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. A simple example of, unexpectedly rich, 'polynomial dynamics' is retraced in detail and illustrated via an animation (available from stacks.iop.org/JPhysA/46/175206/mmedia). (paper)