is not mandatory to use it for our construction). Then the standard rules of quantum mechanics and its probabilistic nature apply.

We will call the eigenvalues of the “clock” operator $T$ and the eigenvalues of the “observables” $O$. We define the projector associated to the measurement of the time variable within the interval $[T_0-\Delta T,T_0+\Delta T]$, $$P_{T_0}(t) =\int_{T_0-\Delta T}^{T_0+\Delta T} dT \sum_k |T,k,t><T,k,t|$$ where $k$ denotes the eigenvalues of the operators that form a complete set with $\hat{T}$ (the eigenvalues can have continuous or discrete spectrum, in the former case the sum should be replaced by an integral). We have assumed a continuous spectrum for $T$ therefore the need for the integral over an interval on the right hand side. The interval $\Delta T$ is assumed to be very small compared to any of the times intervals of interest in the problem, in particular the time separating two successive measurements. Similarly we introduce a projector associated with the measurement of the observable $O$, $$P_{O_0}(t) =\int_{O_0-\Delta O}^{O_0+\Delta O} dO \sum_j |O,j,t><O,j,t|$$ with $j$ the eigenvalues of a set of operators that form a complete set with $\hat{O}$. These projectors have the usual properties, i.e., $P_a(t)^2 = P_a(t)$, $\sum_a P_a(t)=1, \forall t$ 