How many dimensions are there?

When the fall quarter started, there were six papers that I absolutely had to finish by the end of the term. Three have been completed, two are very close, and the last one — sadly, I think the deadline has irrevocably passed, and it’s not going to make it. So here’s the upshot.

About a year ago I gave a talk at the Philosophy of Science Association annual meeting in Austin. The topic of the session was “The Dimensions of Space,” and my talk was on “Why Three Spatial Dimensions Just Aren’t Enough” (pdf slides). I gave an overview of the idea of extra dimensions, how they arose historically and the role they currently play in string theory.

But in retrospect, I didn’t do a very good job with one of the most basic questions: how many dimensions does spacetime really have, according to string theory? The answer used to be easy: ten, with six of them curled up into a tiny manifold that we couldn’t see. But in the 1990’s we saw the “Second Superstring Revolution,” featuring ideas about D-branes, duality, and the unification of what used to be thought of as five distinct versions of string theory.

One of the most important ideas in the second revolution came from Ed Witten. Ordinarily, we like to examine field theories and string theories at weak coupling, where perturbation theory works well (QED, for example, is well-described by perturbation theory because the fine-structure constant α = 1/137 is a small number). Witten figured out that when you take the strong-coupling limit of certain ten-dimensional string theories, new degrees of freedom begin to show up (or more accurately, begin to become light, in the sense of having a low mass). Some of these degrees of freedom form a series of states with increasing masses. This is precisely what happens when you have an extra dimension: modes of ordinary fields that wrap around the extra dimension will have a tower of increasing masses, known as Kaluza-Klein modes.

In other words: the strong-coupling limit of certain ten-dimensional string theories is an eleven-dimensional theory! In fact, at low energies, it’s eleven-dimensional supergravity, which had been studied for years, but whose connection to string theory had been kind of murky. Now we know that 11-d supergravity and the five ten-dimensional string theories are just six different low-energy weakly-coupled limits of some single big theory, which we call M-theory even though we don’t know what it really is. (Even though the 11-d theory can arise as the strong-coupling limit of a 10-d string theory, it is itself weakly coupled in its own right; this is an example of strong-weak coupling duality.)

So … how many dimensions are there really? If one limit of the theory is 11-dimensional, and others are 10-dimensional, which is right?

I’ve heard respected string theorists come down on different sides of the question: it’s really ten-dimensional, it’s really eleven. (Some have plumped for twelve, but that’s obviously crazy.) But it’s more accurate just to say that there is no unique answer to this question. “The dimensionality of spacetime” is not something that has a well-defined value in string theory; it’s an approximate notion that is more or less useful in different circumstances. If you look at spacetime a certain way, it can look ten-dimensional, and another way it can look like eleven. In yet other configurations, thank goodness, it looks like four!

And it only gets worse. According to Juan Maldacena’s famous gravity-gauge theory correspondence (AdS/CFT), we can have a theory that is equally well described as a ten-dimensional theory of gravity, or a four-dimensional gauge theory without any gravity at all. It might sound like the degrees of freedom don’t match up, but ultimately infinity=infinity, so a lot of surprising things can happen.

This story is one of the reasons for both optimism and pessimism about the prospects for connecting string theory to the real world. On the one hand, string theory keeps leading us to discover amazing new things: it wasn’t as if anyone guessed ahead of time that there should be dualities between theories in different dimensions, it was forced on us by pushing the equations as far as they would go. On the other, it’s hard to tell how many more counterintuitive breakthroughs will be required before we can figure out how our four-dimensional observed universe fits into the picture (if ever). But it’s nice to know that the best answer to a seemingly-profound question is sometimes to unask it.

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70 Responses to How many dimensions are there?

  1. Vincent Noel says:

    it’s really ten-dimensional, it’s really eleven. (Some have plumped for twelve, but that’s obviously crazy.)

    Obviously.

  2. erc says:

    Sean, could you please explain the jump from the end of the fourth paragraph to the beginning of the fifth in more detail please? ie what do you mean when you say the Kaluza-Klein modes give rise to an extra dimension.

    Thanks,
    E

  3. Dissident says:

    Sean wrote: “Now we know that 11-d supergravity and the five ten-dimensional string theories are just six different low-energy weakly-coupled limits of some single big theory”

    We know this? Last time I checked, it was a conjecture. Did I miss a new string revolution? One every other day, it seems…

  4. Sean says:

    Well, the usual line of thought is: imagine we have an extra dimension that is relatively large. It gives rise to a tower of modes corresponding to all the fields in the large dimensions. (So, not only we would have a massless photon, but an infinite number of photon-like particles with increasing masses.) As the extra dimension got smaller, all the extra modes get heavier.

