Collective Lorentz invariant dynamics on a single "polynomial" worldline

TL;DR

This paper studies polynomial worldlines and shows that the apparent particles detected by an observer follow Lorentz-invariant collective dynamics with conserved quantities, including a self-quantized rest mass, resembling scattering processes.

Contribution

It introduces a purely algebraic framework for particle dynamics on polynomial worldlines, revealing conserved quantities and a self-quantized rest mass without differential equations.

Findings

Total angular momentum and rest energy are explicitly calculated.

The rest mass takes only integer values, showing self-quantization.

Particles asymptotically form pairs and clusters resembling scattering events.

Abstract

Consider a worldline of a pointlike particle parametrized by polynomial functions, together with the light cone ("retardation") equation of an inertially moving observer. Then a set of apparent copies, R- or C-particles, defined by the (real or complex conjugate) roots of the retardation equation will be detected by the observer. We prove that for any "polynomial" worldline the induced collective dynamics of the system of R-C particles obeys a whole set of canonical conservation laws (for total momentum, angular momentum and the analogue of mechanical energy). Explicit formulas for the values of total angular momentum and the analogue of total rest energy (rest mass) are obtained; the latter is "self-quantized", i.e. for any worldline takes only integer values. The dynamics is Lorentz invariant though different from the canonical relativistic mechanics. Asymptotically, at large values of the observer's proper time, the R-C particles gather themselves into pairs and then assemble into compact incoming/outgoing clusters. As a whole, the evolution resembles the process of (either elastic or inelastic) scattering of a beam of composite particles. Throughout the paper the consideration is purely algebraic, with no resort to differential equations of motion, field equations, etc.