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As a quick disclaimer, I thought this might be a better place to ask than Physics.SE. I already searched there with "kaluza" and "klein" keywords to find an answer, but without luck. As background, I've been reading Walter Isaacson's biography of Einstein, and I reached the part where he briefly mentions the work of Kaluza and Klein. Also, I did my undergraduate degree in theoretical physics, but that was quite a few years ago now...

The way I understand the original work of Kaluza and Klein is that you can construct a theory that looks like a 5-D version of GR that reproduces the Field Equations and the Maxwell Equations. An important bit is that the 4th spatial dimension is the circle group $U(1)$, which we now know as the gauge group of EM. I guess the first part of the question should be whether I've understood that correctly.

Then, if you build a Kaluza-Klein type theory, but use the SM gauge group $U(1)\times SU(2)\times SU(3)$ instead of $U(1)$, what do you get? Is it gravity and the standard model together? If not, then what?

The Wiki article on Kaluza-Klein theory says that this logic "flounders on a number of issues". The only issue is explicitly states is that fermions have to be included by hand. But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?

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6 Answers

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if you build a Kaluza-Klein type theory, but use the SM gauge group U(1)×SU(2)×SU(3) instead of U(1), what do you get?

If you want to use U(1)xSU(2)xSU(3), you get gravity over a 11 dimensional manifold, such the extra seven dimensional manifold is of the kind produced by quotienting $S^3 \times S^5$ by an orbit of U(1). A particular space in this family is $S^3\times CP^2$, you can learn that the group of isometries of $CP^2$ is SU(3) and the group of isometries of $S^3$ is obviously SO(4), so SU(2)xSU(2). More general spaces of this kind can be obtained by using a generic lens space instad of $S^3$; remember that lens spaces interpolate between $S^3$ as fiberes product of $S^1$ and $S^2$ and the plain product $S^2 \times S^1$. (This is already a bit beyond the answer, but I mention it becasue my first question in Physics.SE was about if this interpolation was a kind of Weinberg angle).

The dimension of a Lie Group is equal to its number of generators, so G=U(1)xSU(2)xSU(3) has, as a manifold, dimension 1+3+8=12. Such manifold has an action with GxG, which is overkill. So we can quotient the manifold using a maximal non trivial subgroup of G, in this case H=U(1)xU(1)xSU(2), and use instead the manifold G/H. Thus the number of dimensions that we need is $1+3+8-(1+1+3)=7$.

The ways to map H into G are not unique, and in the particular case of the SM group this creates a 3-parameter family of manifolds, and each of them seems to have, according Salam et al, a 2-parameter family of metrics. In some special cases of this parameter space, as the aforementioned $S^3\times CP^2$, some extra symmetries can appear.

I am not sure, but it seems that before Witten the technique to put "$G$ instead of U(1)" was really to put the whole Lie manifold as compact space, and then act on it with $G\times G$. A particularly intriguing case is when $G$ has the topology of an sphere, and then the maximal possible number of isometries. So $S^1$ and $S^3$ had naturally attracted some attention, and Adam theorem could have pushed some interest on $S^7$.

But even if this (Kaluza-Klein)+(SM gauge) theory only describes interactions, isn't that okay, or at least a big help?

It seems that it does not help, and I am as surprised as you.

The question of fermions "by hand" goes beyond the chirality problem. It was a program, led mainly by Salam, that an analysis of the compactification manifold and its tangent plane should reveal the charge assignments. For the SM-like manifold in 7 dimensions, the program fails; you can not find the charge assigments that the standard model has. It was noticed later, by Bailin and Love, that by going to 8 extra dimensions the problem could be solved, but further research was not pursued.

A reasonable inquiry is how the jump to 8 and eventually 9 dimensions relates to Pati-Salam, SU(5) and SO(10). Of course SO(10) needs nine extra dimensions (It is the isometries of $S^9$), and the projections down to the standard model seem very much as recent work of John Huerta. Other interesting question, to me, is if the extra dimension, from 7 to 8, can really be a local gauge symmetry, given that we have reasons to keep ourselves in D=11 at most. When one notices that the extra dimension is the origin of $B-L$ charge, that is interesting.

History

You can also check SPIRES for the history of Witten involvement in Kaluza Klein: FIND A WITTEN AND K KALUZA-KLEIN (edit:link changed towards inspire)

He has four papers with the Keyword "Kaluza Klein". The first of them is "Realistic Kaluza Klein Theories". It is the start of the KK trend, not the end. All the relevant papers come because of it. Do an

FIND K KALUZA-KLEIN AND TOPCITE 50+ AND DATE BEFORE 1990 AND DATE AFTER 1975

And order by increasing date. You will notice the works of Salam et al, Pope, Duff, all of it. The difference with the previous, and later, research, is that in this timelapse KK was considered seriously as in the original proposal, while generic references about KK in modern literature are really about compactification from higher dimensions; in some cases the fields comming from KK are even a nuissance to avoid.

