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By studying quantum cosmology I was asking myself if the fact that the universe is expanding, so space is expanding and with it I would say that phase space is also expanding, so it's a non-unitary evolution, am I right? If yes, can unitarity be restored in a multiverse picture? Because I am always troubled when I hear that as we have experimentally verified the unitarity of low energy QM, by pushing further we arrive at the concluse that the multiverse can be extended from $t= - \infty$ to $t= + \infty$, sounds a bit dogmatic to me.

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Short answer: no. More complicatedly, it's hard to define what time evolution is cosmologically. For observable consequences, usually "cosmological" really means "only the largest scale degrees of freedom", which is not special as a quantum system, and as long as one has a clock somehow (which need not be tied to this very coarse spacetime structure), evolution will be unitary. Once you include everything, however, the issue of "what's a clock" rears its ugly head. –  genneth Jun 11 '12 at 16:06
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