arXiv subjects

Zurab K. Silagadze

Publications and source records attributed to Zurab K. Silagadze.

At least 19 recordsLinked to original sources

arXiv2026

Pauli equation in spaces of constant curvature and extended Nikiforov-Uvarov method

We apply the extended Nikiforov-Uvarov method to the non-relativistic limit of the Dirac equation with a Coulomb potential in spaces of constant curvature. In this case, the radial equation reduces to the Heun equation, and the extended Nikiforov-Uvarov method easily yields a quantization condition which leads to necessary condition under which the resulting Heun equation can have polynomial solutions. The energy spectrum implied by the quantization condition is virtually identical to the spectrum of a spinless particle obtained using the Schrödinger equation, except for the absence of the ``geometric potential", confirming the non-commutativity of the naive non-relativistic limit with the ``squaring" of the Dirac equation, first discovered on curved surfaces. However, the necessary conditions for the existence of polynomial solutions cannot be met, and this fact undermines the reliability of the results obtained. This circumstance forces us to conclude that the extended Nikiforov-Uvarov method has limited, if any, value when considering similar problems in quantum mechanics.

quant-ph↗

arXiv2026

Chukchi Myths perspective on Special Relativity

The teaching of special relativity still follows Einstein's original two-postulate approach and thus recreates the relativistic revolution in the minds of students again and again, with all its attendant shocking and mysterious aspects. As Hermann Bondi long ago noted, such an approach, which emphasizes the revolutionary aspects of a theory rather than its continuity with earlier thought, "is hardly conducive to easy teaching and good understanding". But what could be a better alternative? In 1923, the distinguished Russian ethnographer, linguist, and anthropologist Tan-Bogoraz described the striking similarities between the special theory of relativity and the mythology of Chukchi shamans. Inspired by this surprising observation, I assume that the basic concepts of relativity are not at all alien to our innate perception of time and space, and I propose an approach to the foundations of relativity that emphasizes absolute concepts such as proper time and causal cones rather than relative ones.

physics.gen-ph↗

arXiv2026

Astrophysical positronium and Dicke superradiance

Dicke superradiance is a fascinating phenomenon in which a large number of atoms cooperate to produce a brief and very intense burst of spontaneous emission. This phenomenon has been well studied in the laboratory, but its astrophysical aspects have only recently attracted the attention of a small number of researchers. Since the phenomenon of Dicke superradiance is relatively little known to the wider astrophysical community, we provide a fairly detailed review of its elementary theory in the appendix and speculate on the significance of superradiance for astrophysical hydrogen and positronium, given the abundant formation of the latter near the galactic center.

astro-ph.HE↗

arXiv2025

Asymptotic potential of a rose-shaped disk

Based on extensive numerical evidence, a recent paper (Sheng Chen et al., Eur. J. Phys. {\bf 45} (2024), 045703) suggested that the potential of a uniformly charged rose-shaped disk in the plane of the disk has a simple asymptotic form. We present an analytical proof of this interesting conjecture.

physics.class-ph↗

arXiv2025

Elementary atoms in spaces of constant curvature by the Nikiforov-Uvarov method

The Nikiforov-Uvarov method is a simple, yet elegant and powerful method for solving second-order differential equations of generalized hypergeometric type. In the past, it has been used to solve many problems in quantum mechanics and elsewhere. We apply this method to the classical problem of hydrogen-like atoms in spaces of constant curvature. Both the spectra of these atoms and their wave functions, including normalization, are easily obtained.

quant-ph↗

arXiv2024

Electrostatic potential of a uniformly charged disk through Green's theorem

All existing derivations of the electrostatic potential of a uniformly charged disk are technically rather involved. In an old and now almost forgotten publication, Duffin and McWhirter proposed a method for calculating the electrostatic potentials of planar bodies by a skillful application of Green's theorem. It is shown that this method significantly simplifies the problem of calculating the electrostatic potential of a uniformly charged disk and makes it an almost trivial task.

