Talk:Kaluza–Klein theory

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This article asserts that all neurinos 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)"

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

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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.

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

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)

[edit] 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 seperation.
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 panalopy 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

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)

[edit] 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

[edit] 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

can be regarded as analagous to the electromagnetic field tensor

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)