Collective Lorentz invariant dynamics on a single ‘polynomial’ worldline

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 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 couple 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.

Three kinds of particles on a single rationally parameterized world line

“…The latter can be either defined implicitly, by a set of algebraic equations containing the time parameter t [3,4], or in a familiar parametric way through consideration of the light cone equation (LCE) (equivalent to the well-known retardation equation) corresponding to an external observer [5]. In both cases, at some fixed value of t, one has a whole set of roots of the considered algebraic system which determine the positions and, consequently, temporal dynamics of the collection of identical particlelike formations.…”

“…Such 'events' can be interpreted as the process of annihilation/creation of a pair of R-particles (precisely, of a particleantiparticle system) [1,3,5]. At the moments of merging, the effective twistor and electromagnetic fields (the latter of Lienard-Wiehert type) become singular on a null straight line connecting the points of observation and merging.…”

“…In the case of a purely polynomial worldline [5] corresponding conserved quantities have been identified with the 4-vector P µ of total energy-momentum and the scalar of total rest mass (rest energy) M of the RC-system, respectively. In this case (for l > k ) it has been proved that these invariants identically satisfy the fundamental relativistic energymomentum relation…”

“…The above presented results can be proved by analyzing the structure of resultants, that is, determinants of corresponding Sylvester matrices. Quite a similar procedure has been used and exposed in details in [5] for the case of purely polynomial worldlines. Note that the second invariant M (identified previously with total rest mass) can be negative: this might be related to the contribution (∼ 2n) of negative interaction energy, a possible relativistic analogue of the potential energy.…”

“…It is especially interesting that all these laws follow solely from the structure of Vieta formulas for the whole system of roots, or from derivations of these formulas w.r.t. the time parameter.In the case of implicitly defined polynomial worldline [3,4] the considered algebraic dynamics is Galilei-invariant and can be compared with Newton's collective N -point dynamics. On the contrary, in the second case of a polynomial worldline implemented by the LCE of an inertially moving observer one obtains a full set of Lorentz-covariant conservation laws for the total set of R-C particles.…”