First, I'll state some background that lead me to the question.

I was thinking about quantization of space-time on and off for a long time but I never really looked into it any deeper (mainly because I am not yet quite fluent in string theory). But the recent discussion about propagation of information in space-time got me thinking about the topic again. Combined with the fact that it has been more or less decided that we should also ask graduate level questions here, I decided I should give it a go.

So, first some of my (certainly very naive) thoughts.

It's no problem to quantize gravitational waves on a curved background. They don't differ much from any other particles we know. But what if we want the background itself to change in response to movement of the matter and quantize these processes? Then I would imagine that space-time itself is built from tiny particles (call them space-timeons) that interact by means of exchanging gravitons. I drawed this conclusion from an analogy with how solid matter is built from atoms in a lattice and interact by means of exchanging phonons.

Now, I am aware that the above picture is completely naive but I think it must in some sense also be correct. That is, if there exists any reasonable notion of quantum gravity, it has to look something like that (at least on the level of QFT).

So, having come this far I decided I should not stop. So let's move one step further and assume string theory is correct description of the nature. Then all the particles above are actually strings. So I decided that space-time must probably arise as condensation of huge number of strings. Now does this make any sense at all? To make it more precise (and also to ask something in case it doesn't make a sense), I have two questions:

  1. In what way does the classical space-time arise as a limit in the string theory? Try to give a clear, conceptual description of this process. I don't mind some equations, but I want mainly ideas.

  2. Is there a good introduction into this subject? If so, what is it? If not, where else can I learn about this stuff?

Regarding 2., I would prefer something not too advanced. But to give you an idea of what kind of literature might be appropriate for my level: I already now something about classical strings, how to quantize them (in different quantization schemes) and some bits about the role of CFT on the worldsheets. Also I have a qualitative overview of different types of string theories and also a little quantitative knowledge about moduli spaces of Calabi-Yau manifolds and their properties.

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    $\begingroup$ Having intuition is great but great discoveries are rarely made solely by intuition. I am not saying this to discourage you, if you try to prepare a mathematical background, you'll stumble upon some facts about your theory which you may not be able to reach solely by thinking. $\endgroup$
    – Cem
    Nov 18 '10 at 17:29
  • $\begingroup$ As for your initial question of is this valid, why not? What you say makes sense to me, if you can prepare a solid mathematical background, we can take this a bit further. $\endgroup$
    – Cem
    Nov 18 '10 at 17:30
  • $\begingroup$ @Cem: I most certainly agree. I gave my thoughts and intuition just to put the problem into some context. But what I am really after in this question is AFAIK a very well established result in the string theory. $\endgroup$
    – Marek
    Nov 18 '10 at 17:33
  • $\begingroup$ The string theories I know of are formulated against a static space-time background. So your idea of many strings forming space-time is not quite correct, although if I remember how GR's equations arise in string theory, there's something to it. But the way you present your ideas now makes me think more of loop quantum gravity than string theory. I learnt the basics of string theory from a primer many years ago, I think it was from Polchinski. I'd advise to avoid anything Polchinski. :p $\endgroup$ Nov 18 '10 at 17:58
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    $\begingroup$ I don't have time to think about how to write a proper answer, so just some comments: in perturbative string theory, spacetime is essentially put in by hand via the choice of worldsheet field theory. AdS/CFT gives a setting where one direction of (a particular curved) space emerges. Matrix theory is a setting where all directions of a flat-space theory emerge. It's probably the most enlightening thing to try to study to get some feeling for how space can emerge. $\endgroup$
    – Matt Reece
    Nov 18 '10 at 18:19

First, you are right in that non-Minkowski solutions to string theory, in which the gravitational field is macroscopic, it should be thought of as a condensate of a huge number of gravitons (which are one of the spacetime particles associated to a degree of freedom of the string). (Aside: a point particle, corresponding to quantum field theory, has no internal degrees of freedom; the different particles come simply from different labels attached to ponits. A string has many degrees of freedom, each of which corresponds to a particle in the spacetime interpretation of string theory, i.e. the effective field theory.)

