Quantum Mechanics as a Classical Theory IX: The Formation of Operators and Quantum Phase-Space Densities

In our previous papers we were interested in making a reconstruction of quantum mechanics according to classical mechanics. In this paper we suspend this program for a while and turn our attention to a theme in the frontier of quantum mechanics itself—that is, the formation of operators. We then investigate all the subtleties involved in forming operators from their classical counterparts. We show, using the formalism of quantum phase-space distributions, that our formation method, which is equivalent to Weyl’s rule, gives the correct answer. Since this method implies that eigenstates are not dispersion-free we argue for modifications in the orthodox view. Many properties of the quantum phase-space distributions are also investigated and discussed in the realm of our classical approach. We then strengthen the conclusions of our previous papers that quantum mechanics is merely an extremely good approximation of classical statistical mechanics performed upon the configuration space. 1 The Formation of Quantum Operators The formation problem of quantum mechanical operators has already been treated by a number of authors [1, 2, 3, 4, 5, 6]. The problem resides basically in forming, from a given classical function f(q, p) of the commuting generalized coordinates and momenta, the related quantum mechanical operator O[f(p, q)]. This problem, as expressed in Shewell’s review article [3], either cannot be solved in a non-ambiguous way (e.g. von Neumann’s and Dirac’s rules) or it does not give the result expected by the orthodox epistemology of quantum mechanics, viz. Weyl’s and Revier’s rules.