arXiv subjects

Vladimir V. Kassandrov

Publications and source records attributed to Vladimir V. Kassandrov.

At least 19 recordsLinked to original sources

arXiv2023

Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields

We briefly present our version of noncommutative analysis over matrix algebras, the algebra of biquaternions ($\mathbb B$) in particular. We demonstrate that any $\mathbb B$-differentiable function gives rise to a null shear-free congruence (NSFC) on the $\mathbb B$-vector space $\mathbb{C}\bf M$ and on its Minkowski subspace $\bf M$. Making use of the Kerr--Penrose correspondence between NSFC and twistor functions, we obtain the general solution to the equations of $\mathbb B$-differentiability and demonstrate that the source of an NSFC is, generically, a world sheet of a string in $\mathbb {C}\bf M$. Any singular point, caustic of an NSFC, is located on the complex null cone of a point on the generating string. Further we describe symmetries and associated gauge and spinor fields, with two electromagnetic types among them. A number of familiar and novel examples of NSFC and their singular loci are described. Finally, we describe a conservative algebraic dynamics of a set of identical particles on the ``Unique Worldline'' and discuss the connections of the theory with the Feynman--Wheeler concept of ``One-Electron Universe''

math-ph↗

arXiv2023

The algebrodynamics: super-conservative collective dynamics on a "Unique Worldline'' and the Hubble's law

We study the properties of roots of a polynomial system of equations which define a set of identical point particles located on a Unique Worldline (UW), in the spirit of the old Wheeler-Feynman's conception. As a consequence of the Vieta's formulas, a great number of conservation laws is fulfilled for collective algebraic dynamics on the UW. These, besides the canonical ones, include the laws with higher derivatives and those containing multi-particle correlation terms as well. On the other hand, such a ``super-conservative'' dynamics turns to be manifestly Lorentz invariant and quite nontrivial. At great values of ``cosmic time'' $t$ roots-particles demonstrate universal recession (resembling that in the Milne's cosmology and simulating ``expansion'' of the Universe) for which the Hubble's law does hold true, with Hubble parameter being inversely proportional to $t$.

gr-qc↗

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

Extended geometry and kinematics induced by biquaternionic and twistor structures

The algebra of biquaternions possess a manifestly Lorentz invariant form and induces an extended space-time geometry. We consider the links between this complex pre-geometry and real geometry of the Minkowski space-time. Twistor structures naturally arise in the framework of biquaternionic analysis. Both together, algebraic and twistor structures impose rigid restriction on the transport of singular points of biquaternion-valued fields identified with particle-like formations.

gr-qc↗

arXiv2022

New symmetries, conserved quantities and gauge nature of a free Dirac field

We present and amplify some of our previous statements on non-canonical interrelations between the solutions to free Dirac equation (DE) and Klein-Gordon equation (KGE). We demonstrate that all the solutions to the DE (possessing point- or string-like singularities) can be obtained via differentiation of a corresponding pair of the KGE solutions for a doublet of scalar fields. On this way we obtain a "spinor analogue" of the mesonic Yukawa potential and previously unknown chains of solutions to DE and KGE, as well as an exceptional solution to the KGE and DE with a finite value of the field charge ("localized" de Broglie wave). The pair of scalar "potentials" is defined up to a gauge transformation under which corresponding solution of the DE remains invariant. Under transformations of Lorentz group, canonical spinor transformations form only a subclass of a more general class of transformations of the solutions to DE upon which the generating scalar potentials undergo transformations of internal symmetry intermixing their components. Under continious turn by one complete revolution the transforming solutions, as a rule, return back to their initial values ("spinor two-valuedness" is absent). With arbitrary solution of the DE one can associate, apart of the standard one, a non-canonical set of conserved quantities, positive definite "energy" density among them, and with any KGE solution -- positive definite "probability density", etc. Finally, we discuss a generalization of the proposed procedure to the case when the external electromagnetic field is present

quant-ph↗

arXiv2021

On a Crucial Role of Gravity in the Formation of Elementary Particles

We consider the model of minimally interacting electromagnetic, gravitational and massive scalar fields free of any additional nonlinearities. In the dimensionless form, the Lagranginan contains only one parameter γ= Gm^2/e^2 which corresponds to the ratio of gravitational and electromagnetic interactions and, for a typical elementary particle, is about 10^-40. However, regular (soliton-like) solutions can exist only for γ\ne 0 so that gravity would be necessary to form the structure of an (extended) elementary particle. Unfortunately (in the stationary spherically symmetrical case), the numerical procedure breaks in the range γ\le 0.9 so that whether the particle-like solutions actually exist in the model remains unclear. Nonetheless, for γ\approx 1, we obtain, making use of the minimal energy requirement, a discrete set of (horizon-free) electrically charged regular solutions of the Planck's range mass and dimensions ("maximons", "planckeons", etc.). In the limit γ\to \infty the model reduces to the well known coupled system of the Einstein and Klein-Gordon equations. We obtain -- to our knowledge -- for the first time, the discrete spectrum of neutral soliton-like solutions ("mini-boson stars", "soliton stars", etc.) %

gr-qc↗

arXiv2021

Conservative relativistic algebrodynamics induced on an implicitly defined worldline

