This post in “Science After Sunclipse” reminded me of one of the most beautiful ideas to come out of the continuing attempts to combine String Theory and Cosmology. The idea of Robert Brandenberger and Cumrun Vafa, dubbed more recently as “String Gas Cosmology”, is a wonderfully creative attempt to explain why our world has three spatial directions. There is no other theory on the market where the dimensionality of space could be determined dynamically, or at least come out as a result of a calculation, rather than being put in as an input, so in String Theory this is a natural question to ask. The idea for an answer, provided by String Gas Cosmology, could be stated naturally and simply, which is what I try doing in this post. As usual, one has to remember that the devil is in the details, and those at the moment provide a real challenge for the idea.

One of the beautiful aspects of this approach is even asking the question: why four dimensions? famously, the dimension of spacetime in string theory is larger than the observed one, in the simplest scenarios 10 or 11 spacetime dimensions, depending on details. We of course only see 3 spatial directions to move in, and can only make sense out of one time direction. Isn’t that a clean Popperian falsification of String Theory?

As you may suspect, the answer to this question is no. There is nothing in the theory which requires all dimensions to be macroscopic, visible to the naked eye, part of our daily experience. Imagine an ant moving on a garden hose, that ant could move in two independent directions. In one of the directions the hose looks infinitely long (at least for an ant), and the other direction closes onto itself, having finite length. The ingenuity of the Brandenberger-Vafa approach is to realize that the natural question is not “how come most of the spatial dimensions in String Theory are so small?”, it is quite the opposite. String Theory has a single length scale, set by the tension of the fundamental strings. In generic situations, any other length scale should be more or less of that order of magnitude. The better question then is, how did some of the dimensions of space came to be so large?

The scenario of string gas cosmology goes as follows: imagine we start in the most natural initial state. In this state we have small dimensions in any direction in space, let’s imagine they are all circular, like in the example of the garden hose. We assume also that all the degrees of freedom of the theory are excited in some thermal equilibrium. Of course, the question of which initial conditions are natural for Cosmology has long and subtle history, but this certainly sounds plausibly generic. The idea is then to concentrate on “winding” strings wrapping those small circular directions (imagine a rubber band wrapping the small direction of the garden hose), which would be initially present. The claim is that those winding strings provide a tension that prevent the corresponding direction from expanding. As long as they are there, the direction stays small. Conversely, to allow for a particular direction to expand, reaching the macroscopic (cosmological, in fact) size we observe today, we need to somehow get rid of the strings wrapping it.

How do we get rid of those strings? this requires some interaction between them. If two winding strings meet each other, they can re-combine ends to create two non-winding strings. The strings that do not wind can then shrink and disappear. The upshot is that the basic interaction between two closed strings can make both of them disappear, allowing then some spatial dimensions to grow. The issue is now transformed to figuring out how many dimensions we can get to expand this way, by eliminating the strings wrapping them.

And here is the punchline: for two strings to have an appreciable change of interacting, they need to intersect each other. The chance of two strings intersecting each other depends very strongly on the number of dimensions. In one spatial dimension any two wrapped strings actually overlap with each other. In two spatial dimensions any two strings (wrapping independent directions) necessarily intersect each other, at each moment in time. Three spatial directions is the magic number: this allows for finite probability of intersection. Any larger number of directions leads to vanishing probability of string intersection. Then, the story goes, the winding strings would not have a chance of interacting with each other. They just stick around, preventing those extra dimensions from growing. The expansion stops at three spatial dimensions.

Isn’t this beautiful? if you are a theorist, the next step is to go and check all kinds of details. Do the winding strings really prevent expansion? is the probability for interaction really finite in three spatial dimensions (after all, the fundamental strings are both relativistic and quantum mechanical, maybe those classical pictures miss the point). How about experimental signature? for example statistics of fluctuations of the cosmic microwave background provides lots of constraints for any model of early universe cosmology. The upshot is, beautiful as it is, so far no cigar (to use the expression favored by my former graduate adviser). Some details don’t seem to work at the moment, which is the fate of the overwhelming majority of beautiful ideas. No other way to go but to keep trying…

on October 20, 2008 at 8:35 pmYooAm I wrong in my understanding that causal dynamical triangulations is an approach that successfully predicts a smooth four-dimensional space-time in the large scale? Or is it related to string gas cosmology?

on October 20, 2008 at 8:40 pmMosheYeah, generally some of my statements would implicitly involve judgment call about the validity of various claims, but in this case, even if the claim is correct the “smooth” part is the surprising result (about which one can be skeptical), the four dimensions were there from the get go.

on October 21, 2008 at 3:02 amBlake StaceyThanks for sending a link my way.

on October 21, 2008 at 3:22 amMosheNo problem Blake, I am kind of amused to see this beautiful yet somewhat arcane physics infiltrating pop culture. Next, the gauges for quantizing non Abelian gauge theories!

on October 21, 2008 at 6:01 amMark RIt is beautiful. It even has a romantic quality.

