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

Abstract

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)

22 pages, 10 figures and 1 animation file. Reorganized, references and citations added, put in partial correspondence with the printed version

References in corpus (4)

• Twistorial Analyticity and Three Stringy Systems of the Kerr Spinning Particle
• The Kerr theorem and multiparticle Kerr-Schild solutions
• Algbrodynamics over complex space and phase extension of the Minkowski geometry
• Complex Structure of the Four-Dimensional Kerr Geometry: Stringy System, Kerr Theorem, and Calabi-Yau Twofold

Cited by in corpus (6)

• Maxwell, Yang-Mills, Weyl and eikonal fields defined by any null shear-free congruence
• Collective Lorentz invariant dynamics on a single "polynomial" worldline
• Double gauge invariance and covariantly-constant vector fields in Weyl geometry
• Three kinds of particles on a single rationally parameterized worldline
• The algebrodynamics: super-conservative collective dynamics on a "Unique Worldline'' and the Hubble's law
• Algebrodynamics: Shear-Free Null Congruences and New Types of Electromagnetic Fields