How many dimensions are there?

By Sean Carroll | December 7, 2005 12:35 pm

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.

CATEGORIZED UNDER: Science
  • http://vnoel.wordpress.com Vincent Noel

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

    Obviously.

  • erc

    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

  • Dissident

    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…

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

    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.

  • FP

    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 8-) http://xxx.lanl.gov/abs/hep-th/0512047

  • Aaron Bergman

    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?

  • George Musser

    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

  • Helix

    String Theory – Ptolemy for the 21st Century.

  • http://arunsmusings.blogspot.com Arun

    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?

  • http://http WL

    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.

  • http://arunsmusings.blogspot.com Arun

    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.

  • http://www.math.coumbia.edu/~woit/blog Peter Woit

    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.

  • http://www.mazepath.com/uncleal/ Uncle Al

    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.

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

    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.

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

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

    Recent Comments
    How many dimensions are there? 15

    :)

  • Moshe

    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.

  • http://eskesthai.blogspot.com/2005/12/general-relativity.html Plato
  • Moshe

    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.

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

    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!

  • Paul Valletta

    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?

  • Aaron Bergman

    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.

  • http://eskesthai.blogspot.com/2005/12/general-relativity.html Plato

    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? :)

  • Doug

    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

  • http://feynman137.tripod.com/ Science

    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.”

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

    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.

  • JC

    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?

  • Moshe

    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.

  • Fyodor

    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.

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  • rof

    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.

  • http://arunsmusings.blogspot.com Arun

    Where is Smolin when we need him?

  • Matt B.

    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!

  • http://nigelcook0.tripod.com/ Science

    “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

  • http://eskesthai.blogspot.com/2005/12/general-relativity.html Plato
  • Daniel Elander

    How does string theory defined in twistor space fit into all this?

  • George Musser

    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

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

    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.

  • Haelfix

    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.

  • http://www.mazepath.com/uncleal/ Uncle Al

    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?

  • Moshe

    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…

  • Moshe

    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.

  • Chris W.

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

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

    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.

  • Moshe

    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.

  • agm

    definition please: plump?

  • 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.”

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

    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

  • JC

    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?

  • http://eskesthai.blogspot.com/2005/12/general-relativity.html Plato

    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.

  • Aaron Bergman

    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….

  • http://Jimmypettyjraol.com Jimbo Petty, Jr.

    There are ten dimensions. There are twelve dimensions. Yes?

  • rof

    Matt B.: Sorry for taking so long to reply. I’ve been away from the computer for a while. Introducing the observer doesn’t change the system being studied because he isn’t introduced into the system. He’s outside it, making occasional measurements and then updating his mathematical description of the system to take into account the results of his measurements.

    Moshe: I can try to explain any part that was confusing; I probably made statements that seem alien to people who don’t already see things my way.

    Moshe and Aaron: The S-matrix is nice if you can indeed sit at infinity and throw strings and branes and so on into the middle and see what comes out. You can do this in terms of calculating scattering amplitudes in a fixed background. Then the S-matrix elements are observables (in the sense that you can experimentally observe something about them) if we live in that background and can produce the ingoing states and can detect the outgoing states. But do we, and can we? The dualities make the problem more explicit. If I live inside the boundary CFT, then the scattering amplitudes inside the bulk of AdS are not what I’m going to see if I do a scattering experiment. The best you can say is that they’re honest experimental observables for hypothetical creatures that live inside the bulk.

    Or take T-duality. A nine-dimensional creature living in R^9 x S^1, where the radius of the S^1 is tiny would say the world is nine-dimensional, but a ten-dimensional creature living in the T-dual space, where the radius of S^1 is much larger would say that the world is ten-dimensional. From the point of view of string theory, those two worlds are “the same”, and before you can answer the question of how many large dimensions there are you have to first know whether it’s a creature in one space or a creature in the T-dual space asking the question.

    This isn’t a criticism of the theory; as I said before, it just underlines the importance of making contact with experiment.

  • Moshe

    JC, normally everything has 4dim features and also extra-dimensional features, there is no such thing as 4dim sector, no more than an “up and down” sector in more familiar physics problems.

    rof, there seem to be a few issues conflated here, but let me take the T-duality picture and phrase it as I would Lorentz invariance; a 9-dimensional observer cannot make any measurments that will distinguish a very small extra circle from a very large one. They can say the total space is mostly 9dim (if they choose the small circle “frame”) or almost 10dim (if they choose the large circle picture). Both are right, and therefore the question “how many dimensions are there *really*?” is a bad question to ask. Similar to an observer that insists on knowing if they are really at rest or not. As Sean said “the best answer to a seemingly-profound question is sometimes to unask it”.

