### Is the Kaluza-Klein idea conceptually compatible with quantum theory?

This post is a continuation of a conversation started at Peter Woit's blog.

For background: String theory is presently the most widely accepted candidate theory for fundamental physics beyond what is already confirmed experimentally. Some see the widespread acceptance of string theory as validation of its correctness. Others claim there are fundamental conceptual problems with string theory.

In this case, the question relates to the Kaluza-Klein idea - the idea that there are extra dimensions to space which wrap around within a tiny distance (often presumed to be close to the Planck length - a tiny length around 10^-33 metres which one can get by combining the fundamental physical constants in a particular way). The original Kaluza-Klein idea was to add a single extra "compactified" dimension to spacetime. When you study Einstein's equation (which describe gravity) in this 5-dimensional spacetime, you find that the equations describing electromagnetism appear automatically. This idea has been adopted by string theorists who claim that this way of tucking extra dimensions away and having them cause the physics we are familiar with can explain the discrepancy between the number of dimensions needed for string theory (or rather, critical string theory, which is more or less what everybody has been calling "string theory" until now) with the number of dimensions we actually see.

String theory says there are 10, 11, or 26 dimensions, depending on which version of string theory we are talking about. We observe fewer than this number experimentally. The explanation given is that the extra dimensions are wrapped up in the way described by Kaluza and Klein, and that the details of the complicated way that they are wrapped up (called a six-dimensional Calabi-Yau manifold for 10 dimensions and a seven dimensional G2 manifold for 11 dimensions) is responsible for the details of the physics we observe.

The suggestion has been made that the Kaluza-Klein idea, while compatible with the classical theory of general relativity, is not compatible with quantum theory, due to subtleties in the difference between diffeomorphism invariance (invariance of the physical state of affairs under continuous, infinitely differentiable, mappings of the universe to itself) and gauge invariance (invariance of the physical state of affairs under changes of the mathematical representation of the configuration of physics fields which leave the field configurations themselves unchanged).

The purpose of this post is to open the comment area for discussion of the topic in the hope of bringing some much needed light to a dark and shady area.

For background: String theory is presently the most widely accepted candidate theory for fundamental physics beyond what is already confirmed experimentally. Some see the widespread acceptance of string theory as validation of its correctness. Others claim there are fundamental conceptual problems with string theory.

In this case, the question relates to the Kaluza-Klein idea - the idea that there are extra dimensions to space which wrap around within a tiny distance (often presumed to be close to the Planck length - a tiny length around 10^-33 metres which one can get by combining the fundamental physical constants in a particular way). The original Kaluza-Klein idea was to add a single extra "compactified" dimension to spacetime. When you study Einstein's equation (which describe gravity) in this 5-dimensional spacetime, you find that the equations describing electromagnetism appear automatically. This idea has been adopted by string theorists who claim that this way of tucking extra dimensions away and having them cause the physics we are familiar with can explain the discrepancy between the number of dimensions needed for string theory (or rather, critical string theory, which is more or less what everybody has been calling "string theory" until now) with the number of dimensions we actually see.

String theory says there are 10, 11, or 26 dimensions, depending on which version of string theory we are talking about. We observe fewer than this number experimentally. The explanation given is that the extra dimensions are wrapped up in the way described by Kaluza and Klein, and that the details of the complicated way that they are wrapped up (called a six-dimensional Calabi-Yau manifold for 10 dimensions and a seven dimensional G2 manifold for 11 dimensions) is responsible for the details of the physics we observe.

The suggestion has been made that the Kaluza-Klein idea, while compatible with the classical theory of general relativity, is not compatible with quantum theory, due to subtleties in the difference between diffeomorphism invariance (invariance of the physical state of affairs under continuous, infinitely differentiable, mappings of the universe to itself) and gauge invariance (invariance of the physical state of affairs under changes of the mathematical representation of the configuration of physics fields which leave the field configurations themselves unchanged).

The purpose of this post is to open the comment area for discussion of the topic in the hope of bringing some much needed light to a dark and shady area.

## 9 Comments:

This is just a test to make sure the comment section works. This is the first time I've done this.

Regards Kaluza-Klein idea, see Lunsford's paper:

http://cdsweb.cern.ch/record/688763

and my analysis of it on the second post at my blog http://nige.wordpress.com/

Lunsford shows that the correct unification is in the spin orthagonal group SO(3,3) where the 3 time dimensions mean that the cosmological constant disappears as experimentally observed and explained by the Yang-Mills exchange radiation redshift.

