Quantum-mechanization of dynamical systems

Abstract

In an earlier paper [l] we presented a set of axioms concerning the concept of wave equation for a dynamical system G with finitely many degrees of freedom, and it was shown there that the familiar equations (Schrodinger’s for the Newtonian case, and the “Klein-Gordon” equation for the Einsteinean case) are the unique equations satisfying those axioms when interpreted for systems of these two types. The axioms also apply to more general systems. The selection of a wave equation for a system 6 is of course only one of the steps that must be taken in order to construct the wave-mechanical counterpart G’ to the system 6. One has also to replace the finite dimensional phase space CD, for each observer w by an infinite-dimensional counterpart @k and show, for each pair w1 , w, of observers, how their quantum-mechanical phase spaces @:, and @:, are to be dynamically related by the wave equation. To do this in the greatest generality is the purpose of the present paper. These quantum-mechanical phase spaces @: are, as one would expect, related to Hilbert spaces of square-integrable functions on the configuration space QU . We must specify, among other things, the volume element in Q,,, in terms of which this square-integrability is to be formulated. These, and other specifications for the construction of the quantummechanical counterpart 6’ of the system 6, are given below. They are given in such a way that when there is an isomorphism between two systems 6, and 6,) then there is immediately induced an isomorphism between the quantum systems S; and 6;. This isomorphism-induction can be regarded as the origin of the representation in the quantum system 6’ of the symmetry group of the original system 6. In order to explain this we have had to make precise the concept of observer. A list of the headings of the sections of this paper is as follows: (2) General