CERN has (almost) promised us to discover the extra dimensions in 2011:

ZDNet, Popular Science, others.That could sound as a piece of crazy populist pseudoscience except that remarkably enough, it is not

*completely*crazy.

My estimate of the likelihood that the extra dimensions are accessible the the LHC has always been around 1-2 percent which is, by the way, hugely higher than the probability that global warming will cause significant O(1) problems to the mankind, the world economy, or the biosphere in this century.

It would be absolutely spectacular if the extra dimensions were discovered. It's unlikely but it's not impossible because there are models with observable extra dimensions which are consistent with all the known observations - and that may even offer some advantages such as the ability to explain certain mysteries. In this text, we will look at some basic possible incarnations and properties of extra dimensions.

**Einstein and his narrow-minded unified theories**

As far as I know, no thinker before the 20th century has ever thought about the possibility that our Universe contains more than three spatial dimensions we know and love. Only when Einstein, after a decade of courageous intellectual struggles, wrote down his theory of gravity, the general theory of relativity, that presented spacetime as something that "lives", something "dynamical", people really began to play with it and to appreciate that it could be much more interesting than they have naively thought.

Once Einstein had discover general relativity, he thought that the only remaining task was to find the unified theory of electromagnetism and gravity - the only two forces he acknowledged. He denied the existence of the strong and weak nuclear forces because they had something to do with quantum mechanics and quantum mechanics was surely just an illusion whose correct predictions would be superseded by his unified theory - and he wouldn't need to care about the details because it simply had to work. ;-)

Well, it didn't quite work for Einstein in this way but many aspects of Einstein's naive reasoning were right.

During the last four decades of his life, Einstein tried hard to construct the unified field theory. As a high school students, I would religiously read his articles (in German) in the Pilsner scientific library. But I can tell you: Most of his attempts were hopelessly flawed - they envisioned various kinds of metric tensors with an antisymmetric part; theories with torsion, and so on. But one theory he worked on, although he didn't initiate it, was both conceptually new and valuable. It was the hypothesis of extra dimensions.

**Kaluza-Klein theory**

Theodor Kaluza was an unknown German mathematician who was born in Silesia. In 1919, inspired by Einstein's new dreams about the unified theory, he found his own approach to crack this problem: a fifth dimension!

He was probably the first person who appreciated that general relativity worked in any number of dimensions and should be studied in any number of dimensions. In particular, in five dimensions, the metric tensor had some additional components

gThe indices "m,n" go over "t,x,y,z", the known dimensions._{mn}(t,u,x,y,z), g_{m5}(t,u,x,y,z), g_{55}(t,u,x,y,z).

Here, I chose "u=5" for the fifth coordinate in order to exploit the portion of the alphabet between "t" and "z" more efficiently. ;-) Kaluza realized that the five-dimensional symmetric tensor decomposes into the four-dimensional symmetric tensor, a vector, and a scalar.

Einstein had previously (and later) tried many other attempts to squeeze either "A_m", the electromagnetic potential, or directly "F_mn" into the metric tensor. For example, he had thought for quite some time that "F_mn" was the antisymmetric part of the metric, or "A_m" was included in some torsion, and so on, and so on.

All of Einstein's (and others') Ansätze were just completely wrong, sometimes even at the level of the counting of the number of derivatives. Only Kaluza's approach was able to pass the basic tests - and it was actually morally correct. Einstein liked it at the beginning, then hesitated and as the referee, he delayed the publication of Kaluza's paper for 2 years, and then he kind of liked it again.

If you decompose the five-dimensional Einstein's equations for the five-dimensional metric in five dimensions to equations for the three pieces listed above, assuming that the fields don't depend on the fifth dimension at all, you essentially get the four-dimensional Einstein's equations for the four-dimensional metric (with some extra sources), Maxwell's equations for the mixed components (the vector), and some wave equation with sources for the scalar (the Kaluza-Klein dilaton whose task is to remember the radius of the extra dimension which is compact, as discussed below).

That's great because Maxwell's equations arise truly naturally in this way - something that wasn't the case of Einstein's other unification attempts.

