Three kinds of particles on a single rationally parameterized world line

We consider the light cone (‘retardation’) equation (LCE) of an inertially moving observer and a single world line parameterized by arbitrary rational functions. Then a set of apparent copies, R- or C-particles, defined by (real or complex conjugate) roots of the LCE will be detected by the observer. For any rational world line 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 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.