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Arkady L. Kholodenko

Publications and source records attributed to Arkady L. Kholodenko.

At least 19 recordsLinked to original sources
arXiv2023

Maxwell-Dirac isomorphism revisited: from foundations of quantum mechanics to geometrodynamics and cosmology

Although electrons (fermions) and photons(bosons) produce the same interference patterns in the two-slit experiments, the description of these patterns is markedly different. Photons are spin one, relativistic and massless while electrons are spin 1/2 massive particles producing the same interference patterns. In spite of these differences, already in early 30ies of 20ieth century the isomorphism between the source-free Maxwell and Dirac equations was established. It permitted us to replace Born probabilistic interpretation of quantum mechanics with the optical. In 1925 Rainich combined source-free Maxwell equations with Einstein's equations for gravity. His results were rediscovered by Misner and Wheeler in their "geometrodynamics". Absence of sources in it remained a problem until Ranada's work. His results required the existence of null electromagnetic fields absent in geometrodynamics. They were added later on by Geroch. Ranada's solutions of source-free Maxwell's equations came out as knots and links. In this work new proof of knotty nature of the electron is established. The obtained result perfectly blends with the descripion of rotating and charged black hole.

physics.gen-ph↗
arXiv2021

Searching for high temperature superconductivity: From Mendeleev to Seiberg-Witten via Madelung and beyond

Recently,a noticeable progress had been achieved in the area of high temperature superconductors. The maximum temperature of 250K for LaH(10) and 288K for CSH(8) were reported at the megabar pressures. The highest possible temperatures were achieved by employing hydrides of chemical elements. Empirically, many of these are made of Madelung-exceptional atoms. Here the theoretical background is provided explaining this observation. The, thus far empirical, Madelung rule is controlling Mendeleev's law of periodicity. Although the majority of elements do obey this rule, there are some exceptions. Thus, it is of interest to derive it and its exceptions theoretically in view of experimental findings. As a by product, such a study yields some plausible explanation of the role of Madelung-exceptional atoms in the design of hightemperature superconductors. Thus far the atoms obeying the Madelung rule and its exceptions were studied with help of the relativistic Hartree-Fock calculations. In this work we reobtain both the rule and the exceptions analytically. The newly developed methods are expected to be of value in quantum many-body theory and, in particular, in the theory of high temperature superconductivity. Ultimately, new methods involve some uses of the Seiberg-Witten (S-W) theory known as the extended Ginzburg-Landau theory of superconductivity. Using results of the S-W theory the difference between the Madelung-regular and Madelung-exceptional atoms is explained in terms of the topological transition. Extension of this, single atom, result to solids of respective elements is also discussed

quant-ph↗
arXiv2020

How the modified Bertrand theorem explains regularities of the periodic table I. From conformal invariance to Hopf mapping

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently extended Bertrand theorem. He designed new static spherically symmetric (Bertrand) spacetimes obeying Einsteins equations and supporting closed orbits. In this work we prove that the topology and geometry of these spacetimes permits to solve quantum many-body problem for any atom of periodic system exactly. The computations of spectrum for any atom of periodic system becomes analogous to that for hydrogen atom. Initially the exact solution of the Schrödinger equation for any multielectron atom was obtained by Tietz in 1956. However, neither himself nor others fully comprehended what actually was obtained. We recalculated Tietz results by applying the methodology consistent with new (different from that developed by Fock in 1936) way of solving Schro"odingers equation for hydrogen atom. In the light of this new result it had become possible to demonstrate rigorously that the Tietz-type Schro"odingers equation is in fact describing the quantum motion in Bertrand spacetime. As a bonus, we obtained the analytical proof of the Madelung rule defined in the text

quant-ph↗
arXiv2020

How the Modified Bertrand Theorem Explains Regularities and Anomalies of the Periodic Table of Elements

