Decoherence without time?

Question

Decoherence is a phenomenon that provides a part of the explanation of why quantum systems and classical systems behave differently.

What I understood from decoherence so far is that it requires time because it simply takes a (somehow very short) while for a large quantum system to transit to classical behaviours through decoherence.

Now, the status of time in quantum gravity is not clear, and it could very well occur that time is an emergent quantity. So I thought decoherence should occur at time scales where time has already emerged. Yet I found this assertion on www.decoherence.de

Decoherence explains also how the Schrödinger equation of general relativity (the Wheeler-DeWitt equation) may describe the appearance of time in spite of being time-less"

I would like to understand how can decoherence be active in the timeless domain of the Wheeler-deWitt equation to allow the emergence of time, as decoherence seems to require time itself to happen. (Handwaving arguments are OK, but references are welcome!)

Answer 1

The webpage you were looking at is run by H. D. Zeh. So if you want to find out what he's talking about the best way to find out is to look up some of his papers, such as:

http://arxiv.org/abs/1012.4708.

He describes decoherence in quantum gravity in Section 5 of the paper. The basic answer is as follows. The Wheeler-DeWitt equation is time-independent in the sense that there is no parameter $t$. Time dependence is recovered by treating some internal system as a clock, as explained in a paper by Page and Wootters. To get the equation of motion for a non-clock observable with respect to clock time $\tau$ you compute the expectation value of that observable in the state $P_{\tau}\rho$ where $\rho$ is the stationary state of the universe and use that to derive a Heisenberg equation of motion for the observable. This idea has given rise to a large literature describing the details of how this procedure works and whether other procedures are better: you might want to look at that literature.