Tuesday, March 28, 2006

Steve Shenker: Inflation in AdS/CFT

The last duality seminar by Steve Shenker was about

Steve is a captivating speaker so we could not miss this one. He brought us some fresh air from the West Coast, including the West Coast thinking where everything is fine, everything goes, just as required by the anthropic principle, and moreover the metric tensor signature is (+---).

One of the dreams of modern, high-energy alchemists was to create a Universe in the bottle. Imagine that there is a scalar field whose potential energy has a local minimum somewhere above our minimum. You prepare a scattering process and try to raise the value of the scalar field to the higher minimum in a large enough region of space inside your bottle. This small seed can start to inflate: it may begin to grow exponentially while its boundary may remain small. A large new Universe with billions of stars may eventually appear inside your bottle.

Is it possible? Although this picture uses physical concepts that are widely believed by theoretical physicists to be correct, the ultimate answer is No. Today we may argue that such a Universe in the bottle violates the entropy bounds and the holographic principle: the entropy and the number of degrees of freedom in a region (inside the bottle) should not exceed the surface area of the region (the bottle).

More materially, Guth and Farhi have proved years ago - using the classical methods of general relativity and the theorems about the existence of singularities - that a few moments or minutes before the inflation starts, the spacetime must develop a singularity. If you want a Universe in the bottle, you must sacrifice all your history except for the last few minutes, pretend that the Big Bang occured five minutes ago, and as you can agree, this is far too high a price to pay. Farhi, Guth, and Guven did not want to give up and they proposed a new way to start the local inflation - using quantum tunneling.

Steve reviewed these things - and he drew the Penrose diagrams in various situations sketched above (for example, an air-conditioning inflating box attached to the filled anti de Sitter cylinder) as he participated in intense debates whether the tunneling breaks the analyticity of various quantities. But his intent was more modern. Today we have alternative definitions of theories of quantum gravity, especially the AdS/CFT correspondence. Can they tell us something new about inflation? While I don't know the answer to this question, Steve could still manage to say some interesting things.

Don't get me wrong: the conclusion remains that the Universe in the bottle is impossible, even if you use quantum tunneling.

But the exact explanation why it is so is somewhat interesting albeit controversial. Steve essentially argued that if there is an inflating region inside anti de Sitter space, it must be described by a mixed state in the full theory. Because pure states can't evolve into mixed states, the evolution that initiates inflation is impossible.

I don't understand the assumptions of this argument (although I agree, of course, with the impossibility to evolve into mixed states) and Steve's answers to my questions confirmed my expectation that I won't ever understand it because it contradicts my understanding of some basic notions of quantum mechanics. We can't ever say that the world is objectively described by a mixed state. A mixed state or a density matrix is nothing else than the quantum counterpart of the classical uncertainty, the classical probability distributions. A mixed state is a mixture of matrices constructed from pure states that is a useful description of reality whenever we only know some features of the pure states (such as some macroscopic or low-energy quantities) but not all of them.

But a representative pure state included in the mixed state must always have the same macroscopic properties as the mixed state itself. And we must always be free to imagine that our system is in a particular pure state - we just don't know which one. If you consider a thermal density matrix, surely you don't think that you can't find particular pure states that behave in the same way. The description of the macroscopic physics in terms of a thermal mixed state may be more convenient than any calculation you could do with any particular pure state that looks thermal, but it is just a matter of convenience, not a matter of the truth.

Moreover, when you derive by some semiclassical methods that a CFT description of your inflating region is traced over some degrees of freedom and it therefore looks as a mixed state, it does not mean that the mixed state is the exact answer. By the same semiclassical methods, you might argue that a black hole is a pure state that has no entropy. That would be, of course, wrong. The opposite case - in which the semiclassical approximation overestimates the entropy - is unlikely because the semiclassical degrees of freedom are subset of all degrees of freedom and an inequality should therefore hold. However, I did not see how one can isolate the "many" degrees of freedom on the boundary that describe the CFT, so I don't see a sharp contradiction with the entropy argument.

At any rate, one can never objectively say whether the Universe is found in a pure state or a mixed state: this question is a subjective question about our complete vs. incomplete knowledge of the physical system. The density matrix is not the same as a classical state of a classical system. And no physically measurable quantities or questions can depend on the answer to this philosophical question about the purity of our Universe because that would be in contradiction with the postulates of quantum mechanics, I think.

Steve was telling me that these basic ideas of "mine" break down in quantum gravity but I don't understand what they have to do with gravity which is why I cannot fully reproduce their argument here.

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