On the structure of phase space
M. Moshińsky, T. H. Seligman · Journal of Mathematical Physics · 1981
We consider phase space as the carrier space of canonical transformations and see that this implies, for nonbijective ones, a much subtler structure than the one commonly assumed. Discussing only problems of one degree of freedom, i.e., a phase plane, we are able to clarify this structure by analogy with the better-known situation of conformal mapping in the complex plane. Apart from the usual Riemann sheet concept, an alternate method is developed that involves the irreducible representations of the ambiguity group, i.e., the group of transformations that connects all points that are mapped on a single point by the conformal or canonical transformation. The algebra of variables then becomes a matrix algebra for which bijectiveness is retrieved.