Half Quantization

A general dynamical system composed by two coupled sectors is considered. The initial time configuration of one of these sectors is described by a set of classical data while the other is described by standard quantum data. These dynamical systems will be named half quantum. The aim of this paper is to derive the dynamical evolution of a general half quantum system from its full quantum formulation. The standard approach would be to use quantum mechanics to make predictions for the time evolution of the half quantum initial data. The main problem is how can quantum mechanics be applied to a dynamical system whose initial time configuration is not described by a set of fully quantum data. A solution to this problem is presented and used, as a guideline to obtain a general formulation of coupled classical-quantum dynamics. Finally, a quantization prescription mapping a given classical theory to the correspondent half quantum one is presented.