Classical and quantum ABCD-transformations and the propagation of coherent and Gaussian beams

We develop the mathematical properties of the ABCD-transformation from the classical and quantum mechanical points of view. First, we list the four types of generating function which generate the ABCD-transformation in classical mechanics. Second, we introduce the unitary operator of the ABCD-transformation. Next, we calculate the normal ordering of this unitary operator and derive the kernels in coordinate-momentum phase space. The kernels are comprised of the generating functions, which generate the ABCD-transformation in classical mechanics. This reveals a new correspondence between classical and quantum mechanics. As an application of these kernels, we show the propagation of coherent and Gaussian beams in the context of quantum optics which corresponds to the ABCD-transformation in matrix optics.