as functions of the observable time $T$. To do this, we recall the expression for the usual probability in the Schrödinger representation of measuring the value $O$ at a time $t$, $${\cal P}\left(O|t\right)\equiv {{\rm Tr}\left(P_{O}(0)\rho(t)\right)
\over {\rm Tr}\left(\rho(t)\right)}\label{ordinary}$$ where the projector is evaluated at $t=0$ since in the Schrödinger picture operators do not evolve. We would like to get a similar expression in terms of the real clock. To do this we consider the conditional probability (\[condprob\]), and make explicit the separation between clock and system, $$\begin{aligned}
 \label{condprob2}
{\cal P}\left( O\in [O_0\pm\Delta O]|T \in 
[T_0\pm\Delta T]\right) &=&
\lim_{\tau\to\infty}
{\int_{-\tau}^{\tau} dt\,{\rm Tr}\left(U_{\rm sys}(t)^\dagger P_O(0)
U_{\rm sys}(t) U_{\rm cl}(t)^\dagger P_T(0) U_{\rm cl}(t)\rho_{\rm sys}\otimes 
\rho_{\rm cl}\right) 
\over 
{\int_{-\tau}^{\tau} dt\,{\rm Tr}\left(P_T(t)\rho_{\rm cl} \right) {\rm Tr}\left(\rho_{\rm sys}\right)}}\\
&=&
\lim_{\tau\to\infty}
{\int_{-\tau}^{\tau} dt\,{\rm Tr}\left(U_{\rm sys}(t)^\dagger P_O(0)
U_{\rm sys}(t) \rho_{\rm sys}\right){\rm Tr}\left(U_{\rm cl}(t)^\dagger P_T(0) U_{\rm cl}(t) 
\rho_{\rm cl}\right) 
\over 
{\int_{-\tau}^{\tau} dt\,
{\rm Tr}\left(P_T(t)\rho_{\rm cl} \right) {\rm Tr}\left(\rho_{\rm sys}\right)}}.\end{aligned}$$

We define the probability that the resulting measurement of the clock variable $T$ correspond to the value $t$, $${\cal P}_t(T) \equiv {{\rm Tr}\left(P_T(0) U_{\rm cl}(t)\rho_{\rm cl} U_{\rm cl}(t)^\dagger\right)\over
\int_{-\infty}^\infty dt\,{\rm Tr}\left(P_T(t) \rho_{\rm cl}\right)},$$ and notice that $\int_{-\infty}^\infty dt {\cal P}_t(T)=1$. We now define the evolution of the density matrix, $$\rho(T) 