In the framework of the Stueckelberg-Wheeler-Feynman concept of a ``one-electron Universe'' we consider a worldline implicitly defined by a system of algebraic (precisely, polynomial) equations. Collection of pointlike ``particles'' of two kinds on the worldline (or its complex extension) is defined by the real (complex conjugate) roots of the polynomial system and detected then by an external inertial observer through the light cone connections. Then the observed collective dynamics of the particles' ensemble is, generally, subject to a number of Lorentz invariant conservation laws. Remarkably, this poperty follows from the Vieta's formulas for the roots of the generating polynomial system. At some discrete moments of the observer's proper time, mergings and subsequent transmutations of a pair of particles-roots take place simulating thus the processes of annihilation/creation of a particle/antiparticle pair

physics.gen-ph↗

arXiv2021

The hydrogen atom: consideration of the electron self-field

We substantiate the need for account of the proper electromagnetic field of the electron in the canonical problem of hydrogen in relativistic quantum mechanics. From mathematical viewpoint, the goal is equivalent to determination of the spectrum of everywhere regular solutions to the self-consistent system of Dirac and Maxwell equations (with external Coulomb potential). We demonstrate that only particular classes of solutions, "nonlinear" analogues of s- and p-states, can be obtained through decomposition of a solution in a series, with respect to the fine structure constant parameter $α$. In the zero approximation at $α\rightarrow 0$ the reduction to the self-consistent non-relativistic system of Schrödinger-Poisson equations takes place. For the latter, using both numerical and variational methods, we obtain the solutions corresponding to the ground and set of excited states. Spectrum of the binding energies with remarkable precision reproduces the "Bohrian" dependence $W_n = W/ n^2$. For this, the ionization energy $W$ proves to be universal yet about two times smaller than its observed value. Possibility of the renormalization procedure and the problem of account for relativistic corrections to the binding energies of order $α^2$ are considered

quant-ph↗

arXiv2020

Relativistic generalization of the Schrodinger-Newton model for the wavefunction reduction

We consider the model of the self-gravity driven spontaneous wavefunction reduction proposed by L. Diosi, R. Penrose et al. and based on a self-consistent system of the Schrodinger and Poisson equations. An analogous system of coupled Dirac and Maxwell-like equations is proposed as a relativization. Regular solutions to the latter form a discrete spectrum in which all the "active" gravitational masses are always positive, and approximately equal to inertial masses and to the mass $m$ of the quanta of Dirac field up to the corrections of order $α^2$. Here $α=(m/M_{pl})^2$ is the gravitational analogue of the fine structure constant negligibly small for nucleons. In the limit $α\to 0$ the model reduces back to the nonrelativistic Schrodinger-Newton one. The equivalence principle is fulfilled with an extremely high precision. The above solutions correspond to various states of the same (free) particle rather than to different particles. These states possess a negligibly small difference in characteristics but essentially differ in the widths of the wavefunctions. For the ground state the latter is $α$ times larger the Compton length, so that a nucleon cannot be sufficiently localized to model the reduction process

gr-qc↗

arXiv2018

Twistor structures and boost-invariant solutions to field equations

We give a brief overview of a non-Lagrangian approach to field theory based on a generalization of the Kerr-Penrose theorem and algebraic twistor equations. Explicit algorithms for obtaining the set of fundamental (Maxwell, SL(2, C)-Yang-Mills, spinor Weyl and curvature) fields associated with every solution of the basic system of algebraic equations are reviewed. The notion of a boost-invariant solution is introduced, and the unique axially-symmetric and boost-invariant solution which can be generated by twistor functions is obtained, together with the associated fields. It is found that this solution possesses a wide variety of point-, string- and membrane-like singularities exhibiting nontrivial dynamics and transmutations.

gr-qc↗

arXiv2018

Self-Dual Connections and the Equations of Fundamental Fields in a Weyl-Cartan Space

Spaces with a Weyl-type connection and torsion of a special type induced by the structure of the differentiability conditions in the algebra of complex quaternions are considered. The consistency of these conditions implies the self-duality of curvature. The Maxwell and SL(2, C) Yang-Mills fields associated with the irreducible components of the connection also turn out to be self-dual, so that the corresponding equations are fulfilled on the solutions of the generating system. Using the twistor structure of the latter, its general solution is obtained. The singular locus has a string-like (particle-like) structure and generates the self-consistent algebraic dynamics of the string system.

