Why 10 or 11?

By Sean Carroll | February 4, 2006 1:44 pm

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!

CATEGORIZED UNDER: Science
  • Moshe

    My god you guys are quick…in fact this came out as elaboration on a comment I made here to George Musser, in the post you wrote. I had no idea at the time he was associated with SciAm, but he asked a concrete question and did not seem to be inclined to tell me precisely what is wrong with me…your post at the time BTW was really nice.

    I did not know about Joe’s route to string theory, but now I am convinced..

  • Paul Valletta

    Surely a one-dimensional “string” embedded into a “two-dimensional” anything?..will always have an energy factor, that is exponentionally sufficient to break/snap, the two-dimnensional ‘brane’, to such an extent that Branes cannot exist?

    Compactification of any “matter”, from any higher dimension, has a default INCREASE in the energy value?

    The only way around this “string-energy” paradox is if the Brane is “one-dimensional” and exists within a “string”..and not be external?

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Here is the question from George that Moshe was nice enough to try to answer. See, it pays to make useful comments on blogs, you could become famous.

  • Moshe

    Sean, you must have meant rich and famous…

    One clarification, despite starting out as question about string theory, the piece is not very specific to string theory. Basically if you want to combine the ideas of supersymmetry and extra dimensions you put an upper bound on the number of those extra dimensions. This by itself does not tell you there is a consistent theory, or of course if it relevant to nature.

  • fh

    This is very nice, and I can almost follow what’s going on…
    So basically the argument is that since we expect the higher dimensions to show up as symmetries at the 4dimensional viewpoint, and to much symmetry makes for vacuous theories, we can at most go to 10/11D if the additional symmetries show up as supersymmetry on a (flat? Wasn’t that an assumption of Weinberg Witten?) background, right?
    I assume there is no nice and easy way to see why we would expect the extra dimensions to manifest as supersymmetry?

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    Sean: Nice string – er, thread. Just how little I grasp of this will be made obvious by the following question: Is it possible to have 6 or 7 extra dimensions locally – that is epiphenomnal to our 4D spacetime – while the whole package exists within a global 5D context? I know that this is probably mathematically paradoxical if not downright silly, but I have to ask.

    I hope the answer is ‘no'; if it’s ‘yes’, I’ll never understand the explanation.

    Thanx.

  • J

    fh: Actually, the explanation isn’t hard. A supersymmetry has a spinor charges. In every dimension, there is a minimal size for a spinor. When we say “N=x” supersymmetry, we mean that you have x spinors of minimal size. In d=4, the minimal size of a spinor is 4 real dimensional, i.e. in 4 dimensions you always have 4x real supercharges that get mixed by rotations. This number grows very rapidly, more or less exponentially in the dimension.

    In 10 dimensions, the smallest spinor you can make has is 16 real dimensional. However, type II superstrings have N=2, so they have 32 supercharges, which from a 4 dimensional point of view looks like N=8 – 4×8=32. Rotations in 10 dimensions mix them together, but rotations in 4 dimensions mix each 4 dimensional spinor seperately.

    In 11 dimensions, you have to have at least 32 real supercharges for N=1, so you’re already at the bound. N>1 in d=11 or N>0 in d>11 requires so many supercharges that you you have to have spin >2 particles.

  • Maynard Handley

    Ignore the stringiness aspect of all this.
    The argument is predicated on supersymmetry being real.
    What is the current experimental status of this?
    Do we ignore ICE CUBE data, or has it just not been running long enough — there was a brief flurry of activity surrounding this about ten days ago which seemed to degenerate into some people saying it proved something or other, others saying no it did not and I lost track of what was actually going on.

    Beyond ICE CUBE, is this something that will actually be resolved to most people’s satisfaction in 1 yr, 5 yrs, 15 yrs? Or is it basically so slippery a concept that people can continue claiming it’s true (or at least possibly true) for the rest of my life.

  • http://countiblis.blogspot.com Count Iblis

    I don’t understand why one expects a quantum theory of gravity to be renormalizable. In principle, one can think of the standard model as a low energy effective theory that is to be obtained from the (unknown) theory of everything by integrating out the unknown high energy physics.

    Adding non renormalizable operators to the fundamental theory has almost no effect on the low energy physics, so I don’t see how one can work ”backward” and theoretically ”derive” the theory of everything from its low energy remnant.

  • fh

    J, thanks. At least on this level this makes sense.