I do not know who invented the late excuse that "Realistic Kaluza Klein" killed the research on KK; it appears very frequently in folk introductions to compactifications in string theory. More rarely, some person notes the contradiction and quotes instead the last paper of Witten on the topic, Shelter Island II, which has a more deeper discussion on the chirality problem, and even hints -or I read between lines- the question of singularity or regularity of the manifold, so that the late proposed solution to the fermion problem (see Moshe answer) is not so surprisingly ironical, really it was there from the start.

The topic of Kaluza Klein, or more properly of using the gauge fields of Kaluza Klein as physical fields, was abandoned in 1984 with the second superstring revolution. Ten dimensions were more interesting that eleven, and then you have not enough room to produce the SM group in a pure way from KK, so why bother? A mix of crowdthinking with "publish or perish" led to the end of the research, as most of the easy topics on KK had been covered in the interval (from 1981 to 1984), and some others were common to any extra dimensional theory: compactification, stability, etc... Nobody was even worried by the estability, because at that time it was believed that an AdS spacetime was reasonable, and then some compactification mechanisms from $M^{11}$ to $AdS \times M^7$ were known. An important role in this mechanism was the 84-component tensor that is added to the 44-component graviton in supergravity; some years later it should be recognised as the starting point of M-theory.

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I'm slightly torn between this answer and @Moshe's. My impression is that you have very well explained what happens when you build a KK-theory of the kind I asked and I gather that the failure (aside from chiral fermions) is that one derives incorrect charge assignments (compared to SM)? –  Warrick Dec 9 '11 at 9:26
    
A separate, simpler question is: why does GR+(U(1)xSU(2)xSU(3)) give an 11-dimensional space? Surely the dimension of U(1)xSU(2)xSU(3) is at most 1+2+3=6? Or am I making a basic misunderstanding of the dimensions of these groups and their product spaces? –  Warrick Dec 9 '11 at 9:28
    
@Warrick. Ok, I am adding further edit to explain why seven is the minimum. And yes, the first failures detected were both chirality AND charge assignments; after them, Moshe discussion follows. –  user135 Dec 9 '11 at 10:23
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Since SPIRES is about to be switched off, I would strongly suggest, in order to future-proof this site, that you link to Inspire (inspirehep.net) instead. –  José Figueroa-O'Farrill Dec 10 '11 at 5:59
    
@JoséFigueroa-O'Farrill done; the guys at INSPIRE have added extra post-1985 papers of Witten to the keyword kaluza-klein, so I was afraid it was more confusing. But it is either this, or not working at all. –  user135 Dec 10 '11 at 9:07
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The original KK motivation is derivation of gauge theory from higher dimensional gravity, so I will assume we are discussing higher dimensional theories of pure gravity (or perhaps supergravity), with no extra ingredients. I will also assume that the higher dimensional space is smooth, for the simple reason that otherwise we cannot make definite statements, at least not without more information. Both these caveats are relevant for string theory, which has higher dimensional gauge fields (not only gravity) and crucially -- in which you can make sense out of singular spaces. Indeed, the singularities give you just the right ingredients to solve the issues below:

Stability of the extra dimensions

The size and shape of the extra dimensions can change from place to place, resulting effectively in light scalar fields (moduli) which are not observed (this is known as the moduli problem). In the simplest examples this typically results in "runaway" behaviour in which the size of the extra dimensions rapidally goes to zero or infinity. To stabilize the size and shape moduli one has to find ways to build a sufficiently complex potential for these moduli, which requires some extra ingredients (such as the ones existing in the KKLT construction in string theory).

Vacuum Decay

Even if the KK compactification is stable to small perturbations, there is a mysterious quantum gravity effect that makes the KK vacuum decay (to "nothing") in non-supersymmetric KK theories (at least ones that, like the original one, contain a circle in the extra dimensions). This is one of the reasons most of the modern work on the subject considers only supersymmetric theories (i.e. supergravity) from the outset.

deSitter Compactifications

Now that we know that we have a small cosmological constant, there is a new argument against higher dimensional supergravity theories. The only known way to circumvent this no-go theorem is using ingredients specific to string theory (e.g. orinetifolds). This argument was not known at the time that the modern KK theories were studied and eventually rejected (roughly the early 1980s).

Chiral Fermions

Historically, the main reason the modern KK program was rejected (or at least slowed down) was this paper by Witten. One of the generic difficulties in constructing any model of fundamental physics is that the standard model has chiral fermions -- fermions of different chirality have different couplings. This is hard to achieve because fermions tend to come and go in pairs of opposite chirality whose couplings are exactly the same. If you do manage to somehow construct chirally asymmetric models, these models have many more possibilities to be inconsistent (anomalous) and therefore many more consistency checks to pass. This is therefore one of the best, most stringent, tests to subject any claim for beyond the standard model physics.

What Witten has shown is that there is no way to get chiral fermions starting with higher dimensional (super)gravity theory on any smooth manifold. This caused a general loss of interest in this research direction. Ironically, it was Witten and various collaborators that demonstrated, about 15 years later, that the problem can be solved (in string theory, using singular manifolds). Turns out that String theory has exactly the right ingredients to make the physics of the required singularities regular, and to pass all the non-trivial consistency checks that accompany any chiral theory.