For the latter, see http://cosmicvariance.com/2006/10/03/the-trouble-with-physics/#comment-123080 and Professor Sean Carroll's comment next. Note there are loads of predictions quantitatively from this mechanism but the editors of Nature and PRL claim any alternative to fu*king string is not needed.

Thanks,

nigel

I would like to state the question more generally. Is K-K idea compatible with what we directly experience.

We experience world as 3-dimensional. Does this reflect the fact that space is really 3-dimensional (that is we are 3-dimensional)? Or does it reflect only the fact that the excitations of various fields are effectively 3-dimensional as K-K philosophy assumes?

If one accepts the first option and wants to explain why Poincare quantum numbers are not the only quantum numbers characterizing elementary particles, only one option seems to remain. Space-time is a surface in some higher D space-time coding the standard model quantum numbers in its geometry (isometries and holonomies). For implications of this choice see my homepage http://www.physics.helsinki.fi/~matpitka/ .

Matti Pitkanen

Egbert,

let us first forget the quantum theoretical situation and look at classical solutions of the Einstein-Hilbert situation. What people actually do in this case is different from what they say.

If you take a classical E-H Lagrangian and only retain the lowest Fourier components in those g-components which you want to compactify, then indeed you observe KK. But should't you do this in an actual solution (which before you do this fulfills higher dimensional local covariance) and at the end fulfills only the lower-dimensional local covariance whereas the remaining spatial directions lost this property which changed to gauge covariance. The question is whether this is possible (i.e. whether doing it on the Lagrangian commutes with doing it on the solution). I doubt that this is possible and apparently Pauli (according to Lunsford) thought about this.

Later I will return to the quantum case.

Bert

For discussing the quantum aspects of KK it is necessary to first get a very good understanding of inner symmetries. Let us agree to exclude “local gauge symmetries” because they are not physical symmetries.

The concept of internal symmetries was since its inception (the SU(2)isospin of nuclear physics introduced by Heisenberg) one of the most mysterious proposals. Whereas it is natural to accept spacetime symmetries (since they accompanied us in the classical setting since the time of Newton) the understanding of internal symmetries is a mysterious concept in QT (in classical physics you can only get them by reading back QT concepts into classical physics i.e. they are classically unnatural). This problem was finally solved in the work of Doplicher, Haag and Roberts during 1970-1990

Their ground-breaking idea was to abstract internal symmetries by taking a dichotomic view about QFT: local observable algebra (bosonic and neutral) which carry all the intuitive physical properties as the basic structural input, and the field algebra (the name for the algebra generated by charge-carrying operators which contains the observable algebra as the fix-point algebra under the action of some compact group) about which such an immediate knowledge is not available but whose structure is preempted in the structure of the local observable net of algebras (indexed by spacetime regions). By a sequence of conceptually extremely interesting and profound steps this unique field algebra (including the concrete inner symmetry group which acts on it) can be constructed. This is similar in spirit but much more subtle in detail than Marc Kac’s “how to hear the shape of a drum”.

The first step is to classify all “representation of physical interest” (local representations)and construct their intrinsically defined statistics. In this way one obtains (for spacetime dimensions bigger or equal to 4) a unitary representation of the infinite symmetric group which belongs to parastatistics of height d (where d is determinded by the structure of the observable algebra). The second step is to realize that these data can be encoded into a field algebra on which a compact symmetry group acts in such a way that precisely the observable algebra is left pointwise invariant. This required the elaboration of a completely new duality theory for groups because the old Tanaka-Krein duality theory was not appropriate for this field theoretic problem. This took Doplicher and Roberts many years and is a magnificent mathematical achievement (highly praised by the mathematician Marc Rieffel) which does not have to hide itself behind Witten’s mathematical achievements.

The representation structure of interests coming from the DHR theory in low dimensions is richer, in this case the observable structure leads to representation sectors which carry a representation of the braid group (a generalization of the symmetric group) and instead of the field algebra you find something which does not permit a clear-cut separation into inner and outer (spacetime) symmetries.

The first step is explained in Haag’s book and the second in the reference to the D=R work cited therein. The mystery of the Heisenberg isospin and its generalization, in fact group representation theory for all compact groups arises from spacetime localization structure of the observable algebras of QFT; which even though being bosonic and neutral, nevertheless preempt the statistics and internal symmetries in particle physics. Having explained something in terms of spacetime localization makes it more palatable.

