The Relativistic Action at a Distance Two Body Problem
Models with action at a distance potentials, such as the Coulomb potential, have been very useful in nonrelativistic mechanics. They provide a simpler framework than the perhaps more fundamental field mediated models for interaction, and are also straight
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The Relativistic Action at a Distance Two Body Problem
Models with action at a distance potentials, such as the Coulomb potential, have been very useful in nonrelativistic mechanics. They provide a simpler framework than the perhaps more fundamental field mediated models for interaction, and are also straightforwardly amenable to rigorous mathematical analysis. In this NewtonianGalilean view, all events directly interacting dynamically occur simultaneously; the dynamical phase space of N particles contains the points xn (t) and pn (t), for n = 1, 2, 3, . . . N ; these points move through the phase space as a function of the parameter t, following some prescribed equations of motion. Two particles may be thought of as interacting through a potential function V (x1 (t), x2 (t)); for Galiliean invariance, V may be a scalar function of the difference, i.e., V (x1 (t) − x2 (t)). It is usually understood that x1 and x2 are taken to be at equal time, corresponding to a correlation between the two particles consistent with the Newtonian-Galilean picture. With the advent of special relativity, it became a challenge to formulate dynamical problems on the same level as that of the nonrelativistic theory. For the relativistic theory, one might think of two world lines with action at a distance interaction, but the correlation that could be used between the two points μ μ x1 and x2 cannot be maintained by the variable t in every frame. Dirac (1932) introduced a “many time” theory to describe the dynamics of an N body system, maintaining the notion of the t component of the four vector position as associated with evolution; later, his lectures at the Belfer School (Dirac 1966) laid the foundations for the constraint dynamics mentioned in Chap. 2, with some details given in the Appendix of that chapter, for which each particle has its own effective invariant evolution parameter based on the canonical transformation properties induced by the constraints of an 8N dimensional phase space; attempts to imbed this classical approach into a quantum theory met some difficulties, but some progress was made in developing a useful scattering theory (Horwitz 1982). As we have pointed out the Stueckelberg [SHP] theory provides an effective and systematic way of dealing with the N body problem, and has been applied in describing relativistic fluid mechanics (Sklarz 2001), the Gibbs ensembles in statistical mechanics and the Boltzmann equation (Horwitz 1981), systems of many identical particles, as described in Chap. 3, and other applications. The essential ingredient in developing these applications is the © Springer Science+Business Media Dordrecht 2015 L.P. Horwitz, Relativistic Quantum Mechanics, Fundamental Theories of Physics 180, DOI 10.1007/978-94-017-7261-7_5
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5 The Relativistic Action at a Distance Two Body Problem
use of a single invariant parameter (Horwitz 1973 and remark of Sudarshan 1981), to define the correlated interactions of a many body system. We study in this chapter the relativistic two body problem with invari
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