**Oskar Klein**

This is where Kaluza was able to get. However, he couldn't even fully articulate the obvious point that the fifth coordinate couldn't be infinite - otherwise we would have observed it. A related problem was that he couldn't explain why the fields should be independent of the fifth coordinate. He couldn't reconcile his theory with quantum mechanics, either - because he didn't really understand quantum mechanics. And there were other problems.

Fortunately, a few years later, in the mid 1920s, one of the brightest physicists of the world, Oskar Klein of Sweden, looked at the problem and was able to fix all the holes that could have been fixed with the existing tools. First of all, Klein has made it totally clear that the extra fifth direction must be compact. If there's just one compact dimension and you don't want any boundaries, the shape must be a circle: the fifth dimension must be periodic.

Second, Klein understood quantum mechanics damn well and he has arguably understood the key conceptual (and many technical) features of quantum field theory more clearly than any of his contemporaries. So first of all, it was trivial for him to see that if the circumference of the circle (periodicity of the fifth coordinate) were "2.pi.R", then the fifth component of the momentum was quantized,

pwhere "N" is integer-valued. Much like with the angular momentum, the particle can only have a quantized value of the momentum. In essense, a single particle species in 5 dimensions will manifest itself as one of many possible species in 4 dimensions that are labeled by "N"._{5}= N/R

In fact, it can be shown that this integer "N" is nothing else than the electric charge in the units of the elementary electric charge. It's not hard to see why. The wave function of the "Kaluza-Klein mode" with momentum "N/R" is proportional to "exp(i.u.N/R)". The electromagnetic U(1) gauge symmetry is nothing else than a "x,y,z,t"-dependent shift of the fifth periodic coordinate "u" by a constant (lambda). Under this shift, the wave function "exp(i.u.N/R)" multiplicatively changes by a phase that proves that the field has charge "N".

The fifth component of the momentum "N/R" is also the lower bound on the energy - i.e. the mass as seen in four dimensions (times c^2) - because the energy (time-like component of the energy-momentum vector) can never be smaller than the momentum. This means that the charged particles had to be massive. In other words, the massless fields or long-range forces are inevitably linked to fields that don't depend on the fifth coordinate, as Kaluza couldn't prove.

Klein has also made some smart but partly obsolete arguments that the new dimension's radius shouldn't be far from the Planck length. However, that would also predict that the mass of the charged particles should be close to the Planck mass as well. That's clearly invalid for the electron which is much (almost 20 orders of magnitude) lighter so Klein was also able to (prematurely) falsify the whole Kaluza-Klein concept.

However, he knew that it behaved well in many respects. And he improved the theory by considering more complicated shapes of the hidden dimensions. If the manifold of extra dimensions has the isometry group "G" (symmetries that don't change the metric), the group "G" will look like a gauge symmetry in 4 dimensions. The mixed components of the metric, "g_{m5}", will transform under the gauge symmetries by changing themselves by "partial_m (lambda)" and everything makes some sense.

In the 1930s, Klein didn't quite complete the formulation and analysis of Yang-Mills theories - which were written down by Yang and Mills in 1953 or so - but he was damn close. If he managed to do the full job, we could say that not just supersymmetry but also Yang-Mills theory itself was discovered within string theory, 40 years before people realized that it was a theory of strings. ;-) Extra-dimensional theories are a damn natural source (or possible "more fundamental origin") of Yang-Mills symmetries.

**Incorporation into string theory**

For a physicist, there is no good reason to study all those ideas outside string theory because all the details that were not described above may be wrong or misleading. So the history of the Kaluza-Klein theory only meaningfully continued in the early 1970s, a few years after string theory was born.

When string theory was just a little baby, people realized that the bosonic one only worked in D=26 while the superstring was only consistent in D=10. String theory is so predictive and constraining that it insists on its right spacetime dimension, too. The special numbers are called the critical dimensions. It was instantly clear that a phenomenologically consistent way to deal with them was to compactify them in the way that Kaluza and Klein taught us half a century earlier. We were in the early 1970s.