  • http://eskesthai.blogspot.com/2005/12/general-relativity.html Plato

    While the discussion is on going here, I just wanted to interject this leading statement, to help those who are coming from the outside, looking in. :)

    Lubos Motl

    Most physicists realize these things – and many fundamental physicists actually use very similar mathematical techniques as Laughlin does in his “emergent” approach.

  • Aaron Bergman

    The S-matrix is contact with (hypothetical) experiments. Most of the things we compute in QFT are S-matrix elements. The fact that we’re not really living in a region with free |in> and |out> states doesn’t stop us from figuring out what happens in a collider.

    The same with T-duality. We can compute scattering. Those are the physical observables. You can interpret them however you like.

  • http://arunsmusings.blogspot.com Arun

    How many dimensions are there? Well, it depends when you look. Just now, the cosmicvariance sidebar answered as follows:

    How many dimensions are there?  67
    Aaron Bergman, Plato, Moshe, rof [...]

    So, as of Dec 10, 2005, 11:16AM US EST, there are 67 dimensions. I note also that Aaron Bergman’s post is numbered 55. It must mean that there are 12 hidden dimensions.

    I wonder how many dimensions we’ll end up with?

  • rof

    Moshe: I’m not proposing that either of the hypothetical creatures should ask how many dimensions there “really” are. I’m just saying that each of them will be in no doubt about how many independent directions he can move his arms around in. R^9xS^1 is compatible with the answer to this being both eight or nine, if the S^1 is small/big enough.

    Aaron; I agree fully. In fact I was going to write almost exactly what you wrote in my previous comment. Asymptotic string states can be identified with results of gedankenexperiments which we do not know how to perform but we can hypothesize that they might be possible in principle (if string theory is correct). That says it all, really.

  • Moshe

    rof, both descriptions of the same model have precisely the same set of answers to precsicely the same sets of questions. The set of things “observers” can do on a small or large circles is precisely identical, they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics. To me that suggests that focusing solely on the geometry is the wrong language to use.

    And back to the observables issue you are discussing with Aaron, the situation is that there are set of observables that are well-defined and are associated with actual measurments that could be performed in principle. Those are S-matrix elements and they depends on having an asymptotic region where the physics simplifies. As we never performed experiments inside a strongly gravitating regions this includes all the experiments we ever performed, and as far as we know all the experiments that could be performed in principle.

    Now there could be other things, and if they make sense (are gauge invariant) and are associated in a clean way with measurments that could be performed in principle, a theory of quantum gravity such as string theory ought to be able to calculate them. Once such things are discovered I’ll be interested to learn about them.

  • http://eskesthai.blogspot.com/2005/12/rayleigh-scattering.html Plato

    I was thinking of Moshe’s relation to circles large and small. The “complexity” of the situation in distinquishing which is which?

    Difficulties in inner and outer?


    How is this possible? Should 3 not be smaller than 2? …

    He goes on to the next shell and finds that 4 is larger than 3, and thus larger than 1. … 5 he finds to be as large as 3 and 1.

    It reminded me of Greene’s statements

    it turns out that within string theory … there is actually an identification, we believe, between the very tiny and the very huge. So it turns out that if you, for instance, take a dimension – imagine its in a circle, imagine its really huge – and then you make it smaller and smaller and smaller, the equations tell us that if you make it smaller than a certain length (its about 10-33 centimeters, the so called ‘Planck Length’) … its exactly identical, from the point of view of physical properties, as making the circle larger. So you’re trying to squeeze it smaller, but actually in reality your efforts are being turned around by the theory and you’re actually making the dimension larger. So in some sense, if you try to squeeze it all the way down to zero size, it would be the same as making it infinitely big. …

    Of course I am in a state of confusion :)

  • rof

    Plato, you’re in a state of confusion because almost all physicists are in a state of confusion. The problem is that the notion of “reality” has been completely reversed in the minds of the theoretical physics community over the last century, so that now they consider the most abstract things (such as quantum fields, elementary particles and so forth) to be the most real, while they view the concrete things (such as experimental phenomena, and objects like cups and bananas, along with concrete itself) as dependent for their existence on the abstract ones. The problems that I’m having explaining what I’m saying to Moshe and Aaron can be traced back to this essential point.

    Moshe, you say: they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics.

    This is precisely the point that I wanted to make, except expressed in a way which supposes that the “real world” is the string-theoretic representation of the world, and that the world in which we wave our arms around is “up to us”, that is, arbitrary. You agree with me that a certain type of observer in the world described by string theory on R^9xS^1 might see nine spatial dimensions around him while another sees eight, and you correctly (from the point of view of string theory) attribute these differences to differing characteristics of the observers, but incorrectly suppose that it is merely a choice that they made.

