Talk:Kaluza–Klein theory

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Unamed section 1

I would like to add updates to show Kaluza's original theory in the index notation, as some have suggested below, and also distinguish the contributions by Klein that tied Kaluza's theory to quantum mechanics. Would anyone object to me making such edits? As it stands, this article is not a very good description of Kaluza or Klein. Is there a primary maintainer?

--Lance W.

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— Preceding unsigned comment added by 130.221.224.7 (talk) 20:49, 31 January 2012 (UTC)

Unamed section 2

This article asserts that all neutrinos appear to be left-handed, "meaning that they are spinning in the direction of the fingers of the left hand when they are moving in the direction of the thumb". However, the Standard Model article implies that, while this may have been believed before, the recent (1998) discovery that neutrinos have mass has changed this. From the "Standard Model" page, "If neutrinos have non-zero mass, they necessarily travel slower than the speed of light. Therefore, it would be possible to "overtake" a neutrino, choosing a reference frame in which its direction of motion is reversed without affecting its spin (making it right-handed)"

If you overtake a neutrino, it will have both left and right handed spin components and it will be a non-relativistic spin 1/2 particle. But in this limit, there would be no such thing as an antineutrino. If you reboost the right handed component to the speed of light again, it would be the anti-neutrino. That's Majorana fermions for ya.

Perhaps right handed neutrinos are now understood to be possible? Could a real physisict confirm/deny/correct this?

The question of what type of fermion a neutrino is represented by (Majorana, Dirac, combination of the two) is an open issue. The See-Saw model employs a combination of a Dirac and Majorana mass term, making the overall mass matrix left-right asymmetric. This is used to explain the extreme small-yet-nonzero'ness of the left neutrino masses; yielding a prediction of large right neutrino masses.

The observation about the ability to overtake a sublight particle applies generally to all the fermions. The relevant attribute relating to handedness is helicity. This is invariant for light speed particles, but relative for sublight particles. Since the weak nuclear force only interacts with the left-handed states of all the fermions, this means that their helicity cannot be relative (since the question of whether an interaction has taken place or not, is not relative).

Consequently, the Standard Model must require all fermion masses to be 0. What you would call the particle's mass is dynamically generated through interaction with a universal energy field (the Higgs). The question of whether neutrinos have mass, translated in this context, is actually rendered as the question of whether the neutrino interacts with the Higgs.(unsigned comment on 19:42, 5 December 2004‎ from 65.57.245.11)

If the neutrino were an ordinary Dirac fermion, like the other fermions, the Higgs interaction would involve an "zig-zagging" between left and right handed states, each zig or zag absorbing or emitting a Higgs. That would be the only interaction a right handed neutrino would be involved in and the only means, ultimately, by which it could ever be observed (other than gravity).

The Standard Model assumed the neutrino masses were zero (that is, that their Higgs interactions were nil). So, technically that model is broken -- hence the various extensions, such as the Standard Model combined with See-Saw; or Standard Model + Dirac neutrinos. ------ (unsigned comment on 15:34, 19 June 2007‎ from 4.159.175.100)

Unamed section 3

This is how I understand Kaluza-Klein theory: General Relativity directly explains gravitational deflection of light, and describes but does not explain gravitational acceleration of matter. Kaluza's insight was that if light propagated in a compacted fourth spatial dimension, its gravitational deflection explained by GR would look exactly like gravitational acceleration of matter in the 3 ordinary spatial dimensions -- so maybe matter is actually electromagnetic energy propagating in a compacted fourth dimension.

If the above explanation has aspects that are correct, I hope that a real physicist can add the information to the article.

Upon further reading, I see that Kaluza's original idea really attempted to unify gravity and electromagnetism in a single geometric theory. That is a really useful thing to do if the result explains things like the equivalence of gravitational mass and inertial mass, or how an electron absorbs or emits a photon. (unsigned comment on 2 June 2003 from 63.198.179.97)

I too have worked through Kaluza's theory and I strongly agree that Kaluza's theory successfully and completely unifies gravity under general relativity and electromagnetism. In particular, I worked through the calculus in Peter Gabriel Bergmann's presentation of Kaluza's work, in _Introduction to the Theory of Relativity_. I have also studied quantum relativity. I feel that this extension does not invalidate Kaluza's work in any way.

Regarding the extra dimension, Kaluza's theory assumes that the universe is INVARIANT in that dimension. So, of course we cannot observe it directly! No curling up is necessary. In fact, no theory has been developed which uses curled up dimensions to succesfully unify electromagnetism and gravity. I believe that this is the source of the myth that gravity has not be unified with the other forces.