Clearly, nothing excessively important happened for another decade, until 1985. In 1985, the next big step was for Candelas, Horowitz, Strominger, and Witten to write a paper with the correct shape of the extra dimensions that simply produced the theory of everything. Well, almost. ;-)

**The 1985 optimism**

In the late 19th century, many people thought that the theory of everything was behind the corner because things have worked pretty well for 3 centuries and some extraordinary recent progress had just occurred, too. They were wrong because two big revolutions were waiting inside or beneath the two Kelvin's clouds looming above the horizon. ;-) The two clouds that Kelvin vaguely saw evolved into relativity and quantum mechanics. However, their wrong expectations were not quite insane.

A similar thing occurred in 1985, at a higher level.

In 1984, Green and Schwarz sparked the first superstring revolution by their proof that the anomalies in type I SO(32) string theory exactly canceled - because of a nontrivial and previously misunderstood mechanism and a huge amount of surprising numerology with a happy end. Edward Witten was also thrilled and string theory was almost instantly transformed from a fringe game pursued by a dozen of heroes into the dominant program in all of high-energy theoretical physics - a position it kept at least until it became to be gradually overtaken by a group of completely incompetent idiots. It was already clear that string theory predicted not only gravity but also all the remaining types of particles as well.

Well, in type I string theory, the detailed gauge groups and quantum numbers didn't quite work.

However, by 1985, it was already realized that the same cancellation also miraculously occurred not just for the SO(32) gauge group but also for E8 x E8 group - both have 496 generators and share many other properties, too. Moreover, the Princeton string quartet has discovered a new kind of string theory - a hybrid of the old D=26 bosonic string theory (for the left-moving excitations) and the D=10 superstring (for the right-movers), the so-called heterotic string.

The heterotic string came in two possible flavors. The 10-dimensional gauge group could have been either spin(32)/Z_2 - a more accurate way to describe a group with the SO(32) algebra - or E8 x E8, following from two possible (26-10=) 16-dimensional even self-dual lattices. In this way, Gross, Harvey, Rohm, and Martinec produced another group with the same miraculous anomaly cancellation as type I and the same gauge group SO(32) - they were found to be S-dual a whole decade later (why such things sometimes took so much time!?). But they also found another possible gauge group in D=10, namely E8 x E8, that could be realized by the heterotic string. And in fact, this new one became the main game changer.

Candelas, Horowitz, Strominger, and Witten have figured out that they could obtain realistic supersymmetric GUT theories from the E8 x E8 heterotic string if it is compactified on a Calabi-Yau manifold. It had to be exciting once they realized the required mathematical properties. Andy Strominger has searched through all the libraries and found exactly one such a Calabi-Yau manifold - the quintic hypersurface in CP^4.

That had to be it, he had thought, as he told me. The theory of everything. It's a top secret and I can't tell you this top secret but Edward Witten would tell him: fine, we're done. So why don't we compute the electron mass to convince others as well? ;-)

Now, this almost sounds as a joke but it was what actually happened and it was pretty reasonable, too. People today may think about some relatively sluggish progress that takes a decade but back in 1985, the rate that was waiting to be extrapolated was very different. Indeed, at some point of 1985, it was reasonable to expect that one needed a few weeks to complete the theory of everything.

Needless to say, it didn't work in this optimistic way. There is not one Calabi-Yau topology in D=6. Instead, there are at least tens of thousands and possibly an infinite number of Calabi-Yau topologies. Each of them has many moduli that have to be stabilized. Since 2000, people have appreciated that one must allow many possible values of generalized magnetic fluxes over all of the manifolds' cycles which is how one produces those proverbial 10^{500} solutions in the landscape etc.

So things are more complex than the naive optimists have expected. However, it's still true that the heterotic string on Calabi-Yau manifolds nontrivially predicts a supersymmetric GUT theory with the right structure of the gauge groups and fermions, aside from gravity - in the same way as the bosonic string theory predicted "just" gravity in the middle 1970s.

