conditional probabilities will be complicated and will bear little resemblance to the usual evolution of probabilities in ordinary quantum mechanics unless we make a “wise” selection of the clock and system variables. What we mean by this is that we would like to choose as clock variables a subsystem that interacts little with the system we want to study and that behaves semiclassically with small quantum fluctuations. Namely, the physical clock will be correlated with the ideal time in such a way to produce the usual notion of time. In such a regime one expects to recover ordinary Schrödinger evolution (plus small corrections) even if one is using a “real” clock. Let us consider such a limit in detail. We will assume that we divide the density matrix of the whole system into a product form between clock and system, $\rho =\rho_{\rm cl}\otimes \rho_{\rm sys}$ and the evolution will be given by a unitary operator also of product type $U=U_{\rm cl}\otimes U_{\rm sys}$.

Up to now we have considered the quantum states as described by a density matrix at a time $t$. Since the latter is unobservable, we would like to shift to a description where we have density matrices 