    There are two types of physical theory, or rather, two attitudes that one can take towards theoretical physics. The first is the “ontological” theory, which is a theory which attempts to say how the world looks “to God”. That is, such theories attempt to describe how the world actually is in itself and considered in isolation from the manner in which we observe it. The attitude associated with those who construct and prefer these types of theories is that the physicist’s job is to describe nature, to say what the universe “really” is, and that the observer should be just a part of the description, since he is just another part of the material world, and the observer certainly shouldn’t play any privileged role in the formulation of the theory. It is this attitude, which I now give the title “physics fundamentalism”, which requires one to regard the most abstract objects as the most real.

    Fundamentalist physicists have produced numerous attempts to remove the observer from quantum mechanics, because of their insistence that measurement is just an interaction like any other and should thus play no special role in the formulation. These attempts have, predictably, always led to assertions about the “real” existence of supersensible undetectable objects, even parallel worlds. For fundamentalist physicists, the idea that quantum mechanics deals with the observer’s knowledge about the system rather than the system itself is not allowed, because it conflicts with the basic tenets of physics fundamentalism. Hence the discontinuous change of the wavefunction, which happens because our knowledge changes discontinuously when we make a measurement, is regarded by them as a problem, which they call the measurement problem.

    The problem with ontological theories in practice is that, since the observer is now supposed to be just a regular object inside the theory, we cannot say, on the basis of the theory, what it is that the observer is going to see, because that depends very much on the structure of the observer. Hence you say that whether he will “describe geometrically” some part of the “real world” is up to him, meaning it depends on his structure.

    The other type of theory, or attitude towards theoretical physics, is one that could be called epistemological, and these theories attempt to describe how the world looks to us (rather than to God). This attitude regards physics as the process of establishing relationships between the results of experiments, or identifying the regularities in the results of experiments. Quantum mechanics is one of these theories, which makes statistical assertions about the results of measurements. It gives a procedure, which the observer can follow, in order to assign probabilities to the results of experiments. To follow the procedure one needs to actually make measurements, construct a set of formal sums of measurement results, fix parameters involved in the mathematical representation by referring to empirical data, and so on.

    This is why I said that string theory is not a quantum theory in the usual sense. It does not construct a set of formal sums of results of actual measurements. It is not an application of the quantum procedure, which is a procedure that can only be applied to empirical data. It is essentially an ontological theory, which is why it suffers from the problem that it cannot predict what the observer will see, because what the observer sees depends on the structure of the observer.

    An example of a genuine quantum theory is the theory of spin measurements with Stern-Gerlach devices. We find that the particle gets deflected either up or down, so we construct the set of expressions b|up>+c|down>, and follow the quantum procedure. When using this theory, we will not find that the predictions about what will be seen depend on the structure of the observer, and we do not need to assign a structure to the observer in order to use the theory to make predictions.

    In string theory, on the other hand, we find that it is insisted that S-matrix elements are observables, regardless of the fact that the structure of the observer needs to be specified before it can even be said how many dimensions he will find himself in, or rather, how many dimensions he will “describe geometrically”. If you will excuse the metaphorical language, as fundamentalist physicists engaged in a war on reality, string theorists have moved beyond the original idea of changing the meaning of “reality” and applying it to abstract mathematical entities, and have now changed the meaning of “observable” so that it applies to merely mathematical entities as well.

    Also, I like your remark that “all the experiments we ever performed, and as far as we know all the experiments that could be performed in principle” are observations of string-theoretic observables. It reminds me of how frequently religious types claim that the size, complexity, or mere existence of the universe provides a proof of the existence of God.

  • Fyodor

    Moshe said:
    “rof, both descriptions of the same model have precisely the same set of answers to precsicely the same sets of questions. The set of things “observers” can do on a small or large circles is precisely identical, they both can wave their arms in the same number of independent ways (9), and it is up to them what portion of it they describe geometrically and what they attribute to “stringy” physics.”

    Perhaps you can justify this apparently strange assertion?

    Let me explain. If you look at the history [say 1930] of Kaluza Klein theory, you will find that there were two schools of thought. One said that the 5th dimension was real, the other that it was just a mathematical formalism. Of course, nobody disputed that the KK equations were *exactly equivalent* mathematically to the Einstein-Maxwell system, but nobody assumed that *exact mathematical equivalence* was the same thing as “equally real”. Similarly, string theorists circa 1985 surely knew that a purely formal interpretation of Calabi-Yau compactifications was possible, but evidently nobody felt moved to attach any importance to this observation.

    Now you want to assert that mathematically indistinguishable situations are *equally real*. Maybe you can persuade us, but you won’t do it by means of analogies with special relativity. You will do it by describing *precisely* how an observer who waves his arms about can possibly describe his experiences in terms of two different dimensionalities. More generally, a string-theoretic analysis of the concept of “reality” would do the trick nicely.

