments provide enough evidence that the underelying microscopic model compatible with Veneziano-type amplitudes is the standard (i.e. non supersymmetric!) QCD. In addition, usefulness of the formalism is illustrated on numerous examples such as physically motivated solution of the saturation conjecture, derivation of the Yang-Baxter and Knizhnik-Zamolodchikov equations as well as Verlinde and Hecke algebras, computation of the Gromov-Witten invariants for small quantum cohomology ring, etc. Finally, we discuss several scattering experiments testing correctness of our calculations and propose some possible new uses of these ideas in condensed matter physics.Comment: 61 pages, 16 figure

Veneziano amplitudes, spin chains and Abelian reduction of QCD

Although QCD can be treated perturbatively in the high energy limit, lower energies requre uses of nonperturbative methods such as ADS/CFT and/or Abelian reduction.These methods are not equivalent. In this paper we provide arguments in favor of the Abelian reduction methods. This is achieved by further developing results of our recent works reanalyzing Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. It is shown, that the obtained new partition function for these amplitudes can be mapped exactly into that for the Polychronakos-Frahm spin chain model recoverable from the Richardson-Gaudin XXX spin chain model used originally for treatments of the BCS-type superconductivity. Because of this, it is demonstrated that the obtained mapping is compatible with the method of Abelian reduction.Comment: 37 pages, 1 figur

Newtonian limit of Einsteinian gravity: from dynamics of Solar system to dynamics of stars in spiral galaxies

Attempts to merge Einsteinian gravity with Newtonian run into inconsistencies because in Newton's gravity time is absolute and the speed of gravity is infinite. Such an assumption was in a focus of attention of scientists in 19th century interested in finding out if the speed of gravity is infinite. By analogy with electrodynamics, some retarded potentials replacing Newtonian were utilized. By using one of such potentials Gerber correctly calculated the perihelion shift for Mercury in 1902. Subsequent attempts at calculation of bending of light using Gerber-style calculations were not successful. Recently Gin\'e (Chaos, Solitons and Fractals 42, 1893 (2009)) reobtained both the perihelion shift and the bending of light using retarded potential. His equations however are not those obtained by Einstein and his results coincide with those by Einstein only at the level of leading order terms of infinite series expansions. The obtained differential equations of motion are of delay-type. When applied to two-body dynamics, such equations lead to orbital quantization. In this work the Einsteinian approach is used to reproduce this quantization. Numerous arguments justifying the superiority of Einsteinian approach, including uses of the Bertrand spacetimes for description of motion of stars around galactic centers are provided. The developed formalism is tested by calculating the number of allowed stable orbits for planets and those for regular satellites of heavy planets resulting in reasonable agreement with observational data. The paper also discusses possible quantum mechanical nature of rings of heavy planets as well as of rotation curves of stars in spiral galaxies.Comment: 52 pages. The replacement is rewritten in accord with available astronomical data (as of year 2015). arXiv admin note: substantial text overlap with arXiv:1003.5696, arXiv:0707.399

Landau's Last Paper and its Impact on Developments in Mathematics, Physics and Other Disciplines in New Millennium

In this paper we discuss the impact of the last Landau paper on physics and mathematics to date and make some predictions about its likely impact on sciences in new millennium.Comment: 42 pages, 3 figures. An invited contribution to a special volume of EJTP on occasion of 100th anniversary of L.D. Landau birthda

Optical knots and contact geometry I. From Arnol'd inequality to Ranada's dyons

Recently there had been a great deal of activity associated with various schemes of designing both analytical and experimental methods describing knotted structures in electrodynamics and in hydrodynamics.The majority of works in electrodynamics were inspired by the influential paper by Ranada (1989) and its subsequent refinements. In this work and in its companion we analyze Ranada's results using methods of contact geometry and topology. Not only our analysis allows us to reproduce his major results but,in addition, it provides opportunities for considerably extending the catalog of known knot types. Furthermore,it allows to reinterpret both the electric and magnetic charges purely topologically thus opening the possibility of treatment of masses and charges in Yang-Mills and gravity theories also topologically. According to(now proven) Thurston's geometrization conjecture complements of all knots/links in S^3 are spaces of positive, zero or negative curvature. This means that spaces around our topological masses/charges are also curved. This fact is essential for design of purely topological theories of gravity, electromagnetism and strong/weak interactionsComment: 37 page