    Count Iblis,
    the point is that if you have a renormalizable theory you can to some degree expect to be able to define it perturbatively, since order by order nothing undefined happens.
    So if you find a renormalizable perturbation expansion of Quantum Gravity you can say you have found a theory. If it’s nonrenormalizable you don’t have a theory.

  • Joe Polchinski

    Hi Sean, thanks for repeating this argument. I’d like to mention that it is just one example. Another starting point (trying to make a minimum length by introducing a position-position uncertainty principle) is followed to its inevitable conclusion in hep-th/9812104 and hep-th/0209105. Yet another (trying to make the graviton as a bound state of two gauge bosons) leads to the same endpoint — this is in a paper with Gary Horowitz that will appear in the next week or so.

  • http://www.pyracantha.com Pyracantha

    When people are mystified by my behavior, I tell them I’m from the “seventh dimension.” That usually ends the conversation.

    Once again, a large question from the ancient high school physics student:
    what is “supersymmetry?” Is it impossible to explain this on just a blog post? Are any books on it up-to-date?

    Yours, Pyracantha

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Joe, I’m glad to hear (at least implicity) that I got the argument right. And the bound-state idea sounds interesting, I’m looking forward to the paper.

    Pyracantha, supersymmetry is a proposed symmetry that relates the two basic kinds of particles, bosons and fermions. Fermions are “matter” particles that take up space (electrons, quarks, neutrinos), while bosons are “force” particles that happily pile on top of each other (photons, gluons, etc). If supersymmetry is true, every kind of boson/fermion we know of has a partner that is a fermion/boson (respectively), but that is so heavy we haven’t seen it yet. Supersymmetry is a key feature of string theory. There’s a good popular-level book on it by Gordon Kane, called simply “Supersymmetry.”

  • anon

    Count Iblis: you raise an interesting point that’s been bugging me lately. It’s true that a nonrenormalizable theory like a naive approach to quantizing gravity is in some sense not a theory at all, as it never makes predictions about high energies (one must measure an infinite number of constants to fix the theory). On the other hand, we don’t really know that the real world doesn’t work that way. It would be depressing, since it would entail a fundamental limitation to the effectiveness of science. But I can’t think of any reason to think it might not be the case. Can anyone else?

    Joe or Moshe: as an interested observer of string theory, I’ve been a little puzzled by the usual story of the critical dimension and compactification. Do we really have any compelling reason to think that our world is decribed by a compactified 10D string instead of a noncritical string with all sorts of nontrivial fields (dilaton, etc….) turned on? I suppose one could try to think of all such things as “compactifications,” where the extra dimensions are not simply a manifold but some more complicated geometric (or noncommutative-geometric or whatnot) object, but it seems like generically this wouldn’t be a useful viewpoint. It seems particularly worth asking since I get the impression that (at least some large number of) people tend to expect a lot of the usual vacua to have SUSY broken at a high scale, so the motivation for wanting to start with 4D space times a Calabi-Yau seems to me to be lost. Do I misunderstand the situation?

  • Thomas Larsson

    The main problem with SUSY is not whether it predicts 11D, but that God doesn’t seem to care.

    The natural prediction of supersymmetry is that all particles have a superpartner of equal mass. Since this is obviously completely wrong, one must assume that SUSY is broken. The first line of defense had to be abandoned immedately.

    But even if SUSY is broken in a natural way, it is still completely wrong, e.g. because it predicts proton decay a million times faster than experimental limits. Thus one had to posit some ugly ad hoc mechanism which suppresses dimension 4 operators, like R-parity conservation. The second line of defense also had to be abandoned.

    But even broken SUSY with R-parity conservation has severe problems with experiments. One would naturally expect to see a light Higgs, proton decay, muon g-2 deviations from the standard model, permanent electric dipole moments, WIMPS, and perhaps even sparticles, at presently available experiments. This kind of SUSY is not yet completely ruled out (it might be at the LHC), but we know that it is not natural – this is expressed by the buzzwords “SUSY requires fine-tuning at the percent level”. The third line of defense is in trouble.

    In view of these problems, some people have proposed so-called split SUSY. This forth line of defense is finally permanently safe from confrontation with nasty experiments, since it makes no predictions at all which can be tested in any experiment that will ever be feasible. (Probably; it is unclear to me if PEDM data could be used to rule out also split SUSY).