To your question (1): certainly there is no great organizing principle of string theory (yet). One practical principle is that the 2-dimensional (quantum) field theory which describes the fluctuations of the string worldsheet should be conformal, i.e. independent of local scale invariance of the metric. This allows us to integrate over all metrics on Riemann surfaces only up to diffeomorphisms and scalings, which is to say only up to a finite number of degrees of freedom. That's an integral we can do. (Were we able to integrate over all metrics in a way that is sensible within quantum field theory, we would already have been able to quantize gravity.) Now, scale invariance imposes constraints on the background spacetime fields used to construct the 2d action (such as the metric, which determines the energy of the map from the worldsheet of the string). These constraints reduce to Einstein's equations.

That's not a very fundamental derivation, but formulating string theory in a way which is independent of the starting point ("background independence") is notoriously tricky.

(2): This goes under the name "strings in background fields," and can be found in Volume 1 of Green, Schwarz and Witten.

  • $\begingroup$ Thank you Eric. This is a nice answer which, although didn't explain my question completely, touched upon the areas I am not comfortable with. In the construction in the second paragraph we assume there is only one string, correct? If so, how does one proceed when having more strings at once? And how are metrics on those independent strings related? Or is this a completely wrong view and instead the one string already describes all the strings (in the same way that e.g. an electron field describes all the electrons)? I am really confused about this and its relation to the first paragraph. $\endgroup$
    – Marek
    Nov 18 '10 at 21:25
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    $\begingroup$ Processes involving say the scattering of two strings still have a world-sheet description. The Riemann surface looks like two cylinders asymptotically at one end, and some other number of cylinders at the other end. The joining of two strings would be described (to leading order) by a "pair of pants." (This is the sense in which interactions are "smoothed out" in string theory and there are no discrete parameters such as coupling constants attached to vertices of Feynman graphs. The graphs are smooth.) $\endgroup$ Nov 18 '10 at 22:02
  • $\begingroup$ Oh, so this is how it works. Thank you very much, it's all starting to fall into place now. I think I more or less got my answer but I'll wait and see whether someone else won't add a different view on the problem. $\endgroup$
    – Marek
    Nov 18 '10 at 22:44

I'm no expert in string theory. However I wasn't satisfied by the above as the proposed answer to this question of mine about how General Relativity is a particular limit of String Theory so I did a little research of my own. I'm interested in this particular question as generally physics pedagogy demonstrates how newer theories are linked to older conceptions: like getting Newtonian gravity from GR - but in String Theory, this step as a physically motivating step is remarkably difficult to pin down. One would have thought, it would be one of the first things taught - even if not demonstrated - in a first course on string theory.

According to Becker, Becker & Schwarz (String and M Theory), super-gravity theories are particular low-energy limits of super-string theories. And then super-symmetry is broken by a super Higgs mechanism to give gravity. This appears to be a classical limit since it's going from the action.

However, the preceding is mostly just jargon to me - for now (and probably later too) - as the mathematics is difficult to follow.

** edit **

There are four related arguments that are outlined in the introduction to Feynman's Lectures on Gravity which begins with the graviton, the hypothetical spin-2 quanta of gravity deduced from the linearised equations of gravity and whose equation of motion was first written down by Fierz & Pauli in 1939:

  • An argument by Suraj Gupta in 1954 deduced that the action of such a theory must obey a non-linear consistency condition which led him to a recursively defined infinite series which summed up yields Einstein's equation. Gupta did not carry out the actual infinite sum, this was done by Deser in 1970.

  • Robert Kraichnan in a an unpublished thesis in 1946 also outlined a similar theory but unlike Gupta did not assume that gravity coupled to the energy-momentum tensor but deduced this from a consistency condition.

  • Weinberg in 1964 beginning with reasonable assumptions on the analycity properties of graviton-gravition scattering he shows that a theory of a graviton can only be Lorentz invariant when it couples to matter, including itself, with a "universal strength". Hence if strong equivalence is satisfied.

  • Feynman in a lecture series 1962-63 showed that a self-consistent theory of the graviton leads to Einstein's gravity. These are the lectures in this book.

I think that these arguments should be better known than they are. They certainly weren't referred to in a couse of string theory that I attended and nor in Schwarz & Becker & Becker's book on string theory. Especially if they are used to argue that if the spectrum of string theory includes a graviton then this means it also is a theorybof gravity.


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