Huge progress. But we know that none of the particular compactifications that have been proposed are "obviously unique" and there are many solutions. In fact, the heterotic E8 x E8 strings no longer represent the only irresistibly attractive "category" of realistic compactifications of string theory. These days, we know that there are at least four additional categories on the market of ideas, including heterotic M-theory (the 11-dimensional strongly coupled limit of the E8 x E8 strings due to Hořava-Witten with two boundaries); M-theory on singular G2 manifolds; type IIA intersecting braneworlds; F-theory vacua and IIB orbifold/orientifolds.

Realistic vacua may be found in many corners and several of them seem to pass all the tests that their proponents are able to check. It's not clear whether any one of them is exactly right.

**Size, shape, and curvature of extra dimensions**

The expectations going back to Klein - that the size of the extra dimensions should naturally be close to the Planck length - were dominating in string theory for a long time, too. However, it was not really necessary. It's natural for all dimensionless numbers to be of order one - and all dimensionful quantities to be close to the "Planck quantities" with the same units.

However, our real world is manifestly not natural in this way. It does produce lots of dimensionless ratios that are very different from one. They have to come from somewhere. And it's plausible that the size of the extra dimensions is unnatural - i.e. very different from the Planck length - as well. And the other unnatural numbers are functions of the unnatural size of the extra dimensions - which could be enough to explain their own unnaturalness.

So for example, Witten has proposed an "intermediate" size of the 11th dimension in heterotic M-theory that improves the gauge coupling unification including gravity. But this "intermediate" dimension - between the Planck scale and the length scale visible by the LHC - is still far from the experiments.

**Observable extra dimensions**

Can there actually be dimensions whose signs we can directly see, e.g. by looking at the Kaluza-Klein modes? The answer is apparently Yes. In 1998, Arkani-Hamed, Dimopoulos, and Dvali (ADD) have proposed extra dimensions that could be as large as 1 millimeter or so.

They would be bigger than many insects so why haven't we seen them yet? The answer is that such scenarios assume that the photons (and other fields and particles of the Standard Model) have to be confined to a brane (3+1-dimensional membrane) that doesn't have any large dimensions. But the large dimensions exist, anyway. And gravity - which was redefined by Einstein as the dynamics of the geometry of space - can always see all of space, by this new definition. So gravitons may move in the extra dimensions.

That also means that the virtual gravitons can see the largest extra dimensions and Newton's law is changed from "1/r^2" to a very different function once the distances between the objects become comparable to or smaller than the size of the extra dimensions. Amazing experiments have measured gravity up to 10 microns or so and no deviations from Newton's law have been found so far. That means that the extra dimensions have to be smaller than 10 microns. They're probably much smaller than that - but there's really no sharp proof that it is so!

Their larger size may be natural in the sense that a large volume of the ADD dimensions - in the fundamental Planck units - contributes (or may contribute) to the explanation why the 4D gravity is so much weaker than other forces: unlike the other forces, gravity gets diluted in the large extra dimensions, leaving much more feeble traces on the brane where the other forces are stuck.

The ADD extra dimensions were proposed by people who would call themselves model builders, not string theorists (even though they know lots of string theory by now), but of course the inspiration by string theory cannot be quite masked. Also, some aspects of the ADD models may be captured by stringy compactifications although the harmony about the details is far from perfect. I would even claim that no realistic stringy model with the nearly right spectrum that truly satisfies the stringy equations and that reduces to an ADD effective field theory is known.

**Randall-Sundrum warped geometry**

An even more realistic - and more explanatory - category of models came in 1999 when Lisa Randall and Raman Sundrum found out that extra dimensions can hide themselves by a different method than by being tiny: they can hide themselves by being appropriately curved.