physics.class-ph↗

arXiv2024

Analytical solution of the Sommerfeld-Page equation

The Sommerfeld-Page equation describes the non-relativistic dynamics of a classical electron modeled by a sphere of finite size with a uniform surface charge density. The Sommerfeld-Page equation is a delay differential equation, and almost no exact results on solutions of this equation was known. However, the progress has been made, especially in the last few years, and recently an analytical solution was found for an almost identical delay differential equation, which arose in the context of mathematical modeling of COVID-19 epidemics. Inspired by this research, we offer a pedagogical exposition of how one can find an analytical solution of the Sommerfeld-Page equation.

physics.class-ph↗

arXiv2024

Transition of dimuonium through foil

This article presents a study of the passage of dimuonium through the foil of ordinary matter. First, we provide an overview of how dimuonium is planned to be produced for such a type of experiment and how it is expected to interact with the ordinary atoms -- predominantly electromagnetically via the screened coulomb potential of the atomic nuclei. Then, we describe the transport equations that represent the evolution of dimuonium states during the passage and their solution methods. Finally, for three different foils (Beryllium, Aluminium and Lead), we present the results of this study. To estimate impact of uncertainties in the potential of a target atom, we study 15 different approximations of the atomic potential and show that the corresponding atomic-potential-model-dependent error in the yields of the low lying states of dimuonium is quite small within the framework of the applied Born approximation. The convergence of the results after truncation of the infinite system of transport equations to the finite number of quantum states of dimuonium is also studied, and good convergence for the yields of low-lying states is demonstrated.

hep-ph↗

arXiv2024

Majorana transformation of the Thomas-Fermi equation demystified

The Majorana transformation makes it possible to reduce the Thomas-Fermi equation to a first-order differential equation. This reduction is possible due to the special scaling property of the Thomas-Fermi equation under homology transformations. Such reductions are well known in the context of stellar astrophysics, where the use of homology-invariant variables has long proved useful. We use homology-invariant variables in the context of the Thomas-Fermi equation to demystify the origin of the otherwise mysterious Majorana transformation.

math.AP↗

arXiv2024

Berry phase of spin-one system in a rotating electric field

We consider in sufficient detail how the Berry phase arises in a rotating electric field in a model system with spin one. The goal is to help the student who first encountered this interesting problem, which is fraught with some subtleties that require attention in order not to go astray.

quant-ph↗

arXiv2024

Lévy-Leblond Equation and Eisenhart-Duval lift in Koopman-von Neumann Mechanics

The Koopman-von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. In Eur. Phys. J. Spec. Top. {\bf 227}, 2195 (2019) it was argued that the Spohn equation [Ann. Phys. {\bf 282}, 420 (2000)] is the correct classical framework for the Koopman-von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond's seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman-von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart-Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

physics.class-ph↗

arXiv2023

Evading Quantum Mechanics à la Sudarshan: quantum-mechanics-free subsystem as a realization of Koopman-von Neumann mechanics

Tsang and Caves suggested the idea of a quantum-mechanics-free subsystem in 2012. We contend that Sudarshan's viewpoint on Koopman-von Neumann mechanics is realized in the quantum-mechanics-free subsystem. Since quantum-mechanics-free subsystems are being experimentally realized, Koopman-von Neumann mechanics is essentially transformed into an engineering science.

quant-ph↗

arXiv2022

Hourglass of constant weight as an illustrative example of a system with a variable mass

We provide an alternative approach to a problem posed in the article "Hourglass of Constant Weight" by Volker Becker and Thorsten Pöschel, which appeared in Granular Matter 10 (2008), 231-232, i.e. 14 years ago. Our goal to return to this article after so much time is purely pedagogical. Although there is a slight and subtle inaccuracy in the derivation of equation (8) in the work of Becker and Pöschel, which we correct, we use this circumstance only to give a detailed pedagogical exposition of an excellent physical problem and to show the pedagogical usefulness of the concept of momentum flux when considering the mechanics of systems with variable mass.

