Canonical transformations to action and angle variables and their representations in quantum mechanics

Abstract

We study the classical canonical transformations to action and angle variables for the repulsive and attractive oscillator and the free particle. These transformations turn out to be nonbijective (not one-to-one onto), and we introduce a sheet structure in phase space to restore bijectiveness. We find the “ambiguity group” for the three problems mentioned which connects points in one phase space that are mapped on a single point in the other. The different irreducible representations of this group can then be used to characterize different components of functions of canonically conjugate variables, thus providing us with an alternative procedure to recover bijectiveness. The above picture—characterized by the “ambiguity spin” components—is readily translated into quantum mechanics. Indeed, by introducing wavefunctions with ambiguity spin, we are able to enlarge our Hilbert spaces in such a way that the spectra of the Hamiltonian and the action variable become the same, and thus a unitary representation of the classical canonical transformation becomes possible. These unitary representations are explicitly obtained for the three problems mentioned above, and we also determine the correspondences between operators in the original and new Hilbert spaces due to the canonical transformation leading to action and angle variables. the suggestions on how to enlarge the Hilbert spaces seem to come entirely from the classical structure in phase space, thus allowing the surmise that some aspects of quantum mechanics (such as the spectra of simple operators) are already implicitly contained in classical mechanics.