In particular, they conjectured one extra dimensions but the resulting five-dimensional space looked like a slice of the anti de Sitter space. This AdS space may be written in coordinates such that the distances in the usual 3+1 dimensions are inflated or deflated by a function of the fifth dimension. This function is known as the warp factor and it exponentially depends on the fifth dimension. So the metric is something like:

dsYou see? The distances in the "t,x,y,z", the usual directions, are multiplied by the exponential factor. At some value of "u", they are larger (many kilometers). At another value of "u", they are shorter (femtometers) for the same coordinate differences. I recycled the letter "u" that has already been used in the story about Kaluza.^{2}= -du^{2}+ exp(2u/R) (dt^{2}-dx^{2}-dy^{2}-dz^{2})

The exponential function wasn't inserted because RS invented it to have some fun. It can actually be extracted as a solution to differential equations that are actually called Einstein's equations in the five-dimensional spacetime.

If the coordinate "u" goes from "0" to "40 R", and the number "40" is not terribly unnatural, you may see that the ordinary distances in the 3+1-dimensional space may be inflated by very different factors at both sides. The ratio of the warp factors is something like "exp(40)" or "10^{15}" or so.

These large numbers may actually explain why gravity is so weak. Most of the graviton's wave function may be concentrated in the part of the spacetime where the proper distances and proper volumes are large - i.e. large "u", large exponential - while we may sit at the place where the proper distances and proper volumes are small - i.e. small "u", small exponential. The interval in the "u" coordinate may be cut by two end-of-the-world branes.

(Aside from this RS1 model with 2 branes, Randall and Sundrum have also proposed the RS2 model with 1 brane - note that the 1/2 numbers are switched. In the RS2 model, the fifth dimension may be infinitely large and we will still be unable to notice that the dimension exists!)

If you ever thought that the insanely large numbers in physics - e.g. the ratio of Planck and proton masses - may be naturally explained as exponentials of more reasonable numbers, you were right except that the exponential, a priori, doesn't follow from anything so its insertion is just a trick to mask the unnaturalness of the large numbers. However, in Randall-Sundrum models, the exponential actually follows from Einstein's equations!

The possible relevance of the Randall-Sundrum dimensions for the hierarchy problem makes their model(s) the most likely scenario(s) by which the LHC could detect extra dimensions.

Once again, it's true that Randall and Sundrum were and are model builders rather than canonical string theorists. But it's again true that the overlap of the ideas with the ideas produced by string theory is clear. In particular, the AdS space became a key player in string theory in 1997, two years earlier. Randall-Sundrum models, however, are not just a phenomenologists' reinterpretation of the AdS/CFT correspondence, at least not according to a dictionary that would be fully understood today. You should definitely view their models as an independent advance in physics.

Both ADD and RS models may be embedded in string theory although string theory inevitably modifies some of their details. The RS geometry becomes just a region in a broader, Calabi-Yau-like geometry. The RS models are arguably more consistent with the details required by string theory than the ADD models.

**Does it make sense to count the dimensions?**

I hope that I have convinced you that theories with extra dimensions are at least as sensible and natural as theories without extra dimensions and they may actually explain some features of the real world that would be hard to understand without taking the extra dimensions into account.

You could ask whether the number and types of extra dimensions is a well-defined, objective question. Well, the answer is that the number of extra dimensions is only "objective" if you can "really see them well". ;-)

In particular, if you had some dimensions whose size would be just one Planck length, or if the curvature radius were one Planck length, the manifestations of these extra dimensions wouldn't be too much different from some "generic Planckian degrees of freedom" or "chaotic quantum physics of quantum gravity".

However, if some dimensions are "much" bigger than the Planck length, and if they're flat enough i.e. if their curvature is much smaller than the inverse Planck length (i.e. if they're much flatter than objects with the Planckian curvature), the Kaluza-Klein modes become dense and predictable and the dimensions become real. If you have an operational definition how to count the number of dimensions, you simply have it. If you don't, you don't: the theory won't offer you any quantification of something that cannot be measured. In fact, there can be many mathematically equivalent definitions of the same physical theory and these definitions may disagree about the properties of the small dimensions.

In particular, if you have dimensions that are small or comparable to the Planck length, string/M-theory provides us with many dualities that often change the shape (or number) of these small extra dimensions. Also, holography is able to add or remove one dimension. Whether the extra holographic dimension is real can only be answered "operationally", as a function of its size or curvature radius. If both of them are much bigger than the relevant higher-dimensional Planck radius, the dimensions gradually become real. There is no sharp turning point.