  • Moshe

    Guys, there are too many high horses around for my comfort. I believe I stated precisely what I mean, let me try to get over some confusions I can identify, and maybe for the rest you can try to read what I actually write.

    First, I emphasize I did not mention string theory at all when talking about observables of quantum gravity, whether in 4 dimensions or in higher dimensions. If you define “observables” as those things that can be observed, where observation includes and actual observer and a measuring device, the only things that we know exist are S-matrix elements. Those include everything we ever measured. Now this was not a statement about string theory at all, and please try to contain your passion and wait for me to actually make an arrogant and ill-informed statement. Yes “string theory observables” is a term you made up yourself rof.

    Second, since the beginning of quantum theory physics is defined as the attempt to describe and predict results of measurments, I have no interest in “reality” beyond that, whatever that may mean. That is precisely why I caution talking about all kinds of comfortable notions one may have, and try to concentrate on precise statements about results of measurments.

    All I had to say about T-duality was in the context of string theory, where this thing exist. It was not a statement about the real world where we don’t have experimental evidence for T-duality. If string thoery describes the world and it has a compact circle, there are no measurements that will distinguish a small circle from a large one. Since I am only interested in results of measurments there is no reason for me to choose.

    I think I will leave it at that, there is too much war on reality to do this week.

  • http://eskesthai.blogspot.com/2005/12/rayleigh-scattering.html Plato
  • Qubit

    IMO, there are no dimensions at all. My existance, along with all of yours, is a virtual one. At some time, me and everybody else will die. From your own point of veiw at that moment, the universe, space-time, everything will come to an end and this will mean that; you will, never have existed.

    My virtual existance, one day reached a singularity. To become real, meant I had to escape from the edge of an horizon and fall into a black hole, at the same time. This was not to live forever, I did this… Just to be born. When I was born, you all! Was already dead. For you all to have existed, would mean I had no choice, other than to escape, I did and I did become real and I was born.

    Teddy bears dont work, Quantum physics is nonsense, there are no magic boxes, CATS SEE WITH TWO EYES and one day I will die!!

  • rof

    Fyodor says: Now you want to assert that mathematically indistinguishable situations are *equally real*. Maybe you can persuade us, but you won’t do it by means of analogies with special relativity. You will do it by describing *precisely* how an observer who waves his arms about can possibly describe his experiences in terms of two different dimensionalities.

    Exactly.

    Moshe: I’m sorry if my tone was too aggressive. I did get carried away on my high horse. Still, I stand by everything I said. Fyodor has expressed it perhaps more succinctly than I did. Good luck with the war on reality.

    Fyodor: I believe that the answer from string theory would be the following. The observer would wave his arms around and count a certain number of dimensions. Then he would learn string theory and do experiments with an LHC and conclude from the particle spectrum that there are extra dimensions that he didn’t count while waving his arms around. Some observers would be able to count all the dimensions by waving their arms and others would have a very difficult time finding them. What is indistinguishable is the final string-theoretic description of the world that the two would reach (which is what string theorists would call the real world), and the differences in the respective worlds that they see around them would be attributed to the fact that the observers would regard themselves as embedded in the stringy world in different ways. An observer imprisoned in a D-brane would be able to wave his arms in fewer directions than an observer who wasn’t so imprisoned.

  • Elliot

    when speaking of physics inventions
    there is none like the extra dimensions
    but how many you query
    well there’s more than one theory
    and discussion increases the tension

    ….oh wait this was supposed to go over on the poetry thread…. ;)

    Elliot

  • http://eskesthai.blogspot.com/2005/12/rayleigh-scattering.html Plato

    In the perfect world, I like circles that are smoothly expressed, but it is not always like that?

  • Claire

    There are 129600 dimensions. Don’t ask why…

  • http://eskesthai.blogspot.com/2006/03/new-search-paradigm.html Plato

    I know I am Physic-ally challenged sometimes, but I can’t just take that for granted Claire. :)

  • Brad Schuman

    It would seem that there should be as many dimensions as there are numbers to denote them. An infinite number that can go on forever in a “positive field”, as well as an infinite number in a “negative” field.
    I see no reason that there need be a finite number of dimensions anymore than there can be a finite numeric system.
    Perhaps “zero” and “infinity” are actually one and the same.
    I’ve heard of the higher dimensions in string theory being spoken of as “existing within a realm that’s smaller than we can measure”…which just sounds like a complicated explanation of describing the attributes of the “infinitely small”.
    Any reason this assumption should be considered “in error”??

  • http://www.facebook.com/vlad.maris Vladymir Mariz-Nobrega

    It could be 26, according to Bosonic string theory – isn’t that so?

    Or could there be infinite dimensions, depending on the perceiver, according to some Epistemologists…

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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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