Squeezing and quantum canonical transformations

In this paper we present an approach to quantum mechanical canonical transformations. Our main result is that time-dependent quantum canonical transformations can always be cast in the form of squeezing operators. We revise the main properties of these operators in regard to its Lie group properties, how two of them can be combined to yield another operator of the same class and how can also be decomposed and fragmented. In the second part of the paper we show how this procedure works extremely well for the time-dependent quantum harmonic oscillator. The issue of the systematic construction of quantum canonical transformations is also discussed along the lines of Dirac, Wigner, and Schwinger ideas and to the more recent work by Lee The main conclusion is that the classical phase space transformation can be maintained in the operator formalism but the construction of the quantum canonical transformation is not clearly related to the classical generating function of a classical canonical transformation. We hereby propose the Much more efficient method given by the squeezing operators. This method has also been proved to be very useful, by one of the authors, in the framework of the dynamical symmetries.