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I was browsing through Foundations of Space and Time, a compilation of essays on various theories of quantum gravity. The following passage in the introduction intrigued me:

Each compactification leads to a different vacuum state.... at least one state should describe our Universe in its entirety.... the enormous number (~10^500 at last count) of solutions, with no perturbative mechanism to select mechanism to select among them, leads some critics to question the predictive power of the theory..Even more worrying is that, while the theory is perturbatively finite order by order, the perturbation series does not seem to converge.

I don't know anything about string theory and so I could not make head or tails this. All I know is that ~$10^{500}$ is a very large number.

  1. What exactly is a 'solution' in string theory? Is it a spacetime metric of some sort or the terms of a S-matrix of some sort?

  2. Why are there so many 'solutions'?

  3. I thought string theory was supposed to be finite, why do perturbative series still diverge?

  4. Is there any experimental technique to limit the number of 'solutions'?

  5. Will experimental techniques be able to pinpoint a solution within present day string theorists' lifetimes too? If not, how long will it take before we can experimentally probe these things?

  6. Are string theorists completely relaxed about these issues? Or are they in anguish?

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Who wrote that passage? It contains some misunderstandings.

All I know is that $10^{500}$ is a very large number.

It is a finite number. How many theories do you know which have a finite number of solutions? Have you tried to count the number of solutions of plain Einstein-Yang-Mills-Dirac-Higgs theory without its string-theoretic UV completion? There are not only infinitely-many solutions, there is a hugely infinite-dimensional space of solutions. This is the usual state of affairs for most every theory of physics ever considered. String theory is special in that it puts many more constraints on the solutions, such as to even leave just a finite number (under some assumptions).

What exactly is a 'solution' in string theory?

A background for perturbative string theory is a choice of 2-dimensional superconformal QFT of central charge -15. This can be interpreted as describing an effective target space geometry which is a solution to a higher dimensional supergravity theory with higher curvature corrections. A "solution" to string theory is a solution of the equations of motion of that. At least without non-perturbative effects taken into account.

See on the nLab: landscape of string theory vacua for more.

I thought string theory was supposed to be finite, why do perturbative series still diverge?

String theory is thought to be loop-wise finite, thus being a renormalized perturbative theory. No sensible remormalized perturbative QFT can have converging perturbation series. The perturbation series must be an asymptotic series to be realistic, and it comes out exactly like this in string perturbation theory.

See at the String Theory FAQ on the nLab the item Isn’t it fatal that the string perturbation series does not converge?

Is there any experimental technique to limit the number of 'solutions'? Will experimental techniques be able to pinpoint a solution within present day string theorists' lifetimes too?

Models that have been and are being built in string phenomenology approximate the standard model to more detail than probably most people are aware the standard model even has. Check out some of the references there. Given the slow but continuous flow of new articles on these matter, one sees that some people are slowly but surely working on improving ever further. Check out the references at string phenomenology.

[edit: I have now added a corresponding item to the nLab String Theory FAQ: What does it mean to say that string theory has a “landscape of solutions”?]

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Thanks. The passage was written by Jeff Murugan, Amanda Weltman and G F R Ellis. So there are only 10^500 completely unique universes possible? But are there any aspects of String theory that give non-perturbated solutions? –  dj_mummy Aug 27 '13 at 16:08
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No, that number 10^500 which has become so famous in public discussion plays no specific role. It is just a generic example of the following counting: 1. IF one assumes that 10 spacetime is compactified on a Calabi-Yau (which one used to be interested in (only) because this makes the effective 4d thory N=1 supersymmetric) and 2. IF one considers type II "flux vacua", then the highr form fields of string theory (the higher electromagnetic fields, if you wish) are quantized/constrained to have integral periods over the cycles of the compact Calabi-Yau. It is this quantization condition, akin... –  Urs Schreiber Aug 27 '13 at 16:37
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...akin to the Dirac charge quantization condition which makes anything finite in this game at all. Moreover, the higher these integral periods are, the more they contribute to some potential energy. Some other constraints say that this energy cannot be too big. So as a rule of thumb on says that one has a choice between 10 different values for each such flux field on each cycle. Similarly, as a rule of thumb, one says that a generic Calabi-Yau has 500 nontrivial cylces, give or take a few hundred. The conclusion of this thumb-counting is that the number of choices for the flux fields is... –  Urs Schreiber Aug 27 '13 at 16:39
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... is 10 per cycle, and 500 times a choice of 10, hence 10^500 choices. As you see, this number is hand-waving incarnate and the only information it carries is this: look, while there are finitely many of choices of internal flux, due to a charge quantization condition, on a typical Calabi-Yau there are still quite a few such choices. –  Urs Schreiber Aug 27 '13 at 16:41
    
I have now added a corresponding item to the nLab String Theory FAQ: ncatlab.org/nlab/show/… –  Urs Schreiber Aug 27 '13 at 17:35
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"Each compactification leads to a different vacuum state.... at least one state should describe our Universe in its entirety.... the enormous number (~10^500 at last count) of solutions, with no perturbative mechanism to select mechanism to select among them, leads some critics to question the predicitive power of the theory..Even more worrying is that, while the theory is perturubatively finite order by order, the pertrubation series does not seem to converge."

Ok.

I don't know anything about string theory and so I could not make head or tails this. All I know is that ~10^500 is a very large number.

That, it is.

What exactly is a 'solution' in string theory? Is it a spacetime metric of some sort or the terms of a S-matrix of some sort?

So, we have a non-peturbative definition of M-theory and string theories on AdS Space through the AdS/.CFT correspondence. Now, these are 10 or 11-dimensional. To get rid of the extra 6 or 7 dimensions, you need to compactify it on a 6-dimensional or 7-dimensional manifold.

A particularly convinient compactifications of 11-dimensional M-theory is on $G(2)$-holonomy manifolds. Particularly convinient compactifications of 10-dimensional string theories, such as Type HE, are on $SU(3)$-holonomy Calabi-Yau manifolds. Of course, it's not necessary; e.g. if the world happens to be something with $\mathcal N=2$ supersymmetry, as opposed to $\mathcal N=1$.

Why are there so many 'solutions'?

Because there are lots of these manifolds!

I thought string theory was supposed to be finite, why do perturbative series still diverge?

Uh... Yes. But it's renormaliable. And there are non-peturbative definitions in AdS spacetime.

Is there any experimental technique to limit the number of 'solutions'? Will experimental techniques be able to pinpoint a solution within present day string theorists' life times too? If not, how long will it take before we can experimentally probe these things?

In principle, it's possible. But in anyone's lifetime... Do you know how big $10^{500} $ is? % See this.

Are string theorists completely relaxed about these issues? Or are they in anguish?

There's the branch of String Phenomenology that attempts to find the correct vacua...

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Thanks. It's good to know that there is a definite way to pinpoint the solution. –  dj_mummy Aug 27 '13 at 16:14
    
@dj_mummy: In fact, I've heard that it can be done so mathematically (as opposed to phenomenologically), too. I don't know how that is done, thougjh,... –  Dimensio1n0 Aug 28 '13 at 1:45
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