Conservative relativistic algebrodynamics induced on an implicitly defined worldline

TL;DR

This paper explores a model where particles are represented as roots of polynomial equations defining a worldline, revealing Lorentz-invariant conservation laws and particle interactions akin to creation and annihilation events.

Contribution

It introduces a novel algebraic approach to relativistic particle dynamics using polynomial roots and demonstrates emergent conservation laws and particle transmutations.

Findings

Roots of polynomial systems define particle positions and types.

Conservation laws follow from Vieta's formulas.

Particle pair creation and annihilation are simulated by root mergers.

Abstract

In the framework of the Stueckelberg-Wheeler-Feynman concept of a ``one-electron Universe'' we consider a worldline implicitly defined by a system of algebraic (precisely, polynomial) equations. Collection of pointlike ``particles'' of two kinds on the worldline (or its complex extension) is defined by the real (complex conjugate) roots of the polynomial system and detected then by an external inertial observer through the light cone connections. Then the observed collective dynamics of the particles' ensemble is, generally, subject to a number of Lorentz invariant conservation laws. Remarkably, this poperty follows from the Vieta's formulas for the roots of the generating polynomial system. At some discrete moments of the observer's proper time, mergings and subsequent transmutations of a pair of particles-roots take place simulating thus the processes of annihilation/creation of a particle/antiparticle pair