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    anon, the main reason people have studied phenomenological scenarios using critical strings as a starting point is because those were are the best understood…. unless I’ve forgotten something. (Progress in this field often follows the path of least resistance….which means that we sometimes take a while to get to important points we were close to a long time before..) Supersymmetry and higher dimensional Lorentz invariance (two key things that put you there in the first place) both get thrown out the window in a short while anyway, “so why were they so important in the first place?” you might ask….. I’ve been on a lonely campaign to remind people that we need to widen the scope a lot. There may be all sorts of riches living in non-critical string, including a real stab at describing Nature. It is time we did the hard work and made a real assault on that area, with our eyes a bit sharper this time around since we know many of the interesting things that show up beyond perturbation theory…..like branes…..

    See my “News from the front” posts on this blog for more on that.

    Best,

    -cvj

  • hackticus

    I’m not sure you’ve really succeeded in “bringing it down to earth” when you have to take as your starting point a theorem by Witten about higher spin particles. “Oh, I see, it’s just a straightforward consequence of the Witten-Weinberg theorem” the average man on the street is saying…

  • http://eskesthai.blogspot.com/2006/02/time.html PLato

    From a layman perspective I like things simplified, and finding model associations that would speak any positon held by the contributors here would help understand this process. But there is something more that needs to be done.

    Would Clifford see diffeently then us, if such a abstract developement would have taken him directly to D-brane analysis? How would he now see, encapsulating such discussion about N dimensional spaces and such. I mean he would had to have been able to see in different ways that most of us wouldn’t he?:)

    Or even a Lubos motl, who having understood these physics processes woud have married thinking of “two sides of a coin” down to the understanding of the dual nature of the blackhole, that such D brane analysis would have made some kind of sense in the physics world.

    So holding Moshe thought and comments on the supersymmetry(Kravstov cosmological computors models) issue this too needs some association, just to help make it easier to understand. Having people like Dvali(D brane effects spreading across surface) throw tidbits in for consideration of conceptual developement is always nice. Fluid flows, and laval nozzles, help greatly, taking vision down to a certain level.

    But the central theme now is where we had gone to such lengths and raising Cerenkov radiation or lagrangian perspective, I thought would be great starting points. along side of He4 or superfluid recognitions?

    As a layman developing concepts, are these starting points wrong having assumed D brane thinking?

  • Moshe

    Clifford, the word “string” is not mentioned in my little piece, it is not necessary…but to your point which IS a point about string theory, Joe’s argument seems relevant: you do want to control UV behavior and for the theory to have at least 4dim, I am not aware of any way of doing that without SUSY, but of course I am one of the students educated after the first stab at the non-critical strings…

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Moshe,

    Fair point. Don’t get me wrong…I like those arguments too…. Nevertheless, (at least) approximately four dimensions and broken susy are experimental facts. I’m just saying that I’m not sure that we understand non-critical strings well enough to know that after all the susy-breaking and dimension changing and strong coupling scenarios that we do once we start with critical strings, we don’t end up some place we could have got to by starting with non-critical strings in the first place…… But given that I don’t have the answer either, this could well be wishful thinking.

    Cheers,

    -cvj

  • Moshe

    Thanks Clifford, I think I see what you mean now, that is an interesting point.

  • Eva Silverstein

    Hi Sean et al,

    I think the reason one has trouble providing an argument that 10 or 11 dimensions is required by string theory is because there is no such argument, as mentioned in the comments by anon and Clifford.

    If you impose low energy SUSY, or if you require *exactly* flat spacetime in the prescribed dimension, then you obtain the critical dimension. These criteria are not required (yet) by either phenomenology/cosmology or top-down consistency.

    In particular, the relevant consistency condition in perturbative string theory is simply that the total Weyl anomaly cancels, which a priori can be arranged in a wide variety of ways. For example, one of the simplest ways around the old “no go” arguments for de Sitter space is to take into account the positive term in the moduli potential that arises in supercritical dimensionality (combined with other ingredients such as orientifolds and RR fluxes).

    I would not particularly advocate the noncritical models for phenomenology in the absence of other motivations, but the question of principle (what is allowed vs what is excluded based on first principles) is an important one and I know of no non-circular top down argument for the critical dimension. Conversely, if someone came up with such a proof it would be new and useful. Of course low energy SUSY has a number of phenomenological virtues and it will be extremely important to see whether or not it is there at the LHC.