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Just to be sure, Freund-Robin solutions are unstable too? My understanding is that they were rejected due to the third point (being AdS instead of dS), but I have not read about its stability and/or decay. –  user135 Dec 8 '11 at 13:14
    
I have defined what I call KK, the reasonable thing to do when the original idea did not work is to try variations, which circumvent some but not all the problems. –  user566 Dec 8 '11 at 16:00
    
Yes, KK in the broad sense of compactification of extra dimensions. But you should read all of the OP question, and not only the header... he is interested on what happens if you try to build exactly the SM in, and that fails. The answers are in Witten 1981, Salam, and then Witten 1984. The classification of isometry groups for 7 dim manifold is also done in this interval (some mathematicians review it later), showing that the SM group appears really between a small set of possibilities. –  user135 Dec 8 '11 at 16:07
    
I did not have enough time, but I will at some point edit it to remove any reference to history (which for some reason tends to start food fights, as we already starting to see), and add something more specific about the SM. –  user566 Dec 8 '11 at 16:10
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Yes, that was the idea. The answer to the questions in the title is no and probably yes. But, before getting into specifics like the gauge group and matter representation, one needs to worry about the basic logic. In the end all the problems can be solved, and higher dimensional theories can work, but they cannot be just gravity theories. For example they need supersymmetry and a whole bunch of other ingredients. which successively leads to structure indistinguishable from string theory (which is the real reason KK was subsumed by ST). –  user566 Dec 9 '11 at 15:43
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Not my domaine of expertise, but I would say that Kaluza-Klein, by itself, can't properly unify GR and EM. If we stick to the 5-dimensional approach, the first studied by KK, we get a few drawbacks right on the starting points. The deviation of the photon, for instance, would be affected by the presence of a 5-dimensional parameter, we can calculate it for a Scwarzchild solution and we would get an extra term that was never measured in any kind of light deviation.

For a metric of the form: $$ \bar{g} = A^a dt^2 - A^{-(a+b)}dr^2 - A^{1-a-b}r^2d\Omega^2 - A^b dl^2.$$, where $A$ is the usual Schwarzchild factor, we would get a photon deviation: $$\delta = (4a+2b)M/r_0$$ if we suppose there is no initial velocity for the fifth coordinate. The experimental data from the 70's would force $b<0,00075$. I guess this is one of the "flounders" mentioned in the article, not actually a real problem, but an annoying one that would force dimensions to be too tiny and making all kind of measurements very difficult.

However, KK theories went well beyond their stating points, and evolved to Yang-Mills theory, so I guess they are still an active topic of research. There is more of it in "Space, Time, Matter : Modern Kaluza-Klein Theory", from P. S. Wesson.

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+1 for the reference to the Wesson book –  lurscher Dec 7 '11 at 19:42
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this is my oldest question at Physics.SE: Measurement of Kaluza-Klein radion field gradients

I get the impression that the main problem why people doesn't continue this venue of research is that there is no known mechanism that explains why the compactified dimensions stay tiny. In that question i argued that not only are the dimensions of those dimensions tiny, but their derivatives as well.

Maybe there is some quantization going on that forces the dimension into well-defined sizes, but thats something the experts on the area should address.

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There are a zillion papers on moduli stabilization out there. –  Aaron Dec 9 '11 at 0:49
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I think K-K is still of much research interest. The K-K theory, which unifies YM fields and gravity are there since a long time.

However, these theories have some consistency problems. In the original K-K theory, it was assumed that the 5D metric functions do not depend on the 5-th coordinate. This was the main reason of inconsistency.

If the 5th dimension (and extra-dimension) dependency is introduced, the extra-dimensions must be compactified to give a discrete spectrum. The size of extra-dimension must be small to give a large mass to the high-order excitations. I don't think the 5th dimension must be a circle to give the gauge symmetry U(1). The original K-K did not have any circle, the gauge symmetry U(1) is still there. The same applies to the YM fields. Of course, you can follow the Witten's program to build 10D SUSY or 11D K-K SUGRA theories, then the gauge symmetries will emerge as extended spacetime symmetries. This is the normal Riemannian geometric approach.

If you follow the noncommutative Riemannian geometric approach a la Connes with the 5th dimension having only two points Z2. You have a consistent K-K theory with a finite spectrum. If you are interested in this direction, let me know, I will talk more.

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Theodor Kaluza Theory successfully unified GR and EM, in 1919.

"The unifying feature of this theory was that it unified Einstein's theory of gravitation and Maxwell's electromagnetic theory.

As Kaku writes

... this unknown scientist was proposing to combine, in one stroke, the two greatest field theories known to science, Maxwell's and Einstein's, by mixing them in the fifth dimension."

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And therein lies the problem with taking popularisations too seriously. –  José Figueroa-O'Farrill Dec 8 '11 at 2:04
    
Jose you are absolutely correct, but who listens. - string theory. –  Terry Giblin Dec 9 '11 at 18:32
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