Canonical Transformations, Entropy and Quantization

This paper considers various aspects of the canonical coordinate transformations in a complex phase space. The main result is given by two theorems which describe two special families of mappings between integrable Hamiltonian systems. The generating function of these transformations is determined by the entropy and a second arbitrary function which we take to be the energy function. For simple integrable systems an algebraic treatment based on the group properties of the canonical transformations is given to calculate the eigenvalue spectrum of the energy.