    In field theory, that’s what it means to have an extra dimension: an infinite tower of particles with certain masses and interactions. So you can play the game in the other direction: if fool around with your coupling constants and you happen to notice a tower of modes, you can interpret it as evidence for an extra dimension you hadn’t noticed before! At least, if you’re Ed Witten you can.

  5. FP says:

    Sean,

    it seems that there are good arguments for F-theory, with 2 time and 10 space dimensions. So you are off by one dimension again. Sorry 😎
    http://xxx.lanl.gov/abs/hep-th/0512047

  6. Aaron Bergman says:

    Even though the 11-d theory can arise as the strong-coupling limit of a 10-d string theory, it is itself weakly coupled in its own right

    In what sense?

  7. George Musser says:

    Sean, can you explain in elementary terms what is special about 10 or 11 dimensions, such that the equations of string theory prefer to live in them? For instance, we might understand a preference for three dimensions in terms of, say, whether knots hang together, or what degree of complexity is allowed by physical laws.
    George

  8. Helix says:

    String Theory – Ptolemy for the 21st Century.

  9. Arun says:

    According to Juan Maldacena’s famous gravity-gauge theory correspondence (AdS/CFT), we can have a theory that is equally well described as a ten-dimensional theory of gravity, or a four-dimensional gauge theory without any gravity at all.

    But since we live in a world that appears to have 3+1 dimensions and certainly has gravity, this correspondence means….?

    If this 4-D gauge theory in the AdS/CFT correspondence has confinement, what does that imply about the ten-d theory?

  10. WL says:

    This topic has been overdue, good that you bring it up.

    “The dimensionality of spacetime is not something that has a well-defined value in string theory; its an approximate notion that is more or less useful in different circumstances.”

    Indeed, dimension (and other geometrical data) is a quite ambiguous quantity in string theory, and only certain 4d theories in certain regions of the paramter space can be viewed as “compactifications” of a 10d (or whatever theory); in general, there isn’t a notion of a “compactification” in the sense there is no energy scale above which a theory looks 10 dimensional (ie, gains 10d Lorentz invariance).

    From such a view-point of “non-geometric” theories (which are known to abundantly exist since exactly 20 years), there is no need to talk about extra dimensions etc, all what counts that there is the right number of “internal” degrees of freedom to make the theory consistent. Often it is possible to interpret these internal degrees of freedom as coming from extra dimensions, but while this can be very useful, it’s not really necessary.

    Here another example in a somewhat different spirit: N=2 supersymmetric gauge theory in 4d. It is known since the work of Seiberg and Witten that this theory is non-perturbatively characterized by a Riemann surface, which seemed ad-hoc at first. Later it was realized that this Riemann surface can be given a concrete geometrical meaning in terms of a higher-dimensional theory (like 6d self-dual strings), which is compactified on that Riemann surface such as to give rise to the 4d gauge theory.

    So, is this theory now intrinsically 4 or 6-dimensional ? All what one can say is that there are different dual formulations of this theory, which involve different numbers of dimensions (and which may be more, or less useful, respectively, for answering certain questions). Often key insights can be gained by switching between such different descriptions of the same theory, and comparing how certain quantities look in each description.

    In view of these well-known facts, I see this fixation of certain string-critical people (incl recent book authors) against extra dimensions all but amusing.

  11. Arun says:

    So, is this theory now intrinsically 4 or 6-dimensional ? All what one can say is that there are different dual formulations of this theory, which involve different numbers of dimensions (and which may be more, or less useful, respectively, for answering certain questions). Often key insights can be gained by switching between such different descriptions of the same theory, and comparing how certain quantities look in each description.

    In view of these well-known facts, I see this fixation of certain string-critical people (incl recent book authors) against extra dimensions all but amusing.

    Well, until you can relate the theory to what we observe, measure, experience, this is a very valid criticism of string theory. Why do we experience a 3+1D world and not its dual, for instance, if both are equivalent? String theory here introduces yet another mystery rather than solving one.

  12. Peter Woit says:

    I don’t think Krauss in his book anywhere objects to “extra dimensions” like the one in AdS/CFT or the two in Seiberg-Witten, where the extra dimensions provide a way of getting a dual, fully equivalent, formulation of a 4d theory. I think if you read his book you’ll see that he is talking specifically about recent brane-world scenarios or the older idea of compactifying 6 or 7 dimensions with very small size, in either case giving these extra dimensions precisely the same dynamics as the 4 standard ones.

