Wigner distribution functions and the representation of a non-bijective canonical transformation in quantum mechanics

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Abstract
In a previous paper, with a similar title, the authors showed that for bijective (i.e. one-to-one onto) canonical transformations they can obtain their quantum mechanical representations in the form of a kernel in the phase space of Wigner distribution functions and recover the classical expectation for this kernel in the limit to 0. For non-bijective canonical transformations the situation is more complex as the phase space acquires either a Riemann surface structure or requires the introduction of the concept of an ambiguity group. By discussing the non-bijective canonical transformation taking them from an oscillator of frequency kappa -1, where kappa is an integer, to one of unit frequency, they see how the kernel is generalised to include the indices associated with the irreducible representations of the ambiguity group, i.e. the ambiguity spins. They obtain the kernel explicitly and show that in the limit to 0 they recover the form that they expect in the Riemann surface picture of their phase space. While they illustrate their analysis through the representation in Wigner distribution phase space of a specific non-bijective canonical transformation, the procedure is clearly extendable to the representations of general canonical transformations of this type.