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A conjecture in quantum mechanics states that any quantum canonicaltransformation can decompose into a sequence of three basic canonicaltransformations; gauge, point and interchange of coordinates and momenta. It isshown that if one attempts to construct the three basic transformations instar-product form, while gauge and point transformations are immediate instar-exponential form, interchange has no correspondent, but it is possible inan ordinary exponential form. As an alternative approach, it is shown that allthree basic transformations can be constructed in the ordinary exponential formand that in some cases this approach provides more useful tools than thestar-exponential form in finding the generating function for given canonicaltransformation or vice versa. It is also shown that transforms of c-numberphase space functions under linear-nonlinear canonical transformations andintertwining method can be treated within this argument.

Quantum canonical transformations in Weyl-Wigner-Groenewold-Moyal formalism