\equiv \int_{-\infty}^\infty U_{\rm sys}(t) \rho_{\rm sys} 
 U_{\rm sys}(t)^\dagger {\cal P}_t(T)$$ where we dropped the “sys” subscript in the left hand side since it is obvious we are ultimately interested in the density matrix of the system under study, not that of the clock. Noting that $${\rm Tr}\left(\rho(T)\right)=\int_{-\infty}^\infty dt\, {\cal P}_t(T) 
{\rm Tr}\left(\rho_{\rm sys}\right)={\rm Tr}\left(\rho_{\rm sys}\right),$$ one can equate the conditional probability (\[condprob2\]) to the ordinary probability of quantum mechanics (\[ordinary\]). We have therefore ended with the standard probability expression with an “effective” density matrix in the Schrödinger picture given by $\rho(T)$. By its very definition, it is immediate to see that in the resulting evolution unitarity is lost, since one ends up with a density matrix that is a superposition of density matrices associated with different $t$’s and that each evolve unitarily according to ordinary quantum mechanics.

Now that we have identified what will play the role of a density matrix in terms of a “real clock” evolution, we would like to see what happens if we assume the “real clock” is behaving semiclassically. To do this we assume that ${\cal P}_t(T) =f(T-T_{\rm
  max}(t))$, where $f$ is a function that decays very rapidly for values of $T$ far from the 