    Obtaining nearly flat space of course requires fine tuning (this is the cc problem), but it is not known whether the tuning required is actually better or worse in the case of high vs low scale SUSY breaking. There has been some preliminary discussion of this in the literature but no firm conclusion.

    Best regards,
    Eva

  • Moshe

    Eva, thanks for your comment. To the extent it relates to anything I have written, I have been careful to state the assumptions and what follows from them. Any theory at all that attempts to use SUSY and extra dimensions, in the regime when the notion of dimension makes sense, is bound to find itself in 11 dimensions or less. In other words the numbers 10/11 are not pulled out of a hat, which may surprise a few people…It is an interesting and independent question whether in string theory these are absolutely necessary, but once again there is no mention of strings in my little piece.

    As for that question, the absence of Weyl anomaly is one way of deriving the critical dimension, in the Green-Schwarz formalism the derivation has much more transparent relation to spacetime supersymmetry, perhaps that is a circular argument…

    In addition, as far as I understand there is no one self-consistent framework that incorporates all the ingredients needed for the super-critical scenario. That does not mean there is none, only that critical strings are under much better control, especially when considered non-perturbatively.

  • Eva Silverstein

    Hi Moshe,

    I was not writing as a criticism of your piece (which if it states an assumption of low energy SUSY is perfectly reasonable). People sometimes claim that string theory essentially predicts both 10d and low energy SUSY; it is this of which I am not convinced.

    I worry that this could be a much more sophisticated version of the following. We all learn mechanics first with spherical symmetry, which makes calculations easier. In that context, it would be ridiculous to elevate the symmetry to a principle of nature. Of course SUSY is a much deeper structure, but it is still not obvious to me whether it is ultimately a deep requirement or closer to a theoretical crutch.

    Regarding supercritical constructions: the statement is that enough independent forces to fix the moduli appear in these theories (self-consistenly in the same model). Of course it is true that this arena has not been much studied, and there could be subtleties. On the non-perturbative formulation: recall that there is no known non-perturbative formulation of string theory on a Calabi-Yau. The backgrounds for which a non-perturbative definition presently exist, while very interesting, are few and far between, so I would not use this as a selection mechanism given our present knowledge.

    Best regards,
    Eva

  • Lee Smolin

    Hi, There is another very elegent way to understand why 10 and 11 are special, which was developed a long time ago by Feza Gursey and others. This has to do with the division algebras: A= R (real), C (complex), Q (quaternions) and O (octonions). There are the only algebras which extend the real numbers in having addition, multiplication and division. It turns out that these are related deeply to supersymmetry. One way to see this is that a supersymmetric extension of Yang-Mills theory must have two component spinors, valued in one of these algebras. The four possibilities give theories in 3,4 6 and 10 dimensions.

    Another way to see this is that the supersymmetry charge is fermionic and this is generated by two component spinors that transform under a representation of SL(2,A). SL(2,R) ~ SO(1,2), SL(2,C) ~ SO(3,1), and SL(2,Q) ~ SO(1,5). There is no SL(2,O), because the octonions are non-associative. But there are close relationships between the representation theory of SO(8)-the transverse directions in 9+1 dim spacetime and the octonions that allow a supersymmetric theory to be defined. There are three 8 dimensional representations-the vector, spinor and conjugate spinor, related by a symmetry called triality that can be seen to underlie the structure of the supersymmetry algebras between 8 and 11 dimensions.

    If I may, one last beautiful fact: These three representations organize themselves into components of a very beautiful algebra-which is the algebra of 3 by 3 hermitian matrices of octonions. Under anticommutators they form a unique structure called the exceptional Jordan algebra. This is a 27 dimensional algebra, and it has 3 SO(8) scalars-so it naturally represents physics in 11=8+3 dimensions

  • Moshe

    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).

  • Lee Smolin

    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.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    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?

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    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

  • http://thebumblebeeblog.blogspot.com/ Poppycock

    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?

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    #31 Poppycock: Thank you.

  • http://eskesthai.blogspot.com/2006/02/phase-transitions.html Plato

    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?

  • http://countiblis.blogspot.com Count Iblis

    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.

  • Lee Smolin

    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.

  • http://eskesthai.blogspot.com/2005/07/b-field-manifestations.html Plato

    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.

  • http://eskesthai.blogspot.com/2006/02/strange-abstract-movements.html Plato

    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?

  • Mat Hunt

    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

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    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.

  • Mat Hunt

    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.

  • Qubit

    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.

  • Marcia L.Neil

    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.

