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Ever since Hubble, it is well known that the universe is expanding from a Big Bang. The size of the universe had gone up by many many orders of magnitude as space expanded. If the dimensionality of the quantum phase space is finite because of spatial cutoffs at the Planck scale, does it go up as space expanded? If yes, how can this be squared with unitarity? If no, would this lead to what Tegmark called the Big Snap where something has got to give sometime. What is that something which gives?

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This is a deep mystery in quantum gravity, and it is discussed at length by Banks, and Susskind, among others, in connection with conjectural ds-CFT constructions. This is the central difficulty in describing expanding universes in string theory.

The quantum mechanical number of bits that a cosmological horizon can include seems heuristically to be limited by the area in Planck units. As the universe exapands, this area goes up, so the phase space itself seems to be getting bigger. This is very difficult to conceive, and it seems to be a true paradox, because quantum mechanical evolution can't make the state space bigger.

One way out was suggested by Susskind--- that deSitter space is unstable, and the correct degrees of freedom live on the space it eventually decays into. There is no agreed upon answer, and solving this problem likely requires a good new idea.

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it does struck as something extremely odd (at least for fellows like myself which have some familiarity with GR and QFT but only a bunch of misconceptions and hearsay about string theory) that you have this theory that 1) is equal to GR in the classical limit 2)encapsulates absurdly many different vacua, but as it turns out, 3)does not know nothing besides static non-expanding cosmologies – lurscher Oct 11 '11 at 19:58
Could I ask for what happens classically? For instance, I know that in general it is possible to put a symplectic structure on space of solutions, even in the presence of an isolated horizon. Do the ideas (e.g. conserved Louiville form) make sense cosmologically? – genneth Oct 11 '11 at 22:10
@lurscher: I agree that this is a big problem, but it is unbelievable enough that there is a solution for the non-cosmological AdS quantum gravity. It is not quite correct that string theory doesn't work in the expanding backgrounds, one should say that the techniques for defining the theory don't extend to these backgrounds. It is certain that there is a way to do it, which is not known yet, perhaps requiring a deep physical insight. I speculated on this endlessly, but never got anything solid and new. – Ron Maimon Oct 12 '11 at 4:26
@genneth: that's a really good question, perhaps you can ask it on theoreticalphysics.stackexchange? It might be answerable in 2+1 dimensions more easily, but I am not sure if you can get an expanding horizon, I was thinking of t'Hooft's version of conincal defect dynamics. – Ron Maimon Oct 12 '11 at 4:28
@RonMaimon: done – genneth Oct 12 '11 at 11:11

I have a feeling the dimensionality of the quantum state space is not conserved in an expanding universe. All our evidence for unitarity comes from experiments conducted at length and time scales far smaller than cosmological timescales. I really don't see any fundamental reason why we should assume unitarity applies to quantum cosmology. We have no right to extrapolate our experimental results to cosmological scales with no good reason. Even if unitarity is not a fundamental feature of our universe, at really short time scales, cosmologically speaking, unitarity might emerge as an approximation just as classical mechanics emerges when the action exceeds Planck's constant, or nonrelativistic mechanics emerges at speeds far less than the speed of light.

As an added bonus, the second law of thermodynamics has a trivial explanation and is no longer a mystery. The big bang was not a highly improbable configuration placed on a knife's edge as Penrose would have you believe. The second law is what comes naturally when the size of phase space goes up.

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In the course of the evolution according to the second law of thermodynamics towards a state of higher entropy, the number of relevant variables needed to describe the state of the system decreases but the irrelevant variables (or degrees of freedom) do NOT vanish; so why should unitarity break down? I must confess that althoug I like parts of Roger Penrose`s "Road to Reality" I strongly disagree with his mamboo-jamboo about the "interpretation of quantum mechanics" ... – Dilaton Oct 13 '11 at 19:04

When the universe expands, the "new" modes that arise can be traced to transplanckian -- i.e. shorter than Planck length -- wavelength modes. What was once transplanckian now becomes redshifted adiabatically to subplanckian wavelengths. I stress adiabatically because this doesn't apply to the nonadiabatic Planck epoch. Of course, many physicists object to the existence of transplanckian modes. Supposedly, the Planck scale acts as an ultraviolet scale cut-off. Anyway, looking at the Wheeler-De Witt constraints, it can be shown the transplanckian modes aren't independent of the subplanckian modes. This ought to sidestep the problem.

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