Joseph D. Rudmin (unsigned comment on 22:15, 10 July 2003‎ from User:Rudminjd)

Yes, there's nothing "wrong" with the theory, other than it misses out on the fermions (no spin structure), and fails to provide a quantum mechanical description, and lots of subtle stuff: the helicities of the neutrinos the KKbar BBbar oscilations, etc. The compactification of the 5th dimension is Klien's addition to the theory (as noted below) in response to the development of QM, an the need to quantize the electric charge. (Note that the electric charge can also be quantized by the purely-classical mechanism of setting it up on a U(1) bundle, and then looking at the Bohm-Aharonov effect as a holonomy on that bundle. In such a case, the charge must be quantized.. interesting! .. and nothing to do with KK.) 67.198.37.16 (talk) 15:56, 29 April 2016 (UTC)

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Can we have a date please to help place in some historical context

Linuxlad 20:36, 26 Nov 2004 (UTC).

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Didn't kaluza and klein ultimatly fail to unify general relativity with electromagnetism?, as I recall Einstein spent the latter part of his life working on this theory and failed to complete an accurate theory Cpl.Luke 04:53, 12 July 2005 (UTC)

It provides a unification,but its not a correct an full desccription of nature. First, it fails to include fermions, and second, its not quantum mechanical. So it is an interesting model that has some of the correct features, but its not the complete model. (Nor is it possible to experimentally invalidate it (beyond these obvious failings)). linas 22:26, 1 October 2005 (UTC)

Issues

I have a few issues with the new mathematical derivation:

1. It is too abstract: in physics, Kaluza-Klein theory is normally derived in terms of components of the metric. Why can't this be done here?
2. It doesn't demonstrate the so-called tower of Kaluza-Klein modes: i.e. that the higher modes have masses that go as Λ⁻² and so scale away in the limit of small separation.
3. The dilaton cannot be set consistently to a constant without some kind of moduli stabilization mechanism. Fierz wrote a paper about this in 1956.

Can you please try to fix up these errors? I'm not confident enough with the formalism to do it. –Joke137 15:25, 3 October 2005 (UTC)

Re: point 1, Hah. I figured the component-less presentation would be much easier to understand, rather than harder. All those indecies floating around everywhere makes a component-based derivation look like a Chinese menu and almost as hard to read and comprehend. Although I suppose a 5x5 matrix for the metric could be snuck in there. (There's a saying to the effect: "component-less if you want to understand, components if you need to calculate"; in the end, one must master both notations. FWIW, I know that physicists are almost never taught the component-less notation (since they're always calculating real quantities), but its not hard to learn. If you already know GR, spending a few weeks with a book on Riemannian geometry would be about all it takes, and you might even walk away with a few "aha, so that's what they meant when they said .." moments).

Re: point 2, The "tower" is a purely quantum effect, an expansion of graviton excitations in terms of Λ. The presentation I was aiming for was a pure geometric, non-quantum version. I suppose ... well, I'd have to study and think about how to present a simple derivation. In the meantime, we could add a section stating that "the next step is to start quantizing, and then one gets gravitons in a tower, etc."

I'm also reminded that the set of purely classical solutions to KK are as rich as the panoply in GR ... e.g. "how does assuming KK change the Kerr solution"? I suppose the tower would also show up as classical perturbative tweaks to ... I dunno, gravity waves coupled to light? Maybe solutions to even non-charged Schwarzchild black holes are altered? Is it even correct to go "perturbative" here? There's potentially a whole can of worms, I guess, and I don't know the survey of this material.

Re: point 3, I have not looked at Fierz's work; I presume this is a statement about the quantization of the theory, and I guess its "well known" too, probably. I really am not an expert on this topic either. I'd have to study up a bit.

I hope these are an adequate set of replies. linas 01:02, 4 October 2005 (UTC)

Thanks for the response. I agree with 1 and 2. As for 3, the way I think of it is imagine you had left the dilaton φ in in your derivation. Then you would have a term that looks like

$-{\frac {1}{2}}(\partial \phi )^{2}-e^{{\sqrt {6}}\phi }F^{2}$

in the Lagrangian. If you set φ = 1, this is not a solution to the equations of motion in the case of non-vanishing F², and so the 4D solution of Einstein's equations is not a 5D solution. You can't consistently set φ to a constant, except by fiat. –Joke137 01:19, 4 October 2005 (UTC)

Dang. OK, time for some pencil-n-paper work. It might take me a few days. linas 03:48, 4 October 2005 (UTC)

I'm going to have to get a book on this, since my naive pencil & paper work is raising more questions than its answering. It seemed so straightforward... linas 03:25, 6 October 2005 (UTC)

