by choosing a “clock”. By this we mean a physical quantity (more precisely a set of quantities, like when one chooses a clock and a calendar to monitor periods of more than a day) that we will use to keep track of the passage of [*time*]{}. An example of such a variable could be the angular position of the hand of an analog watch. Let us denote it by $T(q,p)$. We then identify some physical variables that we wish to study as a function of time. We shall call them generically $O(q,p)$ (“observables”). We then proceed to quantize the system by promoting all the observables and the clock variable to self-adjoint quantum operators acting on a Hilbert space. The latter is defined once a well defined inner product is chosen in the set of all physically allowed states. Usually it consists of squared integrable functions $\psi(q)$.

Notice that we are not in any way modifying quantum mechanics. We assume that the system has an evolution in terms of an external parameter $t$, which is a classical variable, given by a Hamiltonian and with operators evolving with Heisenberg’s equations (it is easier to present things in the Heisenberg picture, though it 