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I'm having trouble understanding something about Susskind's Holographic principle. Susskind speaks about the surface of the universe? In what sense does the universe have an outer surface? I'm a bit of an amateur when it comes to cosmology, so be gentle.

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There is a surface more or less at 13.7 billion light years from us which is where we see back to the big-bang (looking back in time, if you define "now" in a global way, although there is no reason that we should't define "now" by what we are seeing "now", i.e. along a past light cone). This surface is analogous to a black hole horizon, except it surrounds us instead of being localized in a region.

This thing is called the "cosmological horizon", and the general idea of the holographic principle suggests that everything inside the cosmological horizon is described by oscillations of this horizon. This is hard to make precise because the horizon has a finite area and growing, and so has a finite maximum entropy associated with it (which is growing), and this is paradoxical seeming, because it suggests that the Hilbert space for our universe is growing.

The number of states in a quantum mechanical system can't increase, so this leads many people to renounce the idea of string theory in our kind of universe, choosing instead to describe the dynamics in terms of the asymptotic future, where presumably the universe will vacuum decay to a supersymmetric state. This is one approach, another is to try and formulate a real theory with a finite Hilbert space. I think a possible third approach is to consider finite area horizons as somehow density-matrix like, so that they, unlike black holes, have fundamental decoherence. Nobody knows the answer, and this is the major unsolved problem of string theory today.

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Could you elucidate on "the number of states in a quantum mechanical system can't increase"? I thought that Hilbert space is infinite anyway, even for the hydrogen atom. infinite + or - or times anything is infinite. Not to forget bosons which can always appear out of energy. – anna v May 2 '12 at 4:25
Just thought I'd point out that the comoving radius of the observable universe is closer to 47 billion light years, instead of 13.7 billion light years. – kleingordon May 2 '12 at 4:34
@kleingordon: This is not useful for the size of the cosmological horizon, which is defined in the correct way using the "now" slice which goes back in time, in which the radius is 13.7 billion light years (a little less because of cosmological constant making the expansion of the cosmological horizon stop). The comoving radius is nonsense people use to feel superior to people who "mistakenly" use the age of the universe as a distance (which is correct). – Ron Maimon May 2 '12 at 6:43
@annav: Hilbert space is infinite when you have no upper bound on radius or energy, so particle in a box has an infinite hilbert space, as does H-atom (the infinity at n large is just the large radius infinity, there is a second infinity in the continuum states, which you can regulate in a box). The point is that a deSitter space is like a particle in a box with an upper bound on the energy (sort of, it's not like that because it is thermal and lossy, but it's like that in terms of state counting), and this gives a finite number of states, the exponential of the horizon area. – Ron Maimon May 2 '12 at 6:46
@RonMaimon: Don't people usually define the cosmological event horizon as the boundary of the events which the observer will ever be able to see? In fig1 here for example, it looks like it's at 16GLy radius. – twistor59 May 2 '12 at 7:11

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