gr-qc↗

arXiv2017

Gauge origin of the Dirac field and singular solutions to the Dirac equation

We present and enhance our previous statements (arXiv:0907.4736) on the non-canonical interrelations between the solutions to the free Dirac equation (DE) and the Klein-Gordon equation (KGE). We demonstrate that all the solutions to the DE (singular ones among them) can be obtained via differentiation of a pair of the KGE solutions for a doublet of scalar fields. In this way we obtain a "spinor analogue" of the mesonic Yukawa potential and previously unknown chains of solutions to the DE and KGE. The pair of scalar "potentials" is defined up to a gauge transformation under which the corresponding solution of the DE is invariant. Under transformations of the Lorentz group, canonical spinor transformations form only a subclass of more general transformations of the DE solutions, under these the generating scalar potentials undergo transformations of internal symmetry intermixing their components. In particular, under continuous rotation through one full revolution the transforming solutions, as a rule, take their original values ("spinor two-valuedness" is absent). With an arbitrary solution of the DE one can associate, apart from the standard one, a non-canonical set of conserved quantities, a positive definite "energy" density among others, and with any KGE solution -- a positive definite "probability density", etc. Finally, we discuss a generalization of the proposed procedure to the case when an external electromagnetic field is present.

physics.gen-ph↗

arXiv2017

Three kinds of particles on a single rationally parameterized worldline

We consider the light cone (`retardation') equation (LCE) of an inertially moving observer and a single worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter property follows directly from the structure of Vieta formulas for the LCE roots. In particular, two Lorentz invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued. Asymptotically, at large values of the observer's proper time, one distinguishes three types of the LCE roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles can assemble into compact large groups - clusters. Throughout the paper, we make no use of differential equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic

physics.gen-ph↗

arXiv2016

Maxwell, Yang-Mills, Weyl and eikonal fields defined by any null shear-free congruence

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, $SL(2,\mathbb C)$ Yang-Mills and complex Maxwell fields, the latter produced by integer-valued electric charges ("elementary" for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding "particle-like" field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit non-trivial collective dynamics simulating physical interactions

gr-qc↗

arXiv2016

Relativistic Algebra of Space-Time and Algebrodynamics

We consider a manifestly Lorentz invariant form $\mathbb L$ of the biquaternion algebra and its generalization to the case of curved manifold. The conditions of $\mathbb L$-differentiability of $\mathbb L$-functions are formulated and considered as the primary equations for fundamental fields modeled with such functions. The exact form of the effective affine connection induced by $\mathbb L$-differentiability equations is obtained for the flat and curved cases. In the flat case, the integrability conditions of the latter lead to the self-duality of the corresponding curvature, thus ensuring that the source-free Maxwell and $SL(2,\mathbb C)$ Yang-Mills equations hold on the solutions of the $\mathbb L$-differentiability equations

gr-qc↗

arXiv2015

Regularized Lienard-Wiehert fields in a space with torsion

A natural modification of the equations of covariantly-constant vector fields (CCVF) in Weyl geometry leads us to consider a metric compatible geometry possessing conformal curvature and torsion fully determined by its trace. The latter is interpreted as a form of the electromagnetic (EM) 4-potentials and, on a fixed metric background, turns out to be fully determined by the CCVF equations. When the metric is set Minkowskian, the named equations possess two topologically distinct solutions, with the associated EM fields being asymptotically of the Lienard-Wiechert type and having distributed sources, with a fixed (``elementary'') value of the electric charge. One of the solutions is everywhere regular, whereas the other - singular on a 2-dimensional shell. The speed of propagation of EM fields depends on the local charge density and only asymptotically approaches the speed of light

gr-qc↗

arXiv2015

Collective Lorentz invariant dynamics on a single "polynomial" worldline

Consider a worldline of a pointlike particle parametrized by polynomial functions, together with the light cone ("retardation") equation of an inertially moving observer. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the retardation equation will be detected by the observer. We prove that for any "polynomial" worldline the induced collective dynamics of the system of R-C particles obeys a whole set of canonical conservation laws (for total momentum, angular momentum and the analogue of mechanical energy). Explicit formulas for the values of total angular momentum and the analogue of total rest energy (rest mass) are obtained; the latter is "self-quantized", i.e. for any worldline takes only integer values. The dynamics is Lorentz invariant though different from the canonical relativistic mechanics. Asymptotically, at large values of the observer's proper time, the R-C particles gather themselves into pairs and then assemble into compact incoming/outgoing clusters. As a whole, the evolution resembles the process of (either elastic or inelastic) scattering of a beam of composite particles. Throughout the paper the consideration is purely algebraic, with no resort to differential equations of motion, field equations, etc.

physics.gen-ph↗

arXiv2014

Double gauge invariance and covariantly-constant vector fields in Weyl geometry

The wave equation and equations of covariantly-constant vector fields (CCVF) in spaces with Weyl nonmetricity turn out to possess, in addition to the canonical conformal-gauge, a gauge invariance of another type. On a Minkowski metric background, the CCVF system alone allows us to pin down the Weyl 4-metricity vector, identified herein with the electromagnetic potential. The fundamental solution is given by the ordinary Lienard-Wiechert field, in particular, by the Coulomb distribution for a charge at rest. Unlike the latter, however, the magnitude of charge is necessarily unity, "elementary", and charges of opposite signs correspond to retarded and advanced potentials respectively, thus establishing a direct connection between the particle/antiparticle asymmetry and the "arrow of time".

gr-qc↗