physics.ed-ph↗

arXiv2022

Eisenhart lift of Koopman-von Neumann mechanics

The Eisenhart lift establishes a fascinating connection between non-relativistic and relativistic physics, providing a space-time geometric understanding of non-relativistic Newtonian mechanics. What is still little known, however, is the fact that there is a Hilbert space representation of classical mechanics (also called Koopman-von Neumann mechanics) that attempts to give classical mechanics the same mathematical structure that quantum mechanics has. In this article, we geometrize the Koopman-von Newmann (KvN) mechanics using the Eisenhart toolkit. We then use a geometric view of KvN mechanics to find transformations that relate the harmonic oscillator, linear potential, and free particle in the context of KvN mechanics.

gr-qc↗

arXiv2022

Do moving clocks slow down?

The special theory of relativity has fundamentally changed our views of space and time. The relativity of simultaneity in particular, and the theory of relativity as a whole, still presents significant difficulty for beginners in the theory. The difficulty stems from the fact that the usual presentation of special relativity is based on Newtonian concepts, which are relativized and change their meaning in the course of the presentation. A better pedagogical practice, in our opinion, would be to base the presentation from the very beginning on the four-dimensional formulation of Minkowski and thus remove from the theory all the paradoxical connotations that invariably accompany the usual presentation of special relativity.

physics.class-ph↗

arXiv2022

On Finslerian extension of special relativity

We demonstrate that Robb-Geroch's definition of a relativistic interval admits a simple and fairly natural generalization leading to a Finsler extension of special relativity. Another justification for such an extension goes back to the works of Lalan and Alway and, finally, was put on a solid basis and systematically investigated by Bogoslovsky under the name "Special-relativistic theory of locally anisotropic space-time". The isometry group of this space-time, $\mathrm{DISIM}_b(2)$, is a deformation of the Cohen and Glashow's very special relativity symmetry group $\mathrm{ISIM(2)}$. Thus, the deformation parameter b can be regarded as an analog of the cosmological constant characterizing the deformation of the Poincare group into the de Sitter (anti-de Sitter) group. The simplicity and naturalness of Finslerian extension in the context of this article adds weight to the argument that the possibility of a nonzero value of $b$ should be carefully considered.

physics.class-ph↗

arXiv2021

Non-local imprints of gravity on quantum theory

During the last two decades or so much effort has been devoted to the discussion of quantum mechanics (QM) that in some way incorporates the notion of a minimum length. This upsurge of research has been prompted by the modified uncertainty relation brought about in the framework of string theory. In general, the implementation of minimum length in QM can be done either by modification of position and momentum operators or by restriction of their domains. In the former case we have the so called soccer-ball problem when the naive classical limit appears to be drastically different from the usual one. Starting with the latter possibility, an alternative approach was suggested in the form of a band-limited QM. However, applying momentum cutoff to the wave-function, one faces the problem of incompatibility with the Schrödinger equation. One can overcome this problem in a natural fashion by appropriately modifying Schrödinger equation. But incompatibility takes place for boundary conditions as well. Such wave-function cannot have any more a finite support in the coordinate space as it simply follows from the Paley-Wiener theorem. Treating, for instance, the simplest quantum-mechanical problem of a particle in an infinite potential well, one can no longer impose box boundary conditions. In such cases, further modification of the theory is in order. We propose a non-local modification of QM, which has close ties to the band-limited QM, but does not require a hard momentum cutoff. In the framework of this model, one can easily work out the corrections to various processes and discuss further the semi-classical limit of the theory.

quant-ph↗

arXiv2021

What sunspots are whispering about covid-19?

Several studies point to the antimicrobial effects of ELF electromagnetic fields. Such fields have accompanied life from the very beginning, and it is possible that they played a significant role in its emergence and evolution. However, the literature on the biological effects of ELF electromagnetic fields is controversial, and we still lack an understanding of the complex mechanisms that make such effects, observed in many experiments, possible. The Covid-19 pandemic has shown how fragile we are in the face of powerful processes operating in the biosphere. We believe that understanding the role of ELF electromagnetic fields in regulating the biosphere is important in our fight against Covid-19, and research in this direction should be intensified.

physics.gen-ph↗