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.

70 Comments

70 thoughts on “How many dimensions are there?”

  1. Sean,

    Are there any strong rigorous “No-Go” theorems which show that it’s impossible to construct consistent theories with massless particles of spin greater than two? Or is it more like “circumstantial evidence” from many people having tried to make these theories consistent, but failed every single time?

  2. JC, I am sure Sean can add to this, but there is the Weinberg-Witten theorem which under certain assumptions proves that all theories with spin higher than 2 have trivial S-matrix, in other words they are free (consistent but boring).

    The basic idea is that higher spin particles have more modes that could lead to negative norm states, so they need more gauge invariance to get rid of them, at some stage there is so much gauge invariance the theory is trivial.

    Some interesting exceptions to the assumptions: conformal field theories that have no S-matrix so they can exist. Two dimensional theories are an exception and there are 2dim theories of massless higher spin. In higher dimensions there are also some (inconclusive) attempts to have infinitely many higher spin fields (e.g all spins) which evades another condition of the thm.

  3. I must say that I’m a bit appalled by the way people these days are so keen to declare that the number of dimensions is more or less a question of how you look at things. That may sound very sophisticated in a sort of phony post-modern way, but, apart from any technical objections, this is a deeply *boring* way of doing physics. In the admittedly highly unlikely event that extra dimensions are revealed at the LHC, you can bet that none of the post-modern physicists will be saying, “nothing to get excited about folks, these so-called extra dimensions into which our beams are disappearing are just a manner of speaking….” On the contrary, they will all [I hope] be jumping up and down saying, “WOW! Extra dimensions REALLY EXIST!!! That is way cool!!!!!”.

    Bottom line: when people talk about whether the extra dimensions “really exist”, they are discussing something which is meaningful and important. To dispute this you would have to give a deep analysis of what it means to really exist, which of course nobody can do; though presumably even post-modernists are able intuitively to distinguish the real from the non-existent.

    I fear that this sort of “multicultural” attitude, that all ways of looking at the theory must be equally valid, is itself a strong symptom of all that ails string theory at the moment. And I’m a believer.

  4. Pingback: A Crazy Physics Week I - The Syracuse-Cornell Joint Theory Meeting | Cosmic Variance

  5. This is related to a more fundamental question – namely, what are the observables in string theory? It seems that the common answer to this question is that the observables are S-matrix elements. This kind of answer might make sense in quantum field theory, where it is assumed in advance that the spacetime dimension is four, and where S-matrix elements have a direct relationship to the statistics of the out-states (which we already know how to identify) that are produced when we send some in-states (which we already know how to prepare) into a small region of space.

    In string theory, however, where the properties of the background geometry are determined dynamically, the situation is not the same. Specifically, these dualities indicate that one set of physical phenomena in N dimensions is “the same as” another set of physical phenomena in M dimensions, where N is not M. The question is, if the theory is “true”, then how many dimensions will we see around us, N or M? If the S-matrix is the only observable, and is compatible with more than one number of spatial dimensions, then the dimension of space is not an observable.

    This is related to a more general question in quantum theory. If I want to show that a certain quantum system, specified by its Hamiltonian, H1, is equivalent to another, with Hamiltonian H2, then I can check to see if the Hamiltonians have the same spectra and degeneracies. If they do, I can form a mapping from one Hilbert space to the other, mapping eigenspaces to eigenspaces of the corresponding eigenvalue. The resulting map establishes that the two are “equivalent”. A trivial example is that a system of two free particles moving in one dimension is equivalent to a single free particle moving in two dimensions.

    But does that mean that a single free particle moving in two dimensions is “physically” the same as two free particles moving in one dimension? Obviously not; we can easily distinguish the two cases if we are allowed to perform some measurements on the system. The two systems are indistinguishable from the point of view of their quantum representations. The Hilbert space along with all the various symmetries that act on it this way or that, are not a complete characterization of what the system is. We need to know what subspaces of the Hilbert space correspond to the results that are registered on our measuring instruments, if we want to say what the system is “physically”.