Haven't forgotten; have blasted through one book, and it now seems that there is so much to say, its hard to think of what's important enough that it needs to be said. I'll try to pull together a coordinate-based derivation "real soon now", along the lines of the first few pages Duff's PDF, but with added motivation for that particular coordinate choice. linas 06:07, 1 November 2005 (UTC)

Let me rephrase; I just blasted through Paul Wesson's book. It could be interesting to recap the main points of that book, but this would be a small article in and of itself. And it covers only one particular variant of KK; it doesn't deal with quanta at all. linas 06:14, 1 November 2005 (UTC)

P.S. I just added some B.S. about how all this is "experimentally interesting", but I suspect you know more about this than I. Care to expand on this? linas 03:48, 4 October 2005 (UTC)

The Extra Scalar Fields, Charge Screening and Cosmological Constant

Regarding the third issue brought up above: it is, indeed, true that if you force the 5-5 component of the metric to be constant, you run into difficulties. In particular, the 5-5 component of the Einstein tensor will be non-zero. This is best seen by writing the Kaluza-Klein version the Reissner-Nordstrom metric, which adds in a term of the form k (du - U dt)^2, where U is the electrostatic potential and k effectively the 5-5 component of the metric.

If, on the other hand, you force the Einstein tensor components to all be zero, including the 5-5 component, you end up getting a metric whose 5-5 component will be variable. This extra field effectively takes on the form of a scalar field which also adds a contribution to the Vacuum Permittivity proportional to the cube of the scalar field. That is: the extra field provides for none other than a non-trivial dielectric structure for the vacuum, itself!

The static spherically symmetric cylindrical point-source solutions will split into two families that have only a one parameter family of solutions at their intersection. The first family is a two parameter family of solutions that captures the Kaluza-Klein extension of the Reissner-Nordstroem metrics, and has a constant 5-5 metric component, but a non-zero 5-5 Einstein tensor. The second family is a 3-parameter family of solutions corresponding to a charged source with vacuum polarization and charge screening. The 5-5 Einstein tensor is zero, but the 5-5 metric component, k, varies. The potential U will exhibit a complex radial dependence implementing this screening.

The two families intersect in the trivial 5-dimensional extension of the Schwarzschild metric.

In the more general case with larger symmetry groups, the extra metric components will embody extra scalar fields that correspond to the metric native to the gauge group. If one requires the gauge metric to be adjoint invariant (but possibly still dependent on position and time), then for simple Lie groups it will be determined up to scale: the Killing metric. For semi-simple Lie groups, it will be a sum of the metrics corresponding to those of each of the simple groups that make up the semi-simple group.

The same considerations apply to the scale factor as to the 5-5 component, k, for 5-dimensional Kaluza-Klein. It cannot be made constant without the extra components of the Einstein tensor being forced to be non-zero. At the same time, setting all the components of the Einstein tensor to 0 will force these extra fields to become variable.

The scale factor, k, is directly associated with the coupling constant g, through the correspondence k = 1/g^2; there is one for each simple group, when the Lie group is semi-simple. Here, too, a variable gauge metric will represent a kind of classical version of charge screening or anti-screening, as one sees in Quantum Field Theory.

In addition, for non-Abelian gauge groups, an extra term appears in the curvature scalar that is quadratic in the structure coefficients and linear in the gauge metric -- a contribution to the cosmological constant. For a constant gauge metric, it will indeed be a constant. Otherwise, it will inherit whatever variations that the various scale factors, k, have. In particular, on a cosmological scale for an asymptotically constant gauge metric, it is quite possible for the extra "constant" to approach some fixed, small value, while yet avoiding the fine tuning problem that a truly constant cosmological term would run into.

Additionally, it might be possible to link the dark energy phenomenon with the extra terms that arise in the in the curvature scalar from a variable gauge metric. That issue is still open. There is a large amount of research underway to attempt to construct scalar field models for dark energy; and the scalar part(s) of the Kaluza-Klein metric provides a natural place to try and fit these models. -- Mark, 11 November 2006

Arbitrary Scale Factor

Is it not so that, of the five independent terms added to the metric, one can be interpreted solely as due to an arbitrary scale factor? This factor, of course does not need to be constant. Geometry has less freedom than a coordinate system that is not normalized. So, a radion field is an artifact of particular coordinates, it would seem.

-- 67.102.139.25 09:43, 6 July 2007 (UTC)

Momentum Interpretation of Electric Charge

Reading off the components of the energy-stress tensor seems to imply that electric charge is momentum (of a spacetime disturbance) in the fifth dimension, and that the test charge is simply velocity in that dimension. But, is the momentum to rest mass ratio of the electron, then, very high? Does putting this inside of an event horizon help?

