Three kinds of particles on a single rationally parameterized world line

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 . 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. In the above cited papers it was shown that for an arbitrary worldline defined by polynomial functions (except a degenerate case of zero measure) the arising collective dynamics of the system of particles-roots is necessarily conservative. This means that a complete set of conservation laws (for total momentum, angular momentum and the analogue of total energy) holds for the system of two kinds (R-or C-) of particlelike formations represented by real and complex conjugate roots of the 1 e-mail: vkassan@sci.pfu.edu.ru 2 e-mail: n.markova@mail.ru generating set of equations, respectively. 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. Asymptotically, for rather great values of the observer's (proper) time T one encounters, in addition, the effects of 'self-quantization' of the admissible values of total rest mass and 'clusterization' of particle-roots. The possible meaning of the obtained algebrodynamics 3 for realistic relativistic physics requires further investigations.A detailed exposition of the above presented results can also be found in [6]. Below we generalize our consideration of polynomial dynamics to the case of a worldline parameterized by arbitrary rational functions.3 On the so-called algebrodynamical program see, e.g., [...

Conservative Relativistic Algebrodynamics Induced on an Implicitly Defined World Line

“…Thus, we come again to a picture of "shrinking and then expanding Universe" in which the final stage completely reproduces the starting one. Unfortunately, there are no indications for coupling or formation of clusters at late stage of evolution, contrary to the emergence of these phenomena in the corresponding model with an explicitly parameterized polinomial [7] or rational [9] worldline.…”

“…T being the macroscopic monotonically increasing proper time of the observer. Even in this simplest case, for polynomial parameterization (as well as for rationally parameterized worldlines, see [9]) the induced collective dynamics turns out to be conservative. That is, a full set of Lorentz invariant conservation laws holds for the collection of RCparticles-roots defined by (2) as τ = {τ k (T )} → x(T ), y(T ), z(T…”

“…Indeed, under these conditions near some critical value of T = T 0 one encounters the phenomenon of pairing of the roots while at greater values T >> T 0 the arised pairs manifest the effect of clusterization, that is, formation of closely located (and generally large) groups of roots-particles [7,8]. In the case of rationally parameterized worldlines [9] the emergent dynamics is still more diverse.…”

Maxwell, Yang–Mills, Weyl and eikonal fields defined by any null shear-free congruence

“…a specific affine connection. Its particular form follows from a relativistic version of biquaternion algebra introduced orginally in [27] and is considered in [30]. It is, however, worth noting that equations for covariantly constant fields on the background of Weyl geometry [28] or a geometry with torsion determined by its trace [29] can also be used for a transparent geometric treatment of electromagnetism and possess a number of properties closely related to those of the generating system of equations.…”