It is to be noted that such effects will occur in addition to the ones we discuss here.

What is the effect of the extra term? To study this, let us pretend for a moment that $\sigma(T)$ is constant. That is, the distribution in the clock variable has a width that grows linearly with time. In that case, the evolution equation is exactly solvable. If we consider a system with energy levels, the elements of the density matrix in the energy eigenbasis is given by, $$\rho(T)_{nm} = \rho_{nm}(0) e^{-i\omega_{nm} T} e^{-\sigma \omega_{nm}^2 T}$$ where $\omega_{nm}=\omega_n-\omega_m$ is the Bohr frequency corresponding to the levels $n,m$. We therefore see that the off-diagonal elements of the density matrix go to zero exponentially at a rate governed by $\sigma$, i.e. by how badly the clock’s wavefunction spreads. It is clear that a pure state is eventually transformed into a completely mixed state by this process. The origin of the lack of unitarity is the fact that definite statistical predictions are only possible by repeating an experiment. If one uses a real clock, which has thermal and quantum fluctuations, each experimental run will correspond to a different value of the evolution parameter. The statistical prediction 