    What does this mean in the case of string theory? I would say that it underlines the fact that a connection to experimental results is very important, and that while string theory looks like a quantum theory, it is not a quantum theory in the original sense, namely a theory where the state space is constructed as a set of formal sums of measurement results. It is the relation between a measurement result on the one hand, and a subspace of the Hilbert space on the other hand, which provides the link between the mathematical formalism and the experimental phenomena, and this is precisely the link which string theory is having difficulty establishing.

  6. But does that mean that a single free particle moving in two dimensions is “physically” the same as two free particles moving in one dimension? Obviously not; we can easily distinguish the two cases if we are allowed to perform some measurements on the system

    rof,

    Doesn’t introducing an observer implicitely make it a three body problem instead of a two body problem? Is there a name for this kind of effect ( a lowly phys undergrad here)?

    Thanks, everyone, for the great comments here!

  7. “Doesn’t introducing an observer implicitely make it a three body problem instead of a two body problem? Is there a name for this kind of effect ( a lowly phys undergrad here)?”

    Poincare chaos

  8. Sean, thanks. It’s a pity that we can’t (yet) justify the full dimensionality of M-theory in elementary terms.

    We have two scenarios where an extra dimension is conjured up. Why stop with a single dimension? If a dimension can be emergent in this way, why not all of spacetime? As an intermediate step, do you know of any conjectures that map an N-dimensional theory to an N-2-dimensional one?

    George

  9. A lot of issues here that I am not expert enough to clear up. But, just to address George’s last question: there are strict rules about what you’re allowed to do, and what will happen. It’s not just a matter of “we made one new dimension appear, let’s do it again and make some more.”

    You have to be careful about words like “emergent,” because it has pre-existing connotations that may or may not be relevant to how the theory ends up actually working. In some sense spacetime is emergent in string theory, but my own quasi-outsiders view is that we don’t yet understand things well enough to make definitive statements.

  10. What bothers me the most in these sorts of discussions, is what we really mean by the ‘topology’ of spacetime. It doesn’t bother me that manifolds can break and change, it does otoh profoundly bother me that the full global topology of the universe is not fixed. AFAIU, string theory can have local topology change (Riemann surfaces getting twisted and unified and so forth), but the premise was the full spacetime topology of the universe was fixed, perhaps into some gigantic monstrous thing.

    Otoh, I find it hard to reconcile then talking about dimensions of space being ’emergent’, it seems to me it will trivially violate various index theorems if we settle for one number and then poof out pops another one with a higher dimension. Thats where I find the degree of freedom infinity = infinity argument ultimately breaks down mathematically.

  11. To Count Iblis,

    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

    You cannot test for Lorentz violation, CPT violation, a pseudoscalar chiral vacuum background, Equivalence Principle parity violation… if you begin by postulating Lorentz Invariance, CPT conservation, an isotropic vacuum background, the Equivalence Principle…. Your “proof” simply restates your assumptions. Even within your sciolism, reversing parity without going to antimatter introduces an asymmetry. “C” is an internal symmetry. Its observables cannot first order couple to translation or rotation by definition.

    Parity has tremendous impact. Should your math be strictly scalars, axial vectors, and tensors or must it include pseudoscalars, polar vectors, and pseudotensors? If there are ungerade parity-odd terms you’ve got a source for measurable overthrow of orthodox gerade physics at its postulate level with well-defined and calculable parity experiments – with no falsification of prior achiral observaitons. All one need do is challenge the heterodox vacuum left foot with a (calculated tightly-fitting) left and right shoe instead of interminable socks. If it is there, one shoe won’t fit like the other one.

    Significant journal acreage has been published and a number of Eotvos experiments have been performed contrasting properties coupled to internal symmetries via Noether’s theorem,

    http://www.mazepath.com/uncleal/lajos.htmTable I – testsTable II – symmetries and propertiesTable V – property mass-% vs. total test mass

    that must all null by default: Internal symmetries cannot first order couple to translation or rotation. There was nothing to be seen. Even if there were to higher orders, the fraction of (active mass)/(total mass) in each experiment is negligible. Take that delta and multiply by the epsilon of higher order scaling. No outputted net signal was to be had under any circumstances, yet the experiments were performed and published without protest.