    Sure, there are all sorts of interesting ways of thinking about particle physics models using more “dimensions” than four. I would claim that the standard model is best thought of by thinking about a 16 dimensional space (a fiber bundle with fibers SU(3)xSU(2)xU(1) over spacetime). The thing for which there is no evidence is not extra dimensions in general, but extra Riemannian geometry dimensions where the metric now carries many more degrees of freedom with supposedly the same dynamics as the four we know.

  13. Uncle Al says:

    Is the mathematics of spacetime gerade parity-even or ungerade parity-odd? That is the most important fundamental question. Rather a lot of journal acreage could be discarded if it rigorously fell one way or the other.

    Spacetime parity bears heavily on the origin of biological homochirality. Why are there only protein L-amino acids and only D-sugars? A mirror-image organism would not have predators outside its chirality – it could not be digested. Why aren’t there any?

    It bears heavily upon physics. Is the Weak Interaction strictly left-handed, or is it symmetric within an underlying asymmetric background? Are teleparallel theories of gravitation degenerate and unnecessary? Is angular momentum conserved by both of a pair of extremal opposite parity bodies, forcing the arrow of time to point in only one direction?

    Only an observed diastereotopic interaction with the vacuum is diagnostic. A left foot is not detected by sock or a left shoe. A left foot is detected by a right shoe (preferarably a tightly fitting right shoe). Do local left and right hands vacuum free fall identically?

    Somebody should look. Explicit calculation of a mass (e.g., atoms) distribution’s quantitative parity divergence (chirality along all directions without bias) is almost a decade old. It’s in public domain software.

    One cannot count dimensions to predict presence or absence of chirality. (2N+1) odd-dimension counts harbor chirality by default. 2N even-dimension counts also contain chiral objects. A spiral in 2-D and its mirror image cannot be superposed. 2-D scalene triangles are chiral.

    Reality could be shoes – much more interesting than the “obvious” assumption of socks. Somebody must look at the disjoint heterodox case.

  14. Count Iblis says:

    Uncle Al,

    The vacuum can be parity symmetric if there exists a mirror sector. See articles by Mohaptra, Tepliz, Foot, Berezhiani etc. on arXiv about (broken) mirror matter models.

    If that’s the case then, if you introduce an operator M which exchanges mirror fields by ordinary fields and vice versa, you have by the PCT theorem:

    1 = PCT = PM^2CT = (PM)(MC)T

    Now since PM = 1 in mirror matter models, MCT = 1. So, nature is invariant under parity and replacing fields by mirror fields and also under time reversal and replacing fileds by the charge conjugated mirror fields.

  15. Count Iblis says:

    Hmmm, the number of dimensions seems to grow larger and larger:

    Recent Comments
    How many dimensions are there? 15

    🙂

  16. Moshe says:

    Sean, I really like the conclusion about the dimension not being well-defined. In fact many believe that in quantum gravity any geometrical concept in cannot be fundamental (same as the concept of well-defined trajectories in QM) and has to always be an asymptotic, approximate, notion only. The fact that this happens so naturally in string theory is one of the encouraging signs we are on the right track.

    So in string theory this is manifest, when you have different approximations to the same background, often (not always…) you discover geometrical interpretations pop out, but for different asymptotic expansions you get different geometries- those can differ by their dimensions, or can be of same dimension but different topology, etc. etc.

    For the compact dimensions, if their size is small, the geometrical interpretation is ambigous at best, it is better to think about those as “internal” degrees of freedom.

  17. Moshe says:

    OK, I see that WL beat me to most of the points…but there is something even more amazing if we are talking about asymptotic expansions. Similar to the idea of the same model having different geometrical interpretations in different limits, there are models that have different semi-classical limits: in different regions in parameter space an inherently QM model re-arranges itself in a semi-classical approximation, different such arrangements in different limits, in each one of them a different parameter plays the role of h-bar.

  18. Sean says:

    Aaron, I just meant that the 11d theory is weakly-coupled supergravity. No?

    George, I don’t think there’s anything like a simple geometric reasoning behind the numbers 10 and 11 — they pop out of fairly elaborate calculations. For superstrings, 10 is the “critical” dimension, the one in which the theory is truly conformally invariant (local scale transformations). People have done work on non-critical string theory (including Clifford), but my impression is that it’s a murkier subject. For supergravity, 11 is the largest dimension in which you can have supersymmetry without having particles with spins greater than 2, which are hard to make consistent theories from.