Optical knots and contact geometry II.From Ranada dyons to transverse and cosmetic knots

Some time ago Ranada (1989) obtained new nontrivial solutions of the Maxwellian gauge fields without sources. These were reinterpreted in Kholodenko (2015a) (part I) as particle-like (monopoles, dyons, etc.). They were obtained by the method of Abelian reduction of the non-Abelian Yang-Mills functional. The developed method uses instanton-type calculations normally employed for the non-Abelian gauge fields. By invoking the electric-magnetic duality it then becomes possible to replace all known charges/masses by the particle -like solutions of the source-free Abelian gauge fields. To employ these results in high energy physics, it is essential to to extend Ranada's results by carefully analysing and classifying all dynamically generated knoted/linked structures in gauge fields, including those discovered by Ranada. This task is completed in this work. The study is facilitated by the recent progress made in solving the Moffatt conjecture. Its essence is stated as follows: in steady incompressible Euler-type fluids the streamlines could have knots/links of all types. By employing the correspondence between the ideal hydrodynamics and electrodynamics discussed in part I and by superimposing it with the already mentioned method of Abelian reduction, it is demonstrated that in the absence of boundaries only the iterated torus knots and links could be dynamically generated. Obtained results allow to develop further particle-knot/link correspondence studied in Kholodenko (2015b)Comment: 67 pages, 17 figure

New strings for old Veneziano amplitudes IV.Connections with spin chains and other stochastic systems

In a series of published papers we reanalyzed treatments of the Veneziano amplitudes and the models associated with them. In this work we demonstrate that the already obtained partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm spin chain which, in turn, is obtainable from the Richardson-Gaudin XXX model. Reshetikhin and Varchenko demonstrated that such a model is recoverable from their WKB-type analysis of solutions of the Knizhnik-Zamolodchikov (K-Z) equations. The linear independence of solutions of these equations is controlled by determinants whose form (up to a constant) coincides with the Veneziano (or Veneziano-like) amplitudes.In the simplest case, when the K-Z equations are reducible to the Gauss hypergeometric equation, the determinantal conditions coincide with those which were discovered by Kummer in 19th century. Kummer's results admit physical interpretation crucial for providing needed justification associating determinanatal formula(s) with Veneziano-like amplitudes. General results are illustrated by many examples. These include but are not limited to only high energy physics since all high energy physics scattering processes can be looked upon from much broader sctochastic theory of random fragmentation and coagulation processes recently undergoing active development in view of its applications in disciplines ranging from ordering in spin glasses and population genetics to computer science, linguistics and economics,etc. In this theory Veneziano amplitudes play the universal role since they are the Poisson-Dirichlet-type distributions for these processes (analogous to the more familiar Maxwell distribution for gases)Comment: 66 pages 4 figure

Probabilistic vs optical interpretation of quantum mechanics

Although electrons and photons produce the same interference patterns in the two-slit experiments, the description of these patters is markedly different. This difference was analyzed by Bohm. Later on Sanz and Miret-Artes and others were able to squeeze the differences to zero. Fortunately, they left some room for developments presented in this Letter. We noticed that in the absence of sources the electromagnetic field can be represented by the complex scalar field. It is demonstrated that the same fields are being used in the non relativistic Schr\"odinger equation. The connection between the electromagnetic and Schr\"odinger fields allows to study the topology of zero sets (Chladni patterns) of Schr\"odinger eigenfunctions. The existence of these patterns is contingent upon the existence of eigenvalues of multiplicity higher than one. This is permissible only in Schr\"odinger's version of quantum mechanics. Presence of multiplicities is making quantum mechanical and topological entanglements equivalent.Comment: 19 page