Measurement theory for closed quantum systems

We introduce the concept of a "classical observable" as an operator with vanishingly small quantum fluctuations on a set of density matrices. It is shown how to construct them for a time evolved pure state. The study of classical observables provides a natural starting point to analyse the quantum measurement problem. In particular, it allows to identify Schr\"odinger cats and the associated projection operators intrinsically, without the need to invoke an environment. We discuss how our new approach relates to the open system analysis of the quantum measurement problem.