CLASSICAL ANALOGS FOR QUANTUM SYSTEMS

It is shown that suitable classical nalogs for the relativistic quantum systems oftwo particles always exist, and that they are destitute of causal anomalies. The wave mechanics lives in a configuration space; the classical, hamiltonian mechanics in a phase space. This is but one of the difficulties faced by all the attempts of building a coherent interpretation of the quantum phenomena in terms of classical concepts. The prescription of the right extension procedures from one formulation to another will play, in fact, an essential role in the comprehension f the dynamics of the classical analogs. Moreover, from the begin-ning of its history, the causal interpretation of the quantum mechanics for two or more particles is tied to the problem of the direct interactions among them [ 1]. I f this poses no serious challenges in a non-rela-tivistic theory, in a relativistic one we must verify that such a causal description is in fact possible. In this letter we will try to formulate suitable classical analogs for the relativistic quantum mechanics of two particles and verify their internal consistency as a preliminary, essential step for all the future developments. We will start our dicussion with a short and rather intuitive outline of the covariant, canonical formal-ism describing the dynamics of two point-like, clas-sical, interacting particles of identical mass m [ 2]. Let us consider a set of canonical variables qa,U PaU (a = 1, 2;/~=0, 1, 2, 3) for our two-particle system. The evolution in terms of two scalar parameters a, in general not coincident with the usual proper times, is given by means of phase space abstract orbits (sur-faces) qg(zj,z/), p~(z~,z2). This evolution is ruled by the canonical equations Op~a " U " " , Oq~a-- { qU'Hb-~T b-- (p a'l-lb ~ ' (1) written in terms of scalar hamiltonians of the form 1 u Ha ( q~, q ~;p~,p~) = ~ PaPal, + Va ( q~, q ~;p~,p~)