-- 67.102.139.25 09:43, 6 July 2007 (UTC)

A more physical description

While the current gauge- and group-theoretic description is fine, I think that there's definitely a need for a more traditional description of Kaluza-Klein theory, especially in the introduction to the article. In particular, there's very little said about how the model originally arose, i.e., the way in which Kaluza noted that the Christoffel symbols of general relativity

$\Gamma ^{a}{}_{bc}={\frac {1}{2}}g^{ad}\left(\partial _{b}g_{cd}+\partial _{c}g_{db}-\partial _{d}g_{bc}\right),\$

can be regarded as analagous to the electromagnetic field tensor

$F_{ab}=2\partial _{[a}A_{b]}.\$

This really, really needs to be added and, assuming that nobody objects, I will try to find time over the next week to do so. --St Cyrill 05:24, 3 August 2006 (UTC)

Possible Accuracy Problems

Nordstrom's theory only included a scalar gravitational potential, dilaton only, if you like, but I am not sure if Nordstrom had a metrical interpretation. If you take a 5d scalar and put it on a circle, you just get a 4d scalar. So I don't see how he could possibly have discovered EM from 5d Nordstrom gravity. I didn't read the paper though, so I don't feel confident to correct this statement. —Preceding unsigned comment added by 132.236.173.23 (talk) 23:12, August 27, 2007 (UTC)

Spin and Lie Groups

There is a very obvious and large problem with this article. It is obviously written from a mathematics orientation with little or no references to the sub microscopic world in which "spin" is the major defining descriptor of quantum interactions. How can one write this article and fail to observe that Lie groups actually correspond to spin, spin orientation, and angular momentum. Can someone please do this and try to make this article more readable to the lay reader. There's no excuse for the jargon in this article and there's no excuse for not connecting it with spin in quantum mechanics.75.6.255.70 (talk) 05:29, 31 May 2008 (UTC)

Because it would not be historically accurate. When Kaluza wrote, in 1921, QM hadn't really been invented yet. Klien's contribution, in 1926, was to add some QM-style arguments. Since then, knowledge has advanced, but even in the late 1980's when KK resurged, I'm pretty sure that the concept of a spin structure on Riemannian manifolds was unknown to most physicists; the modern concepts of Pin groups and Spin groups had barely percolated from the math to the physics world. Anyway, quantizing KK is a problem. 67.198.37.16 (talk) 16:25, 29 April 2016 (UTC)

Explicit block matrix for of the 5-dimensional value of \hat{g}_{\alpha\beta}

Isn't it the case that the components of ${\hat {g}}$ can be explicitly written in block matrix form in terms of the components of $g$, $A$, the scalar field and some constants? Would someone who knows how that goes add it for ${\hat {g}}_{\alpha \beta }$ and/or ${\hat {g}}^{\alpha \beta }$? Thanks —Quantling (talk | contribs) 14:04, 10 March 2012 (UTC)

Updating references and field theory

I am providing the key references and results of the five-dimensional theory developed by Kaluza, Klein, and other authors. This should satisfy some requests for missing information from the standard description of Kaluza's results. It will also distinguish the classical and quantum results of the theory. The pre-existing group-theory and space-time-matter descriptions are left in place but de-emphasized. There should probably be a separate page for the classical theory, because it spawned a much richer research history after Klein forked the theory with a quantum interpretation. The basic idea was extended to higher, compact dimensions, to get a proper unified quantum field theory, but the five-dimensional quantum interpretation of Klein was never considered seriously after 1930.

Let me know if you have concerns or questions! — Preceding unsigned comment added by ManitouLance (talk • contribs) 03:46, 26 September 2014 (UTC)

Major update completed

I have completed a major update of this Wikipedia page. Before this, it only described some fringe aspects of the Kaluza-Klein theory. With these updates, we now have all the material Kaluza and Klein would have recognized, along with reference citations. It also includes the updated field equations that were discovered only decades after Kaluza and Klein. I did some minor cleanup to the pre-existing sections on group theory and space-time-matter, but mainly left them alone. I would invite someone to contribute a description of Kaluza-Klein theory in a modern quantum field theory framework, or perhaps the extensions of Kaluza's approach done by Bryce Dewitt that went to higher dimensions. I'll try to keep an eye on this page to see if anyone had comments or changes.

