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

TL;DR

This paper explores how algebraic properties of polynomial roots defining a worldline can naturally produce Newtonian mechanics and particle interactions, including creation and annihilation, within a purely algebraic and geometric framework.

Contribution

It demonstrates that Newtonian mechanics and particle correlations emerge from algebraic relations of polynomial roots representing a single worldline.

Findings

Vieta formulas ensure momentum conservation in the algebraic model

Particle creation and annihilation are modeled as root merging events

Polynomial dynamics exhibit rich behaviors, illustrated through examples

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)