Algebraic dynamics on a single worldline: Vieta formulas and conservation laws

TL;DR

This paper develops an algebraic framework for modeling a universe of identical particles on a single worldline, revealing conservation laws through polynomial root relations without using traditional differential equations.

Contribution

It introduces a novel algebraic approach to particle dynamics on a worldline, deriving conservation laws from polynomial root properties instead of differential equations.

Findings

Conservation laws for momentum, angular momentum, and energy are derived algebraically.

Transitions between real and complex roots model particle-antiparticle creation and annihilation.

The approach applies to a wide class of polynomial systems with time-dependent coefficients.

Abstract

In development of the old conjecture of Stuckelberg, Wheeler and Feynman on the so-called "one electron Universe", we elaborate a purely algebraic construction of an ensemble of identical pointlike particles occupying the same worldline and moving in concordance with each other. In the proposed construction one does not make use of any differential equations of motion, Lagrangians, etc. Instead, we define a "unique" worldline implicitly, by a system of nonlinear polynomial equations containing a time-like parameter. Then at each instant there is a whole set of solutions defining the coordinates of particles-copies localized on the unique worldline and moving along it. There naturally arise two different kinds of such particles which correspond to real or complex conjugate roots of the initial system of polynomial equations, respectively. At some particular time instants, one encounters the transitions between these two kinds of particles-roots that model the processes of annihilation or creation of a pair "particle-antiparticle". We restrict by consideration of nonrelativistic collective dynamics of the ensemble of such particles on a plane. Making use of the techniques of resultants of polynomials, the generating system reduces to a pair of polynomial equations for one unknown, with coefficients depending on time. Then the well-known Vieta formulas predetermine the existence of time-independent constraints on the positions of particles-roots and their time derivatives. We demonstrate that for a very wide class of the initial polynomials (with polynomial dependence of the coefficients on time) these constraints always take place and have the form of the conservation laws for total momentum, angular momentum and (the analogue of) total mechanical energy of the "closed" system of particles.