    There is something cute that I left out: 11-d supergravity naturally has two-dimensional branes in it. When you compactify one dimension, these become — superstrings!

  19. Paul Valletta says:

    The fundamental questions re: dimensions is really, :How many dimensions are in expansion, how many are static, and how many are contracting, and how many (if any ) are a mixed and inter-twined of coupled dynamic dimensions?

  20. Aaron Bergman says:

    I just meant that the 11d theory is weakly-coupled supergravity.

    M-theory has 11D SUGRA as a low energy limit, but I don’t know what you mean by weakly coupled. There aren’t any free parameters in M-theory.

  21. Plato says:

    There is something cute that I left out: 11-d supergravity naturally has two-dimensional branes in it. When you compactify one dimension, these become — superstrings!

    Cheez, this sounds like eveything being reduced to a fifth dimensional view within Bekenstein bound, and thus reduced again, to a two dimensional framework.

    Imagine. A string started it all. I would prefer that one saw the upper part of this picture alone, but how can you stop imaginative thinker from addressing perspective that way?

    I won’t mind if you take this post out. 🙂

    Is it to much? 🙂

  22. Doug says:

    Moshe wrote:

    Sean, I really like the conclusion about the dimension not being well-defined. In fact many believe that in quantum gravity any geometrical concept in cannot be fundamental (same as the concept of well-defined trajectories in QM) and has to always be an asymptotic, approximate, notion only. The fact that this happens so naturally in string theory is one of the encouraging signs we are on the right track.

    It is the change in the geometric concept that is at the core of all of this. Sean wrote, “The basic insight is Einstein’s: spacetime is curved and dynamical. Extra dimensions could change in size, curl up, and otherwise hide from our view.” Clearly, however, anything that is dynamical, has size, can curve, curl up, or hide, must exist. Yet, there is no way to measure space without employing motion to do so; that is, without space/time, space cannot be shown to exist. Therefore, it follows that space is only the space aspect of space/time, and cannot exist independently of it any more than the opposite ends of a line segment, the opposite sides of a plane, or the opposite surfaces of a sphere, can exist independently of these. To talk of “extra dimensions” of space makes no more conceptual sense than speaking of extra dimensions of the left end of a line, or the right side of a plane, or the inside of a sphere, independently of the line, plane, or sphere that defines it! It’s nonsense.

    The basis of geometry is motion, not space. As Newton so sagely observed,

    “…the descriptions of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn…and it is the glory of geometry that from those few principles, brought from without, it is able to produce so many things…”

    Once it’s clear that geometry is founded on motion (mechanics), we can view Euclid’s fifth axiom in terms of motion instead of space: motion that is not in one dimension is necessarily in another dimension, and only three distinct dimensions of motion can be observed. Therefore, the space of Euclidean geometry is the space aspect of motion in three dimensions, while the space of non-Euclidean geometry is the space aspect of changing motion in three dimensions. So much for “extra dimensions” of space.

    However, the fact that in string theory we find that 10 or 11 dimensions, or degrees of freedom, are needed to preserve symmetry does not necessarily mean that these requirements have to be extra dimensions of space; they can be ramifications, and thus symptoms, of a mistaken concept: that space is something that can exist independently of space/time.

    Doug

  23. Science says:

    The issue of whether there are 10 or 11 dimensions in ST reminds you of the issue whether there are 8 or 9 gluons in QCD. James Bottomley and John Baez discuss this here
    http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/gluons.html

    Nine types of gluon:

    green-antigreen, green-antired, green-antiblue,
    red-antired, red-antiblue, red-antigreen,
    blue-antiblue, blue-antired, blue-antigreen.

    Why then are there only eight gluons? To make the physics work, you have to subtract one, but you don’t say which particular one you subtract. They concluded:

    “If you are wondering what the hell I am doing subtracting particles from each other, well, that’s quantum mechanics. This may have made things seem more, rather than less, mysterious, but in the long run I’m afraid this is what one needs to think about.”

  24. Sean says:

    Aaron, I just meant in the same sense that GR is weakly coupled in the real world — there’s not a small dimensionless parameter, but there’s a dimensionful parameter (the Planck scale), and at energies that are low compared to that scale perturbation theory works fine. The important point is that the degrees of freedom are organized into a set of almost-freely-propagating fields at low energies.