arXiv subjects

Vladimir V. Kassandrov

Publications and source records attributed to Vladimir V. Kassandrov.

33 records · Page 2Linked to original sources

arXiv2013

Algebraic roots of Newtonian mechanics: correlated dynamics of particles on a unique worldline

In the development of the old ideas of Stueckelberg-Wheeler-Feynman on the "one-electron Universe", we study the purely algebraic dynamics of the ensemble of(two kinds of) identical point-like particles. These are represented by the(real and complex conjugate) roots of a generic polynomial system of equations that implicitly defines a single "worldline". The dynamics includes events of "merging" of a pair of particles modelling the annihilation/creation processes. Correlations in the location and motion of the particles-roots relate, in particular, to the Vieta formulas. After a special choice of the inertial-like reference frame, the linear Vieta formulas guarantee that, for any worldline, the law of (non-relativistic) momentum conservation is identically satisfied. Thus, the general structure of Newtonian mechanics follows from the algebraic properties of a worldline alone. Some considerations on relativization of the scheme are presented. A simple example of, unexpectedly rich, "polynomial dynamics" is retraced in detail and illustrated via an animation(available from ancillary file enclosed)

physics.gen-ph↗

arXiv2011

Electromagnetic Fields in Kerr-Shild Space-Times

Making use of twistor structures and the Kerr theorem for shear-free null geodesic congruences, an infinite family of electromagnetic fields satisfying the homogeneous Maxwell equations in flat Minkowski and the associated curved Kerr-Schild backgrounds is obtained for any such congruence in a purely algebraic way. Simple examples of invariant axisymmetric Maxwell fields are presented.

gr-qc↗

arXiv2011

Dimerous Electron and Quantum Interference beyond the Probability Amplitude Paradigm

We generalize the formerly proposed relationship between a special complex geometry (originating from the structure of biquaternion algebra) and induced real geometry of (extended) space-time. The primordial dynamics in complex space allows for a new realization of the "one electron Universe" of Wheeler-Feynman (the so called "ensemble of duplicons") and leads to a radical concept of "dimerous" (consisting of two identical matter pre-elements, "duplicons") electron. Using this concept, together with an additional phase-like invariant (arising from the complex pre-geometry), we manage to give a visual classical explanation for quantum interference phenomena and, in particular, for the canonical two-slit experiment. Fundamental relativistic condition of quantum interference generalizing the de Broglie relationship is obtained, and an experimentally verifiable distinction in predictions of quantum theory and presented algebrodynamical scheme is established.

physics.gen-ph↗

arXiv2009

A nonstationary generalization of the Kerr congruence

Making use of the Kerr theorem for shear-free null congruences and of Newman's representation for a virtual charge ``moving'' in complex space-time, we obtain an axisymmetric time-dependent generalization of the Kerr congruence, with a singular ring uniformly contracting to a point and expanding then to infinity. Electromagnetic and complex eikonal field distributions are naturally associated with the obtained congruence, with electric charge being necesssarily unit (``elementary''). We conjecture that the corresponding solution to the Einstein-Maxwell equations could describe the process of continious transition of the naked ringlike singularitiy into a rotating black hole and vice versa, under a particular current radius of the singular ring.

gr-qc↗

arXiv2009

On Hierarchy and Equivalence of Relativistic Equations for Massive Fields

A non-canonical correspondence of the complete sets of solutions to the Dirac and Klein-Gordon free equations in Minkowski space-time is established. This allows for a novel viewpoint on the relationship of relativistic equations for different spins and on the origin of spinor transformations. In particular, starting from a solution to the Dirac equation, one obtains a chain of other solutions to both Dirac and Klein-Gordon equations. A comparison with the massless case is performed, and examples of non-trivial singular solutions are presented. A generalization to Riemannian space-time and inclusion of interactions are briefly discussed.