on October 21, 2008 at 1:38 pmKeith CMaye this is a dumb question. Suppose I label the spatial dimensions by 1, 2, …,10. If I understand correctly, in this scenario, at one place spatial dimension 1, 2 and 3 are macroscopic, while in another place spatial dimension, say 4,5,6 are macroscopic, and so on. Can these dimension be knitted together consistently? If so, is it similar to local gauge theory that the phase can be set locally?

on October 22, 2008 at 10:32 amBlake StaceyI don’t think the choice of dimensions to be made macroscopic is a quantity which can vary across spacetime, but perhaps my understanding is oversimple.

on October 22, 2008 at 10:53 amYooMaybe dimensions varying by location could work with fractional dimensions? Although deciding what

kindof fractional dimensions might be a topic for holy wars. :P(Has no idea if the idea is even feasible.)

on October 22, 2008 at 3:25 pmMosheI think the choice of dimensions to be made macroscopic can be a local choice. Moreover, one can imagine having a wavefunction with more or less equal probabilities for all possible choices. In fact, choice of specific 3 directions breaks the initial symmetry spontaneously, and there is no true symmetry breaking in finite volume, the ground state is the symmetric wavefunction. I’m not sure if that issue was ever dealt with, but there are probably more pressing issues.

Yoo: there are not going to be holy wars here, but I do view it as my responsibility to give my opinion, hopefully respectfully. I see no evidence that the concept of fractional dimension plays a role in QG, maybe I am wrong. In the long distance physics we already probed, this is experimentally excluded, in all experiments involving gravity we see three dimensional smooth geometry at long distances.

on October 22, 2008 at 4:37 pmYooI was more referring to the fact that are different definitions for fractional dimensions in a tongue in cheek manner. I don’t even know if you have an opinion on what the One True Definition of fractional dimensions should be, or if there should even be one …

(And I’m still not sure if you just neglected to mention other possible candidates for dynamically derived dimensions or whether it was because you consider all the other candidates to be vastly inferior. All I know is that I am completely unqualified to judge what theories might actually approach reality, so I just wanted to know if I misunderstood something.)

on October 22, 2008 at 5:01 pmYooJust to clarify something: I just brought up fractional dimensions since it sounds odd for one location to have, let’s say, three macroscopic dimensions, while a location right next to it has four macroscopic dimensions with nothing in between. Or having one dimension be macroscopic in one location and it suddenly being microscopic in the next.

Although in retrospect, maybe thinking in terms of how many dimensions are macroscopic is the wrong way to think about it, that the proper way to think about it is how macroscopic each dimension is. I have the feeling that I’m still missing the mark, though …

on October 22, 2008 at 5:57 pmMosheNo worries Yoo, we are just having a quiet conversation here.

For your comments: I am unaware of any successful attempt to derive the number of spacetime dimensions, as I write in the post this cute idea I describe does not quite work. What I meant to say is that if your starting point is an attempt to quantize 4 dimensional GR, using your favorite mathematical trick, you give up from the beginning the possibility of explaining the number 4, it is an input rather than a result. There are also a few claims specific to CDT whose meaning is mysterious to me, but that is besides the point.

In your second comment, I think you got it in the second paragraph.

on October 22, 2008 at 7:12 pmYooThanks! It makes a lot more sense now. :)

on October 23, 2008 at 2:03 amIphigenia“We assume also that all the degrees of freedom of the theory are excited in some thermal equilibrium”

Where exactly did you use this assumption? Certainly you don’t mean that *all* of the degrees of freedom are in thermal equilibrium [there would be no arrow of time in that case] but even the strings don’t need to be in thermal equilibrium, do they? As long as they are moving around in *any* way!

on October 23, 2008 at 4:54 amMosheThe assumption is used in Einstein equations summarizing the response of spacetime to specific distribution of matter, which is assumed to be thermal distribution of winding strings. For the qualitative discussion I gave it is enough to assume there are winding strings present in some distribution not picking any specific direction.

On the larger point, I am completely with you on the “naturalness” of the initial conditions. We know that because of the arrow of time the universe started at some atypical state. Even if the initial state looks kind of natural or familiar, it cannot be generic in any precise sense.

on October 23, 2008 at 5:30 amIphigeniaOh I see. Thanks!

on October 23, 2008 at 8:53 pmBlake StaceyOn the hep-th arXiv today is an M-theoretic attempt to get a 3+1-dimensional universe, which points back to an earlier effort to derive three large spatial dimensions using an entropy maximization principle. This makes me feel a little odd, as I cooked up the same idea for a science-fiction story the year before. Maybe I should go back and revise the story to put in some chatter about gauge/gravity duality and gauges for quantizing non Abelian gauge theories. . . .

on October 23, 2008 at 9:11 pmMosheDo let me know if you manage to work those gauges in, plenty more ideas where this came from.

on October 23, 2008 at 10:39 pmYoYoVip » Predicting the spacetime dimensionality[...] was posted in Military category and has 0 Comments so far. In this text, I would like to expand Moshe Rozali’s comments about string gas cosmology and clarify the difference between legitimate speculative work in [...]

on November 1, 2008 at 7:09 pmmtheorist: 11 dimensions may be a dream in realityfolks, we have come a great diatance in understanding the universe, but still something does say me that we’ve come a long way in assumptions getting ourselves blindfolded in the beauty of mathematics…

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