Why 10 or 11?

Why does string theory require 10 or 11 spacetime dimensions? The answer at a technical level is well-known, but it’s hard to bring it down to earth. By reading economics blogs by people who check out political theory blogs, I stumbled across an attempt at making it clear — by frequent CV commenter Moshe Rozali, writing in Scientific American. After explaining a bit about supersymmetry, Moshe concludes:

A guide in this pursuit is a theorem devised/put forth by physicists Steven Weinberg and Edward Witten, which proves that theories containing particles with spin higher than 2 are trivial. Remember each supersymmetry changes the spin by one half. If we want the spin to be between -2 and 2, we cannot have more than eight supersymmetries. The resulting theory contains a spin -2 boson, which is just what is needed to convey the force of gravitation and thereby unite all physical interactions in a single theory. This theory–called N=8 supergravity–is the maximally symmetric theory possible in four dimensions and it has been a subject of intense research since the 1980s.

Another type of symmetry occurs when an object remains the same despite being rotated in space. Because there is no preferred direction in empty space, rotations in three dimensions are symmetric. Suppose the universe had a few extra dimensions. That would lead to extra symmetries because there would be more ways to rotate an object in this extended space than in our three-dimensional space. Two objects that look different from our vantage point in the three visible dimensions might actually be the same object, rotated to different degrees in the higher-dimensional space. Therefore all properties of these seemingly different objects will be related to each other; once again, simplicity would underlie the complexity of our world.

These two types of symmetry look very different but modern theories treat them as two sides of the same coin. Rotations in a higher-dimensional space can turn one supersymmetry into another. So the limit on the number of supersymmetries puts a limit on the number of extra dimensions. The limit turns out to be 6 or 7 dimensions in addition to the four dimensions of length, width, height and time, both possibilities giving rise to exactly eight supersymmetries (M-theory is a proposal to further unify both cases). Any more dimensions would result in too much supersymmetry and a theoretical structure too simple to explain the complexity of the natural world.

This is reminiscent of Joe Polchinski’s argument (somewhat tongue-in-cheek, somewhat serious) that all attempts to quantize gravity should eventually lead to string theory. According to Joe, whenever you sit around trying to quantize gravity, you will eventually realize that your task is made easier by supersymmetry, which helps cancel divergences. Once you add supersymmetry to your theory, you’ll try to add as much as possible, which leads you to N=8 in four dimensions. Then you’ll figure out that this theory has a natural interpretation as a compactification of maximal supersymmetry in eleven dimensions. Gradually it will dawn on you that 11-dimensional supergravity contains not only fields, but two-dimensional membranes. And then you will ask what happens if you compactify one of those dimensions on a circle, and you’ll see that the membranes become superstrings. Voila!

67 Comments

67 thoughts on “Why 10 or 11?”

  1. Thanks Eva, I think we are in agreement.

    Lee, the facts in your first paragraph are indeed deeply related to supersymmetry, another place this is utilized is in the Green -Schwarz quantization of the string, those are precisly the dimensions the classical GS strings allows (3,4,6,10).

  2. Moshe, yes, thanks, the connection between strings and octonions is recognized but seems a possible clue to a deeper underestanding of string theory that is not yet well explored. Corinne A. Manogue and colleagues have explored it in papers including, Phys.Rev.D40:4073,1989 and hep-th/9807044. Some other attempts are hep-th/0104050, hep-th/0110106, hep-th/0503017.

    The alternative advocated by Eva and Clifford seems very important to explore. It would be good to know whether or not supersymmetry and extra dimensions are essential for string theory. I hope Eva and Clifford are right, but in case SUSY is essential there must be a deeper way to understand it, perhaps afforded by octonions.

  3. Eva et al.– Without being an expert, I completely agree that exploring non-critical string theory is an important idea. This is especially true since we have no detailed experimental data, and what we have isn’t well described by ten-dimensional Minkowski space with unbroken supersymmetry. (And the fact that, not too long ago, we wouldn’t have been talking about 11 dimensions.)

    The point of this discussion, I think, is more about explaining ourselves than about understanding the nuances of string theory or supersymmetry. The fact that there is something special about certain values the number of dimensions of spacetime is surprising, and hard for the person on the street to understand. I think what George Musser was originally looking for was some concrete imagery, similar to the idea that strings only generically intersect in three spatial dimensions. The maximal dimension for susy is not quite so visual, but it’s what we have.