gr-qc↗

arXiv2009

Algbrodynamics over complex space and phase extension of the Minkowski geometry

First principles should predetermine physical geometry and dynamics both together. In the "algebrodynamics" they follow solely from the properties of the biquaternion algebra $\B$ and the analysis over $\B$. We briefly present the algebrodynamics on the Minkowski background based on a nonlinear generalization to $\B$ of the Cauchi-Riemann analyticity conditions. Further, we consider the effective real geometry uniquely resulting from the structure of multiplication in $\B$ which turns out to be of the Minkowski type, with an additional phase invariant. Then we pass to study the primordial dynamics that takes place in the complex $\B$ space and brings into consideration a number of remarkable structures: an ensemble of identical correlated matter pre-elements ("duplicons"), caustic-like signals (interaction carriers), a concept of random complex time resulting in irreversibility of physical time at a macrolevel, etc. In partucular, the concept of "dimerous electron" naturally arises in the framework of complex algebrodynamics and, together with the above-mentioned phase invariant, allows for a novel approach to explanation of quantum interference phenomena alternative to the recently accepted paradigm of wave-particle dualism.

physics.gen-ph↗

arXiv2007

Quaternionic Analysis and the Algebrodynamics

We present the ``algebrodynamical'' approach to field-particle theory based on a nonlinear generalization of the Cauchy-Riemann conditions to non-commutative algebras of quaternion-like type. For complex quaternions the theory is Lorentz invariant and naturally carries some gauge and twistor structures. Point- and string-like singularities are considered as particle-like formations; their electric charge is ``self-quantized''. A novel ``causal Minkowski geometry with additional phase'' is presented that is induced by the structure of primordial biquaternion algebra. On this geometrical background a self-consistent algebraic dynamics of singularities (``ensemble of dublicons'') is briefly discussed.

math-ph↗

arXiv2006

Algebrodynamics in complex space-time and the complex-quaternionic origin of Minkowski geometry

We present a scheme of biquaternionic algebrodymamics based on a nonlinear generalization of the Cauchy-Riemann holomorphy conditions considered therein as fundamental field equations. The automorphism group SO(3,C) of the biquaternion algebra acts as a proper Lorentz group on a real space whose coordinates are bilinear in the complex coordinates of biquaternionic vector space. A new invariant of Lorentz transformations then arises - the geometric phase. This invariant can be responsible for the quantum properties of particles associated in this approach with field singularities. Some new notions are introduced, related to ``hidden'' complex dynamics: ``observable'' space-time, the ensemble of identical correlated particles-singularities (``duplicons'') and others.

gr-qc↗

arXiv2006

Twistor algebraic dynamics in complex space-time and physical meaning of hidden dimensions

We review the algebraic field theory based completely on a nonlinear generalization of the CR complex analiticity conditions to the noncommutative algebra of biquaternions. Any biquaternionic field possesses natural twistor structure and, in the Minkowski space, gives rise to a shear-free null congruence of rays and to a set of gauge fields associated with it. In the paper we develop the algebrodynamical scheme in the complex extension of Minkowski space-time -- in the full vector space of biquaternion algebra. This primodial space dynamically reduces to the 6D ``observable'' space-time of the complex null cone which in turn decomposes into the 4D physical space-time and the 2D internal ``spin space''. A set of identical point charges (``duplicons'') -- focal points of the congruence -- arises in the procedure. Temporal dynamics of individual duplicons is strongly correlated via fundamental twistor field of the congruence. We briefly discuss some new notions inevitably arising in the considered algebrodynamical scheme, namely those of ``complex time'' and of ``evolutionary curve'', as well as their hypothetical links to the quantum uncertainty phenomenon.

gr-qc↗

arXiv2006

On the structure of general solution to the equations of shear-free null congruences

We demonstrate that the source of a shear-free null congruence is, generically, a world sheet of a singular string in the complex Minkowski space. In particular cases the string degenerates into the Newman's point singularity. We describe also a time-dependent deformation of the Kerr's congruence which could indicate the unstability of the Kerr (Kerr-Newman) (electro)vacuum solution. Then we consider a 5D extension of Minkowski space obtained by complexification of the time coordinate. Singular locus of a shear-free null congruence on such an extended space-time consists of a (great) number of point particles (``markeons'') and 2D surfaces -- light-like wave fronts.