But if you try hard to decompactify a maximum number of dimensions in a geometry, in any stringy vacuum with some traces of SUSY-like stability, you will always see that the maximum number of dimensions that can be simultaneously made large enough is either D=10 or D=11, corresponding to the superstring and M-theoretical descriptions, respectively. (F-theory has D=12 in some sense but two of the dimensions are eternally forced to remain infinitesimal.)

**Summary**

Extra dimensions will probably not be discovered by the LHC in 2011 because their size can be anywhere between the current upper bound and the Planck length, and the probability that they will appear right behind the corner is rather small (and the hierarchies may be explained in other ways). But there's no real proof that it's impossible; in fact, it's pretty possible. Even if they're not discovered, they will surely continue to be used by tasteful theoretical physicists and phenomenologists because their role in clarifying the relationships between and the deeper structure behind various kinds of matter and processes has already been established as an important one.

And that's the memo.

## snail feedback (4) :

For the uninteresting time lapse of 1970-1984, and particularly for the last five years of this lapse, the reader should try to locate the collection of reprints "Modern Kaluza-Klein Theories" by Thomas Appelquist, Alan Chodos, and Peter G.O. Freund. It is a hard to find book, out of print and out of bittorrent (afaik), but it is a perfect catalogue of the "unimportant" things happenning before the superstring revolution.

In this retrospective, it is a pity that Witten become involved in string theory too early. Had he continued his work in Kaluza Klein (abandoned after the Shelter Island II preprint), he could have obtained a right formulation of the standard model charges in eleven or twelve dimensions, and then fit it against M-theory ten years later.

Sure, Alejandro, and Feynman would have discovered extraterrestrial aliens if he didn't waste his time on the Manhattan Project.

There is nothing of the sort you mention so Witten couldn't have discovered it. It's a pity that Witten didn't begin to work on string theory when he was a schoolkid - physics could be measurably further.

Still, it is a good piece of science epic. He attacks the question of the symmetry group of Kaluza Klein, causing a mini-revolution in the field, but three years after he is dissapointed by the problem of chirality and he leaves the 11 dimensional world, starting the work in string theory. And ten years later, he discovers a 11th dimension in String Theory itself!

(of course, Duff et al. could have another version of the history)

Dear Alejandro,

those things could have perhaps occurred faster, but the path was pretty much inevitable and Witten's reasoning on these points was totally rational - and still holds.

It is true that the absence of chirality is what *immediately* ruins unification based on smooth 11-dimensional supergravity. This fact is totally trivial to see - 11D SUGRA and its smooth compactifications are simply non-chiral (much like any theory that would resemble the "Lisi" theories in any way). This is a reason to abandon the theories as a phenomenologically possibility *immediately* because chirality is a fundamental feature of the Standard Model and a discrepancy is instantly lethal.

One needs the stringy effects to get realistic vacua from 11D M-theory. There are two classes of realistic descriptions of the Standard Model based on 11 dimensions: heterotic M-theory and M-theory in singular G2-holonomy manifolds.

Hořava-Witten's heterotic M-theory has boundaries with chiral 10D physics: the gauginos on the boundary cancel the anomalies against the projected gravitinos on the space with boundaries. Those things could perhaps be seen from low-energy physics only - but it's just true that they were only discovered once people systematically began to study the strong coupling limit of all string theories, and this was the strong coupling of the E8 x E8 heterotic strings.

The M-theory compactifications on singular G2-holonomy manifolds are also OK phenomenologically and Witten has done lots of things with them, too. However, the singular nature of the manifolds is essential for chirality once again. Because the fermions of the Standard Model (and their couplings) are largely chiral, it's clear that the singularity plays an essential role. In some sense, it's more important than the non-singular rest of the 7-dimensional manifold.

Like any advance in history, it could have been done faster if the people knew what they know today. ;-) But your attempt to paint the history as a contrived one is simply wrong - because there was no "more direct path" here.

Best wishes

Lubos

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