    CTP and Lorentz violation are viable inquiries,

    http://www.physics.indiana.edu/~kostelec/faq.html

    The parity Eotvos experiment in space group P3121 vs. P3221 single crystal quartz and the calorimetric test of /_/_Hfusion of space group P3121 vs. P3221 single crystal benzil are high amplitude tests of external symmetry-coupled properties. Their net active mass fraction exceeds anything else proposed by at least 400-fold.

    We have here a significant discussion of the number of dimensions defining spacetime and even what a dimension constitutes. Nowhere has anybody stopped to question the simplifying assumptions of isotropic and homogeneous spacetime. Newton tacitly assumed lightspeed was infinite, Feynman was thrown for a loop by S-T vs. V-A in beta-decay, Einstein assumed the Equivalence Principle and then spacetime curvature, Yang and Lee got the 1957 Nobel Prize in Physics by demonstrating parity violation in beta-decay. If we have learned anything it is that simplifying assumptions – especially parity transformation symmetries – don’t go very far. String and M-theory are mathematical triumphs and physical disasters. Something we “know” to be true does not obtain in the real world.

    Nobody has examined parity transformation symmetry of the vacuum. You don’t know if there is parity-dependent Lorentz violation, CPT violation, or Equivalence Principle parity violation given a heretofore unexamined pseudoscalar chiral vacuum background. If there is, String Theory et al. is suddenly much smaller and more tractable. Classical gravitation and quantum field theory are still consistent with all prior observations, but must be rewritten to include the parity anomaly case.

    Euclid did good work, but he wasn’t complete. Is it obvious that given a point not on a given straight line only one line may be drawn through that point parallel to the given line? Given the Earth’s surface – surveying and navigation – one is astounded that nobody suspected otherwise until the 1800s. Is it obvious that local left and right hands vacuum free fall identically? Don’t you think somebody should look?

  12. George, let me try to give a simple reasoning of the number 11: if you start with one supersymmetry in 4dim, you have one spin 1/2 fermion for every spin 0 scalar, so you can have SUSY theory with maximal spin 1/2. You can see that to have maximal spin 1 you have at most 4 SUSY, then when you act 4 times on the spin 1 helicity state you get spin 1/2, 0,-1/2,-1. ..Similarly if you want maximal spin to be 2 you can have at most N=8 SUSY. If you have more SUSY than that you necessarily have somewhere spin greater than 2.

    So the most supersymmetric theory you can have in 4dim has 8 supersymmetries, larger symmetry will make the theory trivial. If those supersymmetries come from higher dimension they have to be spinors of the (local) rotation group of that space. By playing with creation and annihilation ops., it is easy to s that the dimension of the spinor representation is 2^{D/2} where D is the dimension for even-dimensional spaces, and (dimension-1) for odd-dimensional ones. So the magic number 8 is obtain for D=6, which means the manifold you compactify on has dimension 6 or 7, resulting in total dimension of 10 or 11.

    As an exercise, if you had 12 dimensional theory, the spinor representation would be 16-dim, which results in N=16 SUSY, and the maximnal spin is 4, too large.

    That turned out pretty long, but hopefully still not too complicated reasoning…

  13. Re: twistor space strings and non-critical strings. Those are two examples of “small” string theories, meaning they do not contain all of the phenomena we want to describe ( gravity, gauge fields, chiral matter etc.) . For example the non-critical strings are formulated at most at 2 dimensions, and the twistor strings don’t contain Einstein gravity. They are nevertheless interesting, for example as toy models for “large” string theories (this could be called the Goldilocks classification…), in fact the non-critical strings are in many ways much better understood than the critical ones, and many interesting phebnomena such as D-branes had their origins there.

  14. Could someone knowledgeable please respond to rof‘s exceptionally lucid comment? Sean, Moshe, Aaron, …?

  15. Like I said above, I’m not expert enough to shed any light on rof’s questions. But I think a lot of them are just good, open questions — I’d certainly like to understand better what the observables of string theory are, or why some certain set of variables appears weakly-coupled rather than some other set.

  16. I am not sure I understand everything rof said, but I’ll have a crack at it.

    Since most of what’s there has to do with quantum gravity generally, let me not even mention string theory. In QG the metric fluctuates like any quantum mechanical degree of freedom, so talking about any geometrical concept at all is same as talking about trajectories of electrons, useful only in some classical limit but generally invalid.