  4. Basic question here: Are ideas like ” a volume contains an infinite number of planes” useless in trying to understand 11D? Does an 11D thing contain an infinite number of 10D things?

  5. I forgot to say: “Joe! Eva! Welcome to the blog!” I don’t think we’ve had comments from you on an earlier thread (unless my memory fails), and it’s really excellent to see you here.

    Eva, we seem to be very much of a similar mind on stamping out the whole “string theory requires D=10/11” business. Excellent!

    Sean, I do think that we understand the perfeclty sensible narrower parameters of the discussion of the post, but I do think that it is appropriate to include what Eva and I am saying in this particular discussion, since the “person on the street” all too often hears (or implicitly gathers from posts like this) the phrase “string theory requires D=10/11”, and it is simply not true and in some years we may well have to be spending a lot of time undoing yet another uncautious claim when/if after doing phenomenology better we find that we don’t need to start in higher D and then “compactify”. We’ll have to go around telling everyone (on the tv shows and radio shows and magazines) “oh…that thing we said about extra dimensions? We were just kidding”…. Just like we’re doing now with the whole “unique vacuum” and “theory of everything” phrases…

    Cheers,

    -cvj

  6. sisyphus: I think the answers to your questions are “no” and “yes” respectively.

    When you think about 3D, you can imagine a set of 3 axes, denoted by a line pointing up-down, a line going left-right and another going backwards-forwards. To imagine more dimensions, I just have to pretend to myself that I can put lines in more directions. For example, if you wanted 3 large dimensions such as you see, and then a fourth compact dimension (I’m just talking spatial dimensions here, ignoring time for now), picture it as there being a little circle at every single point in 3D space.

    Now, you specify the position of a particle in the 4D space by saying where in the 3D space it is (ie give an x-, a y- and a z-coordinate), and then give another number which tells you how far around the circle you have gone, measured from a set point. Since this dimension is compactified to a circle, the fourth coordinate value must be between 0 and 2pi (circumference of a circle of unit radius). But there are still an infinite number of points on this circle.

    Thus, if you want to put objects in the 4D space, you have to specify 4 coordinates, and each of these can take an infinite number of values.

    To work in more dimensions you just have to keep putting in more directions as we did to add the fourth, but the rules for how they work stay the same. Personally, I cannot picture higher dimensional spaces (and I’m not convinced anyone else can really, you just get used to the idea and the maths simplifies things) but in principle they don’t behave any more weirdly than those you are used to (except possibly being compactified!).

    Lee: some years ago I read a bit about graded Lie algebras in the maths literature. Are these related to the division algebras you mentioned? It is all very hazy as I’ve not thought about it in quite some time, but it was certainly related to SUSY some how. It may be that “graded Lie algeba” was just another name for “SUSY algebra”. Is this relevant, or is my memory leading me astray?

  7. Layman scratching head….

    So what is the “right circumstance” that the “analogies(superfluid created)” would speak to symmetry breaking, as “phase transitons?”

    These would be dimensionally linked?

  8. fh# 10, anon # 14

    Yes one can say that you don’t have a theory in that case, but you could also say that Nature doesn’t allow you to fix the theory using measurements made at low energy scales.

    I would agree with anon that there are no valid arguments against a nonrenormalizable theory. The fact that a low energies we have renormalizable theories is i.m.o. just because any nonrenormalizable terms have renormalized to zero when integrating out high energy physics.

    We can’t then use the argument that just because at low energies everything is renormalizable it should always be so.

  9. Poppycock: No, but thanks for reccalling graded Lie algebras, whicih eons ago a few of us studied as a road to a geometry for supergravity. It turned out to be not as useful as super Lie algebras. Division algebras are systems of numbers that share some properties of the reals such as the existence of multiplicaiton, additionn and division, but don’t have others such as commutivity and associativity. A nice intro to them and octonions is John Baez’s http://math.ucr.edu/home/baez/week59.html.

  10. More on name above.

    I think John Baez makes it easier sometimes and I appreciate that kind of material. Banchoff’s discriptions of the computer screen is much different. More indepth? 🙂

    In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number – and Salmon answered, that the number should be exactly 27


    .

