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I'm looking for a treatment of the original basic Kaluza-Klein theory. Can someone recommend a review article or something?

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There is an article "Kaluza-Klein theories" by David Bailin and Alex Love in Rep. Prog. Phys. 50 (1987). You can find it here bookos.org. Search "Kaluza Klein Theories", it should be the first hit. Also, there are several other books on this topic on this site. –  Leonida Mar 8 '13 at 6:57
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Hi Mouse.The.Lucky.Dog, and welcome to Physics Stack Exchange! This isn't the place to ask for recommendations. If there's a particular concept having to do with KK theory that confuses you, you can of course ask about that (and people will refer you to books in the answers if they think it's useful to do so). –  David Z Mar 8 '13 at 9:02
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@DavidZaslavsky this is a reference request about a well defined not too broad topic again !!! It should NOT have been closed as reference requests are allowd as we said again yesterday. Could you at least stop shooting down such reference requests unilaterally and ask what other people think, for example in chat, before and let some close votes accumulate ? –  Dilaton Mar 8 '13 at 10:04
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@DavidZaslavsky he is interested in a review article, not in books ... –  Dilaton Mar 8 '13 at 11:27
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closed as not constructive by David Z Mar 8 '13 at 9:01

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1 Answer

How's this: http://www.weylmann.com/kaluza.pdf ?

Be careful though, because there are a couple very big errors that the author makes. For example in equation (6) there should not be a factor of 1/2 in front of the second term on the right-hand side. It should just be a factor of one. If you do the Christoffel symbol calculations and expand out the 5D geodesic equation, you'll get the following:

$$\frac{d^2 x^\lambda}{ds^2}+\begin{Bmatrix} \lambda \\ \mu \nu \end{Bmatrix} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}=-k\left ( \frac{dx^5}{ds} +A_\nu \frac{dx^\nu}{ds} \right )F^\lambda_{~\mu}\frac{dx^\mu}{ds}$$

In fact, the factor 1/2 that the author has in his paper keeps the equation from being gauge-invariant. Similarly, after equation (6) he says:

This expression is now fully covariant, although the $A_\nu F^\lambda_{~\mu}$ term does not have any classical correspondence.

This is incorrect. It makes no sense to split up the terms, which is why I factored them like you see above. It actually turns out that Noether conserved momentum about the curled-up 5th dimension is:

$$-mk\left ( \frac{dx^5}{ds} +A_\nu \frac{dx^\nu}{ds} \right )$$

So it's a conserved quantity which is gauge-invariant. By comparison with the usual Lorentz Force Law, this conserved momentum is therefore associated with charge.

You should also note that in more modern treatments of KK theory $g_{55}$ isn't restricted to being a constant, and is promoted to a scalar field. Physically this means that the 5th dimension is allowed to have variable size.

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