Interaction contra Classical Relativistic Hamiltonian Particle Mechanics

Abstract

The relativistic Hamiltonian formalism is outlined and discussed for classical particles. The implications of the requirement that the coordinates of an event transform according to the Lorentz transformation law are discussed and expressed in a form, called the world-line conditions, which may be considered in the relativistic Hamiltonian formalism. It is then shown that the world-line conditions imply that there is no interaction in the relativistic Hamiltonian formalism; that is, the motion of any pair of particles described by the relativistic Hamiltonian formalism consists of straightline motion. In other words, if the events which compose the world lines of the particles transform according to the Lorentz transformation law, and the path of the particle is not a straight line, then this phenomena cannot be described in terms of a relativistic two-particle Hamiltonian formalism. The experimental basis for a determination of the transformation properties of an event is considered, and the relationship of the experiment to the applicability of such a formalism is discussed.

Single-particle trajectories and interferences in quantum mechanics

“…The uniqueness of its solution is guaranteed by an initial condition (32) ~(ti) =~i (39) which can also be deduced by determining the random variable {,, from the given initial probability density tO(r, ti)l 2. The unique solution of (34) under the initial conditions (39) will coincide with the process chosen by our stochastic variational principle.…”

“…Of course to a different ~ will be associated a different process. The manner of dependence of the process { on the wave function is also apparent from the remark that, if through (37) we separate (21) into its real and imaginary parts OtR2 + V (R2 ~)=O (40) 8tS+2 ---2m R ~-V=0 (41) we can always cast the continuity equation (40) in the form of a forward Fokker-Planck equation for a density p = R 2 with v(+) given in (36) ~tP = -V(pv(+ )) + vV2p (42) but we must also remember that now (42) is not a Fokker-Planck equation in the usual sense since v(+), as remarked before, is not an a priori given function and in fact it depends in its turn on the solution p of (42) through (39). We can see that even from another point of view: if we fix a solution ~b of (21), the form of (42) (namely v(+)) will also be fixed; however, (42) will have an infinity of solutions p. Among these solutions only one satisfies the stochastic variational principle.…”

Hamiltonian Two-Body System in Special Relativity

“…Classical relativistic dynamics of pointlike bodies has a long story; without claiming to be exhaustive let us mention the Wheeler-Feynman (WF) electrodynamics [1] based upon the Fokker action [2], the three forms of dynamics (front form, point form, instant form) advocated by Dirac [3], and after the discovery of a famous No-Interaction theorem [4][5][6], various efforts made in order to circumvent it; for instance Predictive Mechanics [7][8][9][10], the Singular Lagrangian method [11] and Constraint Dynamics [12,13]. In the last decade there were a few papers along the lines of WF [14,15] and also the work carried out by Lusanna et al [16,17] in order to give a covariant status to the instant form.…”

Covariant equilibrium statistical mechanics

“…The work of Balescu et al [6] strongly enforces possibility 2) but applying the method on non-equilibrium statistical mechanics only. The refusing statement of ter Haar and Wergeland concerning possibility 1) was certainly motivated by the fact that any straight forward attempt to formulate classical dynamics in a covariant fashion runs automatically into conflicts defined by the famous no-interaction theorem [7]. This theorem states that only a collection of non-interacting particles can be described within a Poincare invariant fashion if a straight forward generalization of the usual dynamical description is applied.…”

“…The generalization of this non-relativistic particle dynamics to a manifest covariant particle dynamics is not trivial because, one has to know how to treat action-at-a-distance in a covariant fashion. The conceptual problems in this field are formulated in the no interaction theorem [7], which states that only a collection of free particles can fulfill the following requirements:…”