-ManitouLance — Preceding unsigned comment added by ManitouLance (talk • contribs) 03:37, 29 September 2014 (UTC)

I came to this page as a "newbie" after asking the question of google: "are photons deformations of space-time?" (More generally, does the presence of charges cause a deformation of space-time which in turn is observed in 4 dimensions as acceleration of the particles). These questions led me to a book by Cardone and Mignani; the book-summary mentions the Kaluza-Klein theory, so here I am. The physics and math is over my head so I can't comment on its content, but I found it to be well written and a good introduction with all the links etc. More criticism of the theory, more about where it may fall apart etc would be nice. But that said, it seems quite a bit better than a "C"-level article, e.g. a "B". BillWvbailey (talk)

Thanks for the comments. I am not sure how to change the category -- that seems to have been set automatically ManitouLance (talk) 04:29, 8 January 2015 (UTC)

13D

Is there any papers on 13D KK. I understand this is possible, and my interest relates to 4D (spacetime 3S + 1T) + 3D (chargemassweak 2S + 1T) + 6D (velocities (rates of change) of phase space). The last 6 give inertia to the other 7 minus real time, as the rate of time is not observable outside time. A 7D cross product is the expected neutrino handed and baryogenisis mechanism. This 13D model has a unique complex representation needing 26 independent tensor freedoms, and equivalently 26 co-ordinates (13 complex numbers) in a phase space.

188.29.164.49 (talk) 23:16, 18 February 2015 (UTC)

Which problems arise when trying to use KK for unification?

From the article:

As an approach to the unification of the forces, it is straightforward to apply the Kaluza–Klein theory in an attempt to unify gravity with the strong and electroweak forces by using the symmetry group of the Standard Model, SU(3) × SU(2) × U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders on a number of issues...

It is obvious that issues exist, otherwise KK would be acknowledged as a successfull unifying theory. Can some expert provide some more details about such issues? 78.15.181.49 (talk) 23:13, 26 July 2015 (UTC)

Yes, please someone add a section describing its shortcomings. I think http://mafija.fmf.uni-lj.si/seminar/files/2011_2012/KaluzaKlein_theory.pdf is a good place for some info.Vishal0123 (talk) 02:08, 27 April 2017 (UTC)

Yang-Mills?

I'm not an expert, but I'm wondering whether or not K-K is a Yang-Mills theory, and whether or not that should be mentioned here. Certainly, it would be relativistic, if it can embed General Relativity, but I'm less sure about the nature of what constitutes a Y-M theory. 75.139.254.117 (talk) 06:12, 18 December 2016 (UTC)

Minor updates October 2019

I provided the requested clarifications. I also cleaned up a few instances of garbled text.

Regarding some questions on the TALK page about the group theory sections, I cannot speak to that. I could only provide the original theory as understood by Kaluza and Klein. I would invite someone to add modern viewpoints, involving group theory. Otherwise, I would take the later stuff in the article with a grain of salt. I was tempted to delete it, but did not have the heart. ManitouLance (talk) 00:16, 28 October 2019 (UTC)

4D & 5D Abbreviations

Non-mathematicians and non-scientists are very comfortable with the abbreviation of 3D. Therefore, to make this article simpler, I've inserted the abbreviations of 4D & 5D. Math & science strives for simplification. 73.85.202.218 (talk) 14:52, 22 February 2020 (UTC)

Update to references May 2020

A little bit of housekeeping on the references, further reading, and citations. A couple of key references were added to the intro section. The intro is not meant to be a full review, but just to give the main lines of development. References which were already in the citations were removed. Also removed some recommended reading which was off topic. These changes were made in a series of small changes on 16 May 2020. I failed to mark all as minor and give descriptions, but they are. The historical stuff in the group theory section was moved to the history section, and I corrected the reference to Nordstrom, and added in the correct citations. ManitouLance (talk) 21:10, 16 May 2020 (UTC)

Is KK theory inconsistent?

"Kaluza–Klein theory was too beset with inconsistencies to be a viable theory." from Non-relativistic spacetime

Please discuss at Talk:Non-relativistic_spacetime, Yodo9000 (talk) 18:54, 11 May 2022 (UTC)

Modern generalizations of Kaluza-Klein theory

The section on Space-Time-Matter represents just one of several modern generalizations of Kaluza–Klein_theory. It may be worth including a new section Modern generalizations of Kaluza-Klein theory, which includes at least (1) Space-Time-Matter, which interprets the extra spatial dimension as matter; (2) complex-time (kime) generalization of positive real time, which leads to a 5D space-kime statistical interpretation where each point object in the observable 4D Minkowski spacetime corresponds to a distribution in the 5D spacekime ; and (3) Thermal-space-time, which describes quantum particles in space-time as classical 1D threads embedded in a 5D thermal-space-time manifold . VodnaTopka (talk) 01:15, 13 March 2023 (UTC)