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I would like to understand how Physicists think of space-time in the context of String Theory. I understand that there are $3$ large space dimensions, a time dimension, and $6$ or $7$ (or $22$) extra dimensions, and all these dimensions need to fit together in a way such that the extra dimensions are compactified (with a Calabi-Yau or $G_2$ structure).

My question, however is not about the possible $10$, $11$ or $26$ dimensional manifolds that may be possible, but about whether string theorists consider space-time as somehow quantized (or discrete), or rather as a continuous manifold, or are both options possible? In other words, can strings move continuously through space, or are there a discrete set of locations where strings can be, and does string theory rule out one of the options?

How about the same question in loop quantum gravity (LQG)? Should I think of the spin networks in LQG as describing a discrete space-time?

Thanks for your insight, or any references you may be able to provide!

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I hope an expert replies to your question to the education of all. Off hand I know from wikipedia articles that string theory views space time as continuous but LQG quantizes it. –  anna v Apr 24 '12 at 4:18
My understanding is that in ST spacetime is continuous. In LQG it is NOT discrete, it is quantized in the sense that geometric observables (like length area volume) have discrete spectrum. –  MBN Apr 24 '12 at 8:43
Additionally, and a little more one the technical side, I'd like to know how the appearent "get rid of space" idea in pure S-Matrix theory works out with its realization in string theory, where afaik (i.e. in the textbook aproaches to string theory I know) the spacetime is not much different than in QFTs. Spacetime in LQG seems a little cooler (for someone like me who has a huge problem with time), because different possibilities of all-of-space configurations are packaged up in different states. –  NikolajK Apr 24 '12 at 9:07
Thank you all for the replies. @MBN, the geometric observables have a discrete spectrum... but the geometric observables of what? Does this mean that every possible observable length, area, volume (say length of a string, or area of a membrane) is a multiple of a fixed quantity? –  Álvaro Lozano-Robledo Apr 24 '12 at 19:27
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3 Answers

I think Anna s comment is correct, in LQG spacetime consists of discrete atoms and in ST it is continuous.

In addition, This article contains an interesting and quite accessible Nima talk related to the topic. Therein Nima explains why the present notions of spacetime are doomed and introduces the recent cutting edge ideas about how spacetime could emerge from a newly discovered and not yet fully explored structure called T-theory.

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Thank you for your answer, and thank you for the link. I'll watch the video soon. –  Álvaro Lozano-Robledo Apr 24 '12 at 19:29
How is T-duality newly discovered? It's thirty years old already. –  Ron Maimon Jun 22 '12 at 1:08
@RonMaimon Hm, if I remember and understood it correctly T-theory in this context should be something different from T-duality. Some kind of an underlying structure unifying string theory and twistor theory or something (?) ... –  Dilaton Jun 23 '12 at 16:55
@Dilaton: I see, it's not T-duality. "T theory" is nothing at all. –  Ron Maimon Jun 23 '12 at 18:45
@Dilaton: No, I just mean that it's twistors and perturbation theory (the stuff AH is working on now), which is great and interesting and useful, but it isn't a new physical theory, but a bag of (useful and important) calculation techniques. There might be some new idea coming out of this, but I don't see a reconstruction of space time in any of the recent literature on twistors and peturbation relations (although it is very active). The last progress on this question was in the AdS/CFT era. But to advertize this technically challenging work, somebody made up the term "T-theory" to sell it. –  Ron Maimon Jun 23 '12 at 20:04
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This is a very good question, because no one knows the answer. In a recent talk I asked the very same question to Dr. Brian Greene. I also asked him why we don’t' see many papers dealing for example with the quantum dynamics of branes in M-Theory, just many low energy solitonic semiclassical physics from some low energy 11-D supergravity lagrangian. Your questions about quantum spacetime are deeply related with the nature of physical quantum branes. His answer was straight foward he said "you aren't missing anything, we just don't know". In string theory, in principle, space time can be a fully quantum membrane in some dimension with open string excitations and a bulk probably another space filling brane with also closed quantum strings degrees of freedom. But brane quantization is still not well known. At the present state of string theory most calculations assume a space time continuum. But its very difficult to reconcile this notion with space time producing close string states, or a string state warping space. Maybe in the future it will be possible to make calculations or formulate string theory in quantized backgrounds that maybe branes themselves

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Thanks for your answer! –  Álvaro Lozano-Robledo Jun 20 '12 at 21:05
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So far only a perturbative formulation of string theory is known, despite the fact that there are some hints for what a non-perturbative formulation should contain. As far as I understand, it is expected that the geometry of the background spacetime in which the string propagates in the perturbative formulation, will ultimately be encoded in some other way in a non-perturbative formulation.

Roughly you can think about it as follows: Instead of quantizing General Relativity directly, which fails in a naive perturbative approach, perturbative string theory contains a field, that would arise in a perturbative quantization of General Relativity, too. In string theory this field is the massless part of a whole tower of massive fields. This together with the fact that a consistency condition gives you the vacuum equations $R_{\mu\nu} = 0$ of General Relativity (this is true at least in the bosonic sigma model, without $B$ field or Dilaton) are two reasons to believe that perturbative string theory contains a perturbative quantization of gravity at least in 26 or 10 dimensions. Contrary to the naive quantization, it yields (some) finite results at loop level (for superstring theory, this is actually only known up to two loops I believe).

In a sense that can be made somewhat precise: Certain 2-dimensional QFTs should be thought of as generalized (semi-)Riemannian manifolds.

Since there is actually no known non perturbative formulation one studies instead low-energy effective theories (supergravity theories), compactifications (here is where the calabi-yau manifolds come in), F-Theory and so on. Always in the hope that they might give a clue to what a non perturbative formulation should contain. In that way the fact that there is a 12-dimensional supergravity theory that dimensionally reduces to 11-dimensional theories leads to the idea that there should be an $M$-Theory or the presence of $p$-Form fields leads to the idea that there should be charged "branes".

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