    Thing is, quantum mechanics seems to require having some classical object as the measuring device, at least the usual way it is interpreted, but since gravity is universal it is difficult to arrange some subsystem which is non-fluctuating and set it as a measuring device.

    The objects that are known to make sense are S-matrix elements: you stay away from the strongly gravitating region of spacetime (at “infinity”) send out signals into the chaotic soup and measure what comes out. Since you are in non-fluctuating region you can use all the usual machinery and interpretation of QM, You stay out of all the troubles rof mentions *precisely* because you look at S-matrix elements and not at more general things that do exist in QFT (correlation functions).

    On the other hand this S-matrix can sometimes be consistent with more than (or sometimes less than) one interpratation of “what was going on” inside, if you insist on telling such stories and phrase them in geometrical language. I tend to think about this as a good thing.

    If you want to have other well-defined things in the theory, in addition to S-matrix elements, you may have to work harder, and it is not clear to me these things have any right to exist, that is a point where opinions diverge and therefore discussions become interesting. rof does a good job of making explicit some of the technical and intepretational issues that may arise.

    And let me emphasize again, I did not have to mention string theory at all in all of that, this is just gravity and quantum mechanics.

  17. Torbjorn Larsson

    Uncle says,

    “Spacetime parity bears heavily on the origin of biological homochirality.”

    Why? Are there _any_ explanation yet for biological homochirality?

    “Why are there only protein L-amino acids and only D-sugars?”

    Wikipedia says that the above is wrong. There is 20 standard amino acids, but over 100 found in nature (including nonbiological ones from meteorites). They are used in other important biological roles in addition to protein synthesis.

    “The L amino acids represent the vast majority of amino acids found in proteins. D amino acids are found in some proteins produced by exotic sea-dwelling organisms, such as cone snails. They are also abundant components of the cell walls of bacteria.”

  18. Uncle Al, I agree that my claim follows from the assumptions I put in. I was merely giving you an example of how Nature could be exactly invariant under parity given the known fact that P and CP (appear) to be broken.

    P non-invariance could be a fundamental property of space-time and perhaps that could lead to the effects you are looking for in your experiment.

    Theorists often have to lobby a lot to get even cheap experiments done. I recently read an article about axions and it was claimed that a simple experiment could be done to rule out or confirm a recent positive result, see here:
    http://arxiv.org/abs/hep-ph/0511184

  19. Moshe,

    Are there any sensible theories in 4+n dimensions, where the SUSY operators are coming from both the n-dim higher dimensional sector and the 4-d sector?

  20. Sean:

    You have to be careful about words like “emergent,” because it has pre-existing connotations that may or may not be relevant to how the theory ends up actually working.

    I would have thought the modifications to GR might have signalled some truth to what was emergent(although this would ask us what that quantum geometry is?) from a condense matter perspective, as Witten saids below.

    I also heard Robert Laughlin say, it didn’t matter if you use bricks or sargeant majors?

    I had trouble with this ,and looking at CFT on the horizon, it made me think of string as a fifth dimensional component within the blackhole. Is this wrong and misleading, not to have looked at the spacetime fabric a a graviton constituent since these modifications were made to GR?

    Witten:

    One thing I can tell you, though, is that most string theorist’s suspect that spacetime is a emergent Phenomena in the language of condensed matter physics.

  21. In regards to the other statements rof made, I’m not sure how apposite they are. String (perturbation) theory is still a first quantized theory. We know what the asymptotic states are, and we can compute S-matrix elements in a fair fraction of backgrounds. These are honest experimental observables. The problem is that we don’t know the right background, and it’s unlikely that we’ll soon or ever reach the energy levels where distinctive properties of stringy scattering become apparent.

    Discussions of observables in quantum gravity often end with one participant throwing something at another — usually saying “observe this!” — so I’ll just echo Moshe’s comments in saying that the only thing I know how to do is to sit in my nice semiclassical world (either at weakly coupled asymtopia or on the boundary of AdS), throw stuff into the strongly gravitating mess and see what comes out. One hopes along the way that one doesn’t get eaten. There be dragons….

Comments are closed.

Scroll to Top