Bertrand theorem permits closed orbits in 3d Euclidean space only for 2 types of central potentials. These are of Kepler-Coulomb and harmonic oscillator type. Volker Perlick recently designed new static spherically symmetric (Bertrand) spacetimes obeying Einstein's equations and supporting closed orbits. In this work we demonstrate that the topology and geometry of these spacetimes permits us to solve quantum many-body problem for any atom of periodic system exactly. The computations of spectrum for any atom are analogous to that for hydrogen atom. Initially, the exact solution of the Schrödinger equation for any multielectron atom (without reference to Bertrand theorem) was obtained by Tietz in 1956. We recalculated Tietz results by applying the methodology consistent with new (different from that developed by Fock in 1936) way of solving Schrödinger's equation for hydrogen atom. By using this new methodology it had become possible to demonstrate that the Tietz-type Schrödinger's equation is in fact describing the quantum motion in Bertrand spacetimes. As a bonus, we solved analytically the Löwdin's challenge problem. Obtained solution is not universal though since there are exceptions of the Madelung rule in transition metals and among lanthanides and actinides. Quantum mechanically these exceptions as well as the rule itself are treated thus far with help of relativistic Hartree-Fock calculations. The obtained results do not describe the exceptions in detail yet. However, studies outlined in this paper indicate that developed methods are capable of describing exceptions as well. The paper ends with some remarks about usefulness of problems of atomic physics for development of quantum mechanics, quantum field theory and (teleparallel) gravity.

physics.gen-ph↗
arXiv2017

Huygens triviality of the time-independent Schrödinger equation. Applications to atomic and high energy physics

Huygens triviality-a concept invented by Jacques Hadamard-describes an equivalence class connecting those 2nd order partial differential equations which are transformable into the wave equation. In this work it is demonstrated, that the Schrödinger equation with the time-independent Hamiltonian belongs to such an equivalence class. The wave equation is the equation for which Huygens' principle (HP) holds. The HP was a subject of confusion in both physics and mathematics literature for a long time. Not surprisingly, the role of this principle was obscured from the beginnings of quantum mechanics causing some theoretical and experimental misunderstandings. The purpose of this work is to bring the full clarity into this topic. By doing so, we obtained a large amount of new results related to uses of Lie sphere geometry, of twistors, of Dupin cyclides, of null electromagnetic fields, of AdS-CFT correspondence, of Penrose limits, of geometric algebra, etc. in physical problems ranging from the atomic to high energy physics and cosmology.

physics.gen-ph↗
arXiv2017

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 Schrödinger equation. The connection between the electromagnetic and Schrödinger fields allows to study the topology of zero sets (Chladni patterns) of Schrödinger eigenfunctions. The existence of these patterns is contingent upon the existence of eigenvalues of multiplicity higher than one. This is permissible only in Schrödinger's version of quantum mechanics. Presence of multiplicities is making quantum mechanical and topological entanglements equivalent.

quant-ph↗
arXiv2015

Black magic session of concordance: Regge mass spectrum from Casson's invariant

Recently, there had been a great deal of interest in obtaining and describing of all kinds of knots in links in hydrodynamics, electrodynamics, non Abelian gauge field theories and gravity. Although knots and links are observables of the Chern-Simons (C-S) functional, the dynamical conditions for their generation lie outside of the scope of the C-S theory. The nontriviality of dynamical generation of knotted structures is caused by the fact that the complements of all knots/links, say, in S^3 are 3-manifolds which have positive, negative or zero curvature. The ability to curve the ambient space thus far is attributed to masses. The mass theorem of general relativity requires the ambient 3-manifolds to be of non negative curvature. Recently, we established that, in the absence of boundaries, complements of dynamically generated knots/links are represented by 3-manifolds of non negative curvature. This fact opens the possibility to discuss masses in terms of dynamically generated knotted/ linked structures. The key tool is the notion of knot/link concordance. The concept of concordance is an adaptation of the concept of cobordism to knots and links. The logic of implementation of the concordance concept to physical masses results in new interpretation of Casson's surgery formula in terms of the Regge trajectories. The latest thoroughly examined Chew-Frautschi (C-F) plots associated with these trajectories demonstrate that the hadron mass spectrum for both mesons and baryons is nicely described by the data on the corresponding C-F plots. The physics behind Casson's surgery formula is similar but not identical to that described purely phenomenologically by Keith Moffatt in 1990. The developed topological treatment is fully consistent with available rigorous mathematical and experimentally observed results related to physics of hadrons

hep-th↗
arXiv2015

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)

math-ph↗
arXiv2015

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 Giné (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.

