Canonical transformations in quantum mechanics

This paper presents the general theory of canonical transformations of coordinates in quantum mechanics. First, the theory is developed in the formalism of phase space quantum mechanics. It is shown that by transforming a star-product, when passing to a new coordinate system, observables and states transform as in classical mechanics, i.e., by composing them with a transformation of coordinates. Then the developed formalism of coordinate transformations is transferred to a standard formulation of quantum mechanics. In addition, the developed theory is illustrated on examples of particular classes of quantum canonical transformations.

► A general theory of canonical transformations in quantum mechanics is presented.
► The theory is developed using a quantum mechanics on phase space.
► A systematic passage to standard description of quantum mechanics is given.
► The theory is illustrated on particular classes of canonical transformations.