  • 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.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    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

  • http://golem.ph.utexas.edu/~distler/blog/ Jacques Distler

    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.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    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

  • http://eskesthai.blogspot.com/2006/02/evidence-for-extra-dimensions-and.html Plato
  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    #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?

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Hi,

    Sorry….I don’t know what the spacetime intercal formula is. Help!

    -cvj

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    The uh.. the s*2 = x*2 + y*2 + z*2 – (ct)*2 thing. Did I use the wrong terminology?

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    sisyphys, that formula is valid in flat spacetime, where Lorentz invariance is a symmetry of the entire universe. In any curved spacetime, the interval will be a different function of the coordinates. Curved spacetimes are generally less symmetric than flat ones, although certain special examples might be equally symmetric, just in a different way.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    Oh! The invariant interval formula! Sorry sisyphus, your terminology was fine….I was distracted by other things and did not make the connection, stupidly enough. Looks like Sean has answered you.

    -cvj

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    Thanks, Sean. I’ll work on it.

    Apropos of something else: Feyerabend may have been a tad radical, but maybe he got it right when he suggested that the slavish observation of a particular methodology can stifle discovery.

    Thanks again.

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    Thanks, Clifford. I’m aware of your loss.

  • http://blogs.discovermagazine.com/cosmicvariance/clifford/ Clifford

    sisyphus:- in addition to the possibility that the spacetime has a non-trivial metric, mentioned by Sean, another common way of changing things quite drastically is to have other dynamical fields switched on in some directions, but not others. The classic example is a linear dilaton…. i.e. a spacetime field generated by the string, the “dilaton”, has a non-trivial dependence (linear in this case) on just one direction in spacetime….. the rather singles out that direction as being rather special as compared to the others, a possiblity not allowed for in the critical string constructions. This is one example of how you can have string theories that live in dimensions other than the “critical” ones.

    -cvj

  • http://1034:Incorrectkeyfilefortableusers;trytorepairit sisyphus

    Thanks, Clifford. This is a bit to chew on. I’ll print it out and do a little research.

  • http://www.canonicalscience.com Juan R.

    This is a fascinating post!

    There is not string theory just a program named “string theory” devoted to search of anything that can be called “string theory”.

    I find particulary interesting the above claim of string theorists that old “axiom” “string theory requires 10D” is not rigorous (i would wait they agree that nothing in string theory research can be considered rigorous).

    I find really amazing the dozens of talks, books and articles devoted to critize Loop quantum gravity and others aproaches because they work in the “old” 4D framework. Once i read from a string theorist that LQG was wrong because was 4D and non-supersimmetric!!!!

    Many string theorists popularly stated that universe WAS 10-11D and NOT 4D. Public is completely misinformed about the real status of string theory in modern science. It is a kind of cancer…

    I find also amazing that many of my predictions in the recent past begin to see accepted by some people. For example, In my Oct 21 post “String theory is not a TOE” in moderated newsgroup sci.physics.strings i did a joke about the infinite malleability of string theory and how even with that infinite malleability string theory was an dead way. By dead way i mean dead for doing science, real science; of course, “string theory” and the Landscape and all stuff will survive during decades as a kind of postmodern religion or metaphysics, as it has done in last decade or so.

    I said in page 2 of my April non-technical article cited in the newsgroup:

    As an illustration of the malleability of the string project, think during an instant on spacetime dimensions, the most characteristic piece of the [string, M) theory for public. Initially, the theory was developed for 4D; after it was for 26D; next “reformulated” on 10D, and the last decade, after Ed Witten last revolution, dimensionality is fixed on 11D. Is it really fixed? There are people working in the possibility of more than one time dimension. 12D? Perhaps 13D? What is more, some theoretician has recently claimed that we would investigate 4D string models due to the failure of compactification for extracting real physics!

    The notation “[string, M)” means the “sequence” of dozens of different versions of “string theory” proposed as the Final theory during the last 3-4 decades. Note that i leave open “)” the sequence by the right hand!!!

    Yes, public reading my post may be skeptic because theorists as Witten or B. Greene newer said them that different versions of string theory were proposed in literature and none of them worked even minimally.

    Please, by the sanity of science, do not forget the real history of field which is not covered in Witten popular essays in Nature or in Greene’s The Elegant Universe!

    Robert B. Laughlin, 1998 Nobel Prize-winning physicist said last year

    People have been changing string theory in wild ways because it has never worked.