    What is interesting is the idea of Sylvestor surfaces and early Cayley references, on how we can shape our thinking in context of these dimensional attributes.

    Is this right that such “spin valuations” would have signalled, one phase transition held in dimensional consideration, to the next? I apologize if I have confused the situation.

  11. It seems there always is this need to try and explain it better? 🙂 Being at a loss for words, one tends to keep trying?

    The summation to D brane consideration, encompasses this view? CY perspective arise from it?

  12. Apologies for being somewhat off topic, I couldn’t figure how to start my own blog and this was the first one I could hijack.

    My question refers to the prize given out by the I believe American Institute of Aeronautics and Astronautics regarding a novel form of space propulsion, namely that of antigravity. I have a BSc(Hons) in mathematics and an MSc in geometry, mathematical physics and analysis where I did lots of courses in general relativity. Their idea is based around something called ‘Heim theory’, which is as far as I can tell is a high dimensional general relativity. The claim further goes to to say that Heim himself managed to couple his theory to quantum mechanics therefore getting some form of quantum gravity.

    I have a gut feeling that this is a complete joke but my colleagues and a well known scientific establishment in the UK feel otherwise. I would like to hear peoples opinion on the matter.

    Once again apologies for hijacking the blog.

    Regards

    Mat

  13. Mat, I don’t know a lot about it, but what I’ve seen seems to not make much sense. I wouldn’t take it seriously if I were you.

  14. That was my thought, it seemed like another crackpot theory to me but senior colleagues were saying not to ignore it as apparently it ‘predicts’ the correct masses of the elementary particles.

    In short, it just sounds too good to be true.

  15. Mat, there is no Pinocchio theory! All the puppets here need strings. I mean.. Come on! Who wants to use a imaginary solar sail anyway, fall through the universe at Twice the speed of light, don’t be silly. No! Am afraid it’s chasing the rainbow, the dazzle in the distance, the light fantastic and the great big mushroom thats always on the Horizon, always.

  16. String theory may be applicable to the the new ‘Lost World’ described in this weeks’ news articles, which outline a team investigation “in boggy clearings” found in the western half of New Guinea, Indonesia. The ‘strings’ would be the atmospheric linkages formed among all the unique organisms of that ecosphere, with resultant “pristine zone”. It can be deduced that an abundance of water is necessary to eliminate friction among the unique species varieties discovered, and facilitate the dynamics of island gravity.

    Just don’t go busting in — the articles state that the region is usually off-limits to foreigners. Infra-red testing and photography of atmospheric pulse might also reveal those linkages which are necessary to hold the creatures within island environmental niches. Most natural biochemistry perpetuates linkages in terms of six or hexagonal groupings.

  17. naive experimentalist

    Perhaps I’m missing something, but I have heard that if you start with superstring theory and quantise it, then you are forced to use 10 dimensions. If you don’t you end up with either negative norms or tachyons.

  18. Well, that is simply not true. This is said a lot, but it is not true, as I say above.

    It is only true if (for example) you assume that you have Lorentz invariance in all dimensions. This was done for simplicity’s sake only. But we know that’s not true. We know only that it is true in 3+1.

    -cvj

  19. But it’s more than than that.

    It’s not just that, for the supercritical strings Eva and collaborators have been considering, you never have D-dimensional Lorentz-invariance (for some D > 10). You never have a lower-dimensional Lorentz-invariance either.

    So it’s not 100% clear what the observables of such theories are.

    I think it’s still an open question whether such theories can be made sense of.

  20. Very true…nevertheless, I find it disturbing that we have not explored non-critical strings as much as we should have, since learning so much about strong coupling physics. I just have a gut feeling that we’re missing a trick here.

    Until we’ve really done that job, I’m not comfortable with the trumpeting of the “stirngs only live in ten dimensions” twaddle that we keep telling everyone, often including our students…. We’ve got to remember what we assumed in order to get to the cirtical dimensions, and then revisit those assumptions every time we learn something new about the whole story. I might be overstating things…. it’s late here and I should get to bed.

    -cvj

  21. #44, Clifford: Just a dumb layman’s question here: Am I right in assuming that if Lorentz invariance doesn’t necessarily survive > 3+1, then, generally, the application of the spacetime interval formula to Multi-D is meaningless?

Comments are closed.

Scroll to Top