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Naive reasoning suggests that a string theory vacuum with cosmological constant Lambda1 is always unstable as long as there is a string theory vacuum with cosmological constant Lambda2 < Lambda1 since the later is a state with lower energy density. If it's true, there should be very few stable vacua since these would be the vacua with the lowest possible cosmological constant. These vacua would probably have a Planckian-size cosmological constant and therefore be highly non-classical.

However, I suspect there is something wrong with this line of reasoning. Probably I am thoroughly confused about something.

Also, how to compare stability of vacua with different number of large dimensions? Comparing cosmological constants doesn't feel to make sense.

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I’ll wait for the community’s feedback, but I personally feel this question is not well formed as it stands. There is a long history of work on spacetime instabilities, which the OP does not seem to be aware of, and which would be the starting point of any technical discussion of the issue. Also, as it stands the answer is simply infinite — more interesting question would be the number of (meta)stable vacua with certain cuts. –  user566 Oct 26 '11 at 20:53
    
Well, I'm more interested to understand what's wrong with the "naive" argument above (if indeed it's wrong) then to known the precise number of stable vacua. –  Squark Oct 26 '11 at 21:05
    
OK - maybe a revision would make this point clear. I’d recommend looking at Coleman - de Luccia classic paper - just because some space has higher cosmological constant, does not mean it will necessarily decay. It is all explained there, but maybe there are other interesting answers. –  user566 Oct 26 '11 at 21:09
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up vote 12 down vote accepted

OK, I will bite on this one.

Suppose you have a quantum mechanical model with a metastable state: it is a local but not a global minimum of the potential. If your wavefunction is initially approximately localized near this so-called "false" vacuum, it will tunnel to the true vacuum sooner or later. This is a probabilistic event, the rate per unit time depends on details of the potential barrier, but if you wait long enough you'll eventually find yourself in the true vacuum.

In field theory this is slightly more complicated because you have to take into account the spatial dependence of the process. Suppose you start in the false vacuum everywhere, then there will be some probability amplitude anywhere in space to tunnel to the true vacuum, exactly as in quantum mechanics. This will result in formation of bubbles of true vacuum suspended in the false vacuum.

The fate of these bubbles depends on their size. The energetics of this is as follows: you gain energy by converting volume from false to true vacuum, but you lose energy because the surface of the bubble has surface tension (in other words interpolating from true to false vacuum costs some gradient energy). So, in flat space bubbles large enough expand out, percolate and eventually eat out the false vacuum, and small bubbles shrink and die. The calculation of tunnelling probability is explained by this classic Coleman paper.

Now, enter gravity. This is not only a quantum gravity process, it is a non-perturbative quantum gravity process, so in all honesty we do not know how to calculate anything about it with complete confidence. But for some intents and purposes, the extent of which is not entirely clear to anyone, you can use semi-classical methods. The standard reference for how to calculate this process semi-classically is this beautiful paper by Coleman and de Luccia.

The value of the potential at the minimum acts as a cosmological constant, so your intuition is that if two local minima exist with different cosmological constants, the higher one is metastable, it decays to the lower one. But this is not correct:

Relevant to your question is repeating the above energetics analysis. You still have volume versus surface competition, but now they are calculated in curved rather than flat spacetime, because you have to include backreaction on the geometry. This makes all the difference in the world: in flat space volume always scales faster than surface, so large enough bubbles always want to expand, resulting in complete conversion of false into true vacuum. Once we take curved spacetime effects into account you compare volume in one geometry to surface in another, and all kinds of things can happen; the detailed picture is quite complicated. In particular, it is no longer always true that the false vacuum necessarily is converted into true vacuum! In other words, some expected decays (for example decays into negative cosmological constant) do not actually happen once you take gravitational effects into account.

For more details I recommend you immerse yourself in these papers, they are beautiful and are known for their style as well as their content. I also have to apologize for not referring directly to string theory. None of this has to do directly with string theory, only with expectations we have from the semi-classical theory.

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Great answer! I'm certainly going to read the Coleman - de Luccia article. However, there's another issue here they don't tackle as far as I understand, namely transitions between different number of uncompactified dimensions. –  Squark Oct 28 '11 at 18:17
    
I am assuming you mean the number of non-compact dimensions. If you think about the size of the extra dimensions as a scalar field, this is not conceptually much different. This is discussed for example in arxiv.org/abs/hep-th/0404220. –  user566 Oct 28 '11 at 20:08
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