    Return to the history of dimensions: 4D,… 26D, 10D, 11D, 12D?, 4D again. It appears to be a syntom of being doing research in circles, Right?

    Now, i am preparing a formal rebuttal to last Witten Essay in Nature, which i consider completely distort the reality of the field and is based in beliefs.

    I am also considering the possibility for an open letter (signed by specialists in many fields) explaining to public last “advances” in string theory and why string theory is completely wrong as candidate to TOE since have a wrong mathematical structure.

    In fact, i personally doubt that even in the restricted field of particle physics and quantum gravity we can wait some good from string M-theory in a future.

    Juan R.

    Center for CANONICAL |SCIENCE)

  • Thomas Larsson

    Once i read from a string theorist that LQG was wrong because was 4D and non-supersimmetric!!!!

    gr-qc/9710008, footnote on page 2.

    Btw, it’s supersymmetry.

  • http://www.canonicalscience.com Juan R.

    Tanks by reference and by the correction!!!

    I did several other errors in the post: “newer” instead of “never”, “have” instead of “has”…

    —

    Juan R.

    Center for CANONICAL |SCIENCE)

  • http://thebumblebeeblog.blogspot.com/ Poppycock

    There are more fundamental errors than just typos.

    For example, I believe Brian Greene did discuss different string theories (type I, IIA, IIB etc) in “The Elegant Universe”.

    Critical string theory in 26D is a bosonic theory only, and I’m sure no-one has ever suggested that this represented a “theory of everything”.

    As for the footnote referenced, since that is not credited to anyone in particular, it does not seem to me to be good evidence that this was ever put forward as a serious argument against LQG.

  • http://www.canonicalscience.com Juan R.

    Poppycock,

    There are more fundamental errors than just typos.

    It is good to know that. I believe that the perfect post has been still not invented. It is good and needed to make errors. My criticism ot string theorists is not by making errors, just by ignoring the last near four-decades criticism and by their obvious attempt to distort the history of the field.

    I would emphasize “near four-decades” because in your blog you write

    String theory has been researched with a lesser or greater intensity for twenty-something years. Initially it was introduced in the context of understanding the strong force, but then QCD came along and worked much better, and thus ousted this idea.

    In a pair of years, string theory will be 40 years-old. Somewhat as many people is misinformed about real status of string theory, I believe that many people would be perplexed when know that your “twenty-sometimes” means near the double of years: 40.

    Now, let us see if it is true you are claiming. You said

    For example, I believe Brian Greene did discuss different string theories (type I, IIA, IIB etc) in “The Elegant Universe”.

    I never said the contrary! That i said was that The Elegant Universe presented to the public a distorted version of the history of string theory research. Moreover, by different versions of string theory i was not refering just to the five (2, 10) versions of string theories usually named by the collective name of “superstring theory”. By different versions of string theory i refer to dozens of versions published in literature you apparently have not read the list of different versions i cited in my April work.

    However, your appeal to superstring theory is good, because now i can add some i said not in my previous post: that some time ago many physicists claimed that ONE of the five known superstring versions WAS the correct final theory.

    In fact, the community of string theorists (“stringers”) thought then that our universe was just described by, and only by, (2, 10) points in the “branescan”. Now, they admit that were wrong in the past which reinforces my thesis history of string theory is the history of sucessive failure. It is more even now it is admited that string is not fundamental…

    It is amazing to write at one hand all past claims of stringers and, at the other hand, our current understanding of topics. Allmost all was claimed in the past has been shown to be wrong.

    Critical string theory in 26D is a bosonic theory only, and I’m sure no-one has ever suggested that this represented a “theory of everything”.

    I never said or even suggested that!

    I said that in the past it WAS suggested that universe WAS 26D. Of course this idea was inspired in critical dimension of bosonic theory. Nobody claimed that a theory of 26D lacking fermionic modes could be considered a candidate to TOE. However, stringers believed that a fermionic corrected theory would be also 26D, because in the contrary case they would say: “NO, universe is not 26D; 16D are just an artifact of current formulation of the theory“.

    el of classical theory, but on attempting to promote it to a quantum theory, researchers discovered that the total number of spacetime dimensions is fixed uniquely to be 26. So, quantum strings could exist only in a world with 25 (rather than 3) spatial dimensions, plus time. The excitement of finding — for the first time — a mathematical consistency condition that determines the number of spacetime dimensions, rather than treating this number as an experimental input, was somewhat tempered by the absurd value predicted for this number.

