The ambiguity group for canonical transformations in classical mechanics and its role in their representation in quantum mechanics

We discuss, through the example of the canonical transformations leading to action and angle variables of the two dimensional isotropic oscillator, the representation in quantum mechanics of non-bijective canonical transformations. We stress in particular the importance of a classical concept, the ambiguity group, both in suggesting the quantization rules for the example just mentioned as well as in achieving the representation in quantum mechanics through the corresponding concept of ambiguity spin. (orig.)