A theory that couples quantum degrees of freedom to classical degrees of freedom is of relevance to quantum measurement theory and, more generally, of interest whenever it may be appropriate to assume that some components of a physical system can be modelled classically. However, despite many attempts, fundamental difficulties have always arisen in trying to implement such a ~Tmixed~T description. In my talk, I will discuss a new approach to solving this problem based on a canonical formalism that describes statistical ensembles on configuration space. Applications include quantum systems interacting with classical measuring apparatuses and equations for coupling quantum fields to a classical space-time.