physics.gen-ph↗
arXiv2014

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 interactions

math-ph↗
arXiv2010

Gravity assisted solution of the mass gap problem for pure Yang-Mills fields

In 1979 Louis Witten demonstrated that stationary axially symmetric Einstein field equations and those for static axially symmetric self-dual SU(2) gauge fields can both be reduced to the same (Ernst) equation. In this paper we use this result as point of departure to prove the existence of the mass gap for quantum source-free Yang-Mills (Y-M) fields. The proof is facilitated by results of our recently published paper, JGP 59 (2009) 600-619. Since both pure gravity, the Einstein-Maxwell and pure Y-M fields are described for axially symmetric configurations by the Ernst equation classically, their quantum descriptions are likely to be interrelated. Correctness of this conjecture is successfully checked by reproducing (by different methods) results of Korotkin and Nicolai, Nucl.Phys.B475 (1996) 397-439, on dimensionally reduced quantum gravity. Consequently, numerous new results supporting the Faddeev-Skyrme (F-S) -type models are obtained. We found that the F-S-like model is best suited for description of electroweak interactions while strong interactions require extension of Witten's results to the SU(3) gauge group. Such an extension is nontrivial. It is linked with the symmetry group SU(3)xSU(2)xU(1) of the Standard Model. This result is quite rigid and should be taken into account in development of all grand unified theories. Also, the alternative (to the F-S-like) model emerges as by-product of such an extension. Both models are related to each other via known symmetry transformation. Both models possess gap in their excitation spectrum and are capable of producing knotted/linked configurations of gauge/gravity fields. In addition, the paper discusses relevance of the obtained results to heterotic strings and to scattering processes involving topology change. It ends with discussion about usefulness of this information for searches of Higgs boson.

hep-th↗
arXiv2010

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.

gr-qc↗
arXiv2009

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.

hep-th↗
arXiv2008

Veneziano Amplitudes, Spin Chains and String Models

In a series of recently published papers we reanalyzed the existing treatments of Veneziano and Veneziano-like amplitudes and the models associated with these amplitudes. In this work we demonstrate that the already obtained new partition function for these amplitudes can be exactly mapped into that for the Polychronakos-Frahm (P-F) spin chain model. This observation allows us to recover many of the existing string-theoretic models, including the most recent ones.

hep-th↗
arXiv2008

Quantum Signatures of Solar System Dynamics

Let w(i) be a period of rotation of the i-th planet around the Sun (or w(j;i) be a period of rotation of j-th satellite around the i-th planet). From empirical observations it is known that the sum of n(i)w(i)=0 (or the sum of n(j)w(j;i)=0) for some integers n(i) (or n(j)), different for different satellite systems. These conditions, known as resonance conditions, make uses of theories such as KAM difficult to implement. The resonances in Solar System are similar to those encountered in old quantum mechanics where applications of methods of celestial mechanics to atomic and molecular physics were highly sucsessful. With such a success, the birth of new quantum mechanics is difficult to understand. In short, the rationale for its birth lies in simplicity with which the same type of calculations are done using new methods capable of taking care of resonances. The solution of quantization puzzle was found by Heisenberg. In this paper new uses of Heisenberg's ideas are found. When superimposed with the equivalence principle of general relativity, they lead to quantum mechanical tratment of observed resonances in Solar System. To test correctness of our theoretical predictions the number of allowed stable orbits for planets and for equatorial stable orbits of satellites of heavy planets is calculated resulting in good agreement with observational data. In addition, the paper briefly discusses quantum mechanical nature of rings of heavy planets and potential usefulness of the obtained results for cosmology.

physics.gen-ph↗
arXiv2008

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.

hep-th↗
arXiv2008

Topological character of hydrodynamic screening in suspensions of hard spheres: an example of universal phenomenon

Although in the case of polymer solutions the existence of hydrodynamic screening is considered as established, use of the same methods for suspensions of hard spheres so far have failed to produce similar results. In this work we reconsider this problem. Using superposition of topological, combinatorial and London-style qualitative arguments, we prove the existence of screening in suspensions. We show that the nature of hydrodynamic screening in suspensions is analogous to that known for the Meissner effect in superconductors. The extent of screening depends on volume fraction of hard spheres. The zero volume fraction limit corresponds to the normal state. The case of finite volume fractions-to the mixed state typical for superconductors of the second kind. Such a state is becoming fully "superconducting" at some critical volume fraction for which the (zero frequency) relative viscosity diverges. Our analytical results describing this divergence are in accord with known scaling results obtained by Brady and Bicerano et al which are well supported by experimental data. We provide theoretical explanation of the divergence of relative viscosity in terms of a topological-type transition which mathematically can be made isomorphic to the more familiar Bose-Einstein condensation transition. Because of this, the methods developed in this work are not limited to suspensions only. In concluding section we mention other applications of the developed formalism ranging from turbulence and magnetohydrodynamics to high temperature superconductors, QCD, string models, etc.

cond-mat.soft↗