Canonical transformations to action and angle variables and their representations in quantum mechanics: III. The general problem

The representation in quantum mechanics of canonical transformations to action and angle variables is discussed for a general class of Hamiltonians including some that have both bounded and unbounded orbits where, in the latter case, the definition of the transformation is suitably extended. Again, as in the particular examples discussed in the previous papers of this series, the canonical transformations are nonlinear and nonbijective. We can recover though the bijectiveness, i.e., the one-to-one onto mapping, either by introducing a sheet structure in the original phase space or using the concepts of ambiguity group and ambiguity spin. With the help of these concepts we obtain an expression for the representation in quantum mechanics of the canonical transformation and recover the latter when we pass to the classical limit with the help of the WKB approximation. Furthermore, we establish in this paper a one-to-one correspondance between the arbitrariness in the phase of the representation and in the choice of the variable conjugate to the action.