Suppose now that you have gone through this analysis in a higher dimensional QFT and you compactify certain spatial dimensions and make them small. Do the correlation functions of such a QFT in the limit approach those of a QFT with smaller spacetime dimensions and a larger symmetry? Of course not, the vacuum fluctuations will prevent any convergence of such a limit.

This is also not what KK aficionados do. What they actually do is to look at the classical action (having a functional integral representation in mind) and perform the limit retaining the lowest Fourier-component in the classical Lagrangian before they quantize (i.e. before they do the functional integral and compute correlation functions) So what they do is not identical with what they say. That KK in the sense of what they do works is clear but it is a completely childish game and does not justify what they say.

What Lunsford attributed to Pauli is something slightly different (this was the reason for my interest). If I understood it correctly it says that you cannot apply KK to a higher dimensional Einstein-Hilbert theory (diff-covariant) into a lower dimensional E-H (lower dim. diff-covariance) and remaining gauge part. This one should understand first classically (see my remarks in my first blog).

Bert

Thanks, guys. It'll take me a while to get through all this, but I think it's certainly worth the effort.

Egbert

here is a more direct criticism of KK in he quantum context.

Imagine that you have a higher dimensional QFT and convert one of the spatial coordinates into a circle (impose periodicity). Then as a consequence of the Nelson-Symanzik duality you can change that particular spatial coordinate with time and obtain a spatially infinitely extendedt system in a thermal state whose temperature depends on the radius of the circle. The KK limit (decreasing the circle) corresponds then to infinite temperature which diverges. The reason is clear, the vacuum fluctuations in the original spatial interpretation become uncontrollably large which becomes more manifest after the N-S duality converts this into the thermal interpretation. Even if they would have succeeded (i.e. if the process would not be divergent) there is still the previous structural argument that spatial extension can never be converted into internal symmetries.

Those who advocate KK in QFT (or string theory) of course don't do that. The manipulate the classical Lagrangian (retaining only the lowest Fourier component) and afterwards contemplate to compute.

It is obvious that there is a huge conceptual discrepancy between what they do and that what they think they do (i.e. what they actually say).

I do not know any statement coming from string theory which whose metaphoric nature can be salvaged as an intrinsic property.

I still have to get through DRL's paper (and Nigel's analysis), but I'm working on it. It is indeed alarming if the editors of Nature and PRL are actually suppressing this. I think the concept of "mainstream speculation" is doing immense harm to the scientific debate.

The classical argument about only keeping the lowest Fourier component makes sense; I suspect that it doesn't commute exactly, but that the commutator (if we can speak of one) will be small and will go to zero as the size of the compactified dimension goes to zero.

The time-temperature argument is something that I hadn't considered before as a criticism of KK. This would give a temperature of the Planck temperature if there are compactified dimensions of that size. This is certainly very hot, probably hot enough to alarm a sensible string theorist.

But do string theorists really only keep the lowest Fourier component? I thought they kept this infinite tower of higher excitations corresponding to the higher Fourier components. Maybe they approximate away the higher excitations because they're so energetic.

Thanks again for pointing this out. I'll have to research the DHR stuff before I have even a vague understanding of what's going on there.

Dear Egbert

The DHR stuff and the final touch in the 1990 DR work is not something which one can absorb rapidly because it is conceptually and mathematically very demanding. The lecture notes "Superselection sectors" in the lecture note section of http://unith.desy.de/research/aqft/may be of some help.

The thermal argument based on the fact that imaginary time can be interchanged with space (N-S duality) is completely obvious in the Feynman-Kac functional integral representation; the rigorous mathematics (including the proof of existence of the theory by its construction) has only been possible for superrenormalizable models which only exist in d=1+1

http://br.arxiv.org/abs/math-ph/0403048

The KK issue is only one topic where string theory gets stuck in metaphors which have no autonomous counterpart. I do not know a single argument in ST which is free from this disease. Maybe you should open a separate blog under the headlines Facts and Fictions in ST. The other controversial issues are "does holography require gravity" and "are strings if ST really string-localized in any intrinsic quantum sense?", "is the S-matrix of string theory really compatible with the S-matrix of scattering in particle physics?"

This are all issues which have not been addressed before and there is a lot to say about them.

it would be wrong to think that that in these matters I am representing my own ideosyncratic ideas; I am really representing the thinking of most QFT experts:

Fredenhagen, Brunetti, Hollands, Wald, Buchholz, Rehren, Verch Summers,...

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