    Nobody predicted a reduction 26D –> 10D via generalization to fermionic modes of the bosonic somewhat as nobody recently predicted the adittion of a new dimension outside the (2, 10) “traditional” regime.

    In fact, i have seen some early work and ideas about compactification 26D –> 4D which now are considered outdated since the number of dimension is assumed to be 10D in superstring (CY) and 11D in M-theory (G2).

    And YES, stringers claimed in public that superstring theory was the searched TOE and universe WAS 10D already in the 80s (i.e. 30 years ago). But 10D was not a correct number again.

    I repeat the history of dimensions: 4D, 26D, 10D, 11D, 12D?, 4D again…

    As for the footnote referenced, since that is not credited to anyone in particular, it does not seem to me to be good evidence that this was ever put forward as a serious argument against LQG.

    If i write E= mc^2 in this blog i think that i do not need to reference a 1905 paper for suporting it. We do not usually reference material is well-known for readers we are focusing. Above preprint was focused to people who already know (you apparently do not) the history of the field.

    Juan R.

    Center for CANONICAL |SCIENCE)

  • http://eskesthai.blogspot.com/2006/02/evidence-for-extra-dimensions-and.html Plato

    Sure is nice to have the opportunities to rebuff the current state of affairs Juan R. :)

    While these models appear abstract, how is it that you could marry such thoughts to “proposed experimental processes” with which to deal with the consequences of such abstract thoughts?

    Was it wrong to go down certain avenues, then find that such experimental processes “stupid” after the fact, or was there real science being looked at?

    Taking in the current state of affairs, to such analogistics levels of scientific validation, is just one more aspect of model assumption?

    The contributors here are providing a service instead of the ole rant(yada yada), yet, we do hear of the 10 or 11 with some clarifications?:) Your contribution is helpful? Thanks. I’m learning a lot from a historical perspective, and after the fact.:)

    Your thoughts on ICECUBE, in a weak field and it’s manifestation?

  • http://www.canonicalscience.com Juan R.

    Beautiful and deep reply Plato ;-)

    During a brief instant (perhaps at Planck scale) your reply resembled the deep vision launched by the classical philosophers that your nick evokes.

    Yes, during that brief instant, I was tempted to become a believer.

    Unfortunately for the believers, the history of the field is very different of is said in talks focused to young and impressionable minds or in books dealing with laymen. Unfortunately for believers, it is a hard task to try to convince to real scientists that there is some serious below the stringy hype far from three decades arrogant claims of being working with the Final Theory, a theory that could explain everything.

    Unfortunately, today string theory continues being so far from any realistic TOE as was 40 years ago. Unfortunately, each little time a new “prediction” of string theory is shown to be false. Last, I remember was the observation some days ago of the so called cosmological strings that were not so string after all…

    Unfortunately, even with the infinite malleability of string theory, in which you can say anything and the contrary of anything (each claim founding a new version of string theory) all of us would recognize (even believers!) that after of all Nature does not like beauty :-)

    Stringy of course…

    About the ICECUBE i prefer to follow “solid” Witten philosphy in gravity and talk of postdictions. Therefore, when you obtain some data contact with me.

    Juan R.

    Center for CANONICAL |SCIENCE)

  • http://eskesthai.blogspot.com/2006/02/warm-dark-matter.html Plato

    Juan R. (1R?)

    hmmmm….

    each little time a new “prediction” of string theory is shown to be false.

    How is this possible? As a lay person such an example was important to me, that if such predictions are made and falsified, this was being handled properly?

    Postdiction? Ah so you are saying any model assumption, or effort to use this model currently, is postdiction?

    So lets say on mathematics alone then, is there any consistancy?

    A side note? War on SeaGull: Any of you know? :) Beer and Present Danger, was actually a good one.

  • http://eskesthai.blogspot.com/2006/02/history-of-superfluid-new-physics.html Plato

    Layman ponders.

    Anyway, what is this “new physics” that might house dimensional perspective?

    Multiverses. Really! How would such “funnels” provide for new beginnings? A geometrodynamical view of bubble technologies? :)Ronald Mallet’s dream of travelling back in time and making way for new things?

  • http://eskesthai.blogspot.com/2006/02/cosmic-variances-very-own-strangelets.html Plato
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Cosmic Variance

Random samplings from a universe of ideas.

About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] cosmicvariance.com .

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