gr-qc↗

arXiv2004

Gauge fields and particle-like formations associated with shear-free null congruences

We consider Maxwell fields associated with any shear-free null geodesic congruence on Minkowski or Riemannian background space-time. Bounded singular loci of these fields are treated as particle-like formations, possess "self-quantized" electric charge and undergo self-consistent time dynamics. Complicated singular solutions of Maxwell (as well as related complex Yang-Mills) equations can be obtained in a purely algebraic way using the Kerr theorem.

gr-qc↗

arXiv2003

Nature of Time and particles-caustics: physical World in algebrodynamics and in twistor theory

In the field theories with twistor structure particles can be identified with (spacially bounded) caustics of null geodesic congruences defined by the twistor field. As a realization, we consider the ``algebrodynamical'' approach based on the field equations which originate from noncommutative analysis (over the algebra of biquaternions) and lead to the complex eikonal field and to the set of gauge fields associated with solutions of the eikonal equation. Particle-like formations represented by singularities of these fields possess ``elementary'' electric charge and other realistic ``quantum numbers'' and manifest self-consistent time evolution including transmutations. Related concepts of generating ``World Function'' and of multivalued physical fields are discussed. The picture of Lorentz invariant light-formed aether and of matter born from light arises then quite naturally. The notion of the Time Flow identified with the flow of primodial light (``pre-Light'') is introduced in the context. Popularization and development of the paper math-ph/0311006.

hep-th↗

arXiv2003

General Solution of the Complex 4-Eikonal Equation and the "Algebrodynamical" Field Theory

We explicitly demonstrate the existence of twistor and ambitwistor structure for the 4-dimensional complexified eikonal equation (CEE) and present its general solution consisting of two different classes. For both, every solution can be obtained from a generating twistor function in a purely algebraic way, via the procedure similar to that used in the Kerr theorem for shear-free null congruences. Bounded singularities of eikonal or of its gradient define some particle-like objects with nontrivial characteristics and dynamics. Example of a new static solution to CEE with a ring-like singularity is presented, and general principles of algebraic field theory - algebrodynamics - closely related to CEE are briefly discussed.

math-ph↗

arXiv2003

Singular Sources of Maxwell Fields with Self-Quantized Electric Charge

Single- and multi-valued solutions of homogeneous Maxwell equations in vacuum are considered, with ''sources'' formed by the (point- or string-like) singularities of the field strengths and, generally, irreducible to any delta-functions' distribution. Maxwell equations themselves are treated as consequences (say, integrability conditions) of a primary ``superpotential'' field subject to some nonlinear and over-determined constraints (related, in particular, to twistor structures). As the result, we obtain (in explicit or implicit algebraic form) a distinguished class of Maxwell fields, with singular sources necessarily carrying a ``self-quantized'' electric charge integer multiple to a minimal ``elementary'' one. Particle-like singular objects are subject to the dynamics consistent with homogeneous Maxwell equations and undergo transmutations -- bifurcations of different types. The presented scheme originates from the ``algebrodynamical'' approach developed by the author and reviewed in the last section. Incidentally, fundamental equivalence relations between the solutions of Maxwell equations, complex self-dual conditions and of Weyl ``neutrino'' equations are established, and the problem of magnetic monopole is briefly discussed.

physics.class-ph↗

arXiv2000

Twistor and "weak" gauge structures in the framework of quaternionic analysis

The earlier proposed conditions of (bi)quaternionic differentiability are nonlinear, give rize to the 2-spinor and the self-dual gauge structures and may be considered as the it generating system of equations (GSE) with respect to the source-free Maxwell, Yang-Mills and eikonal equations. We present the general solution of the GSE in terms of twistor variables, analize its rather specific gauge symmetry and demonstrate the relation of GSE to the equations of shear-free null congruences and, consequently, - to effective metrics of Kerr- Shild type. The concept of particles as singularities of physical fields associated with the solutions of GSE is developed

gr-qc↗