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)