# Three kinds of particles on a single rationally parameterized worldline

## TL;DR

This paper introduces a Lorentz-invariant algebraic model of particles derived from roots of the light cone equation for a rational worldline, revealing three particle types with conserved, integer-valued invariants.

## Contribution

It presents a novel algebraic framework for particle dynamics based solely on roots of the light cone equation, without differential equations or field theories.

## Key findings

- The R-C particle dynamics is Lorentz-invariant and conservative.
- Total 4-momentum and rest mass are integer-valued invariants.
- Three distinct types of particles emerge asymptotically with specific behaviors.

## Abstract

We consider the light cone (`retardation') equation (LCE) of an inertially moving observer and a single worldline parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational worldline the collective R-C dynamics is manifestly Lorentz-invariant and conservative; the latter property follows directly from the structure of Vieta formulas for the LCE roots. In particular, two Lorentz invariants, the square of total 4-momentum and total rest mass, are distinct and both integer-valued. Asymptotically, at large values of the observer's proper time, one distinguishes three types of the LCE roots and associated R-C particles, with specific locations and evolutions; each of three kinds of particles can assemble into compact large groups - clusters. Throughout the paper, we make no use of differential equations of motion, field equations, etc.: the collective R-C dynamics is purely algebraic