Kaluza Klein charge If I take a $(d+1)$ dimensional Einstein Hilbert Lagrangian $L_{d+1}=\sqrt{-\hat{g}} \hat{R}$ and perform a standard Kaluza Klein dimensional reduction by periodically identifying one direction, let's say $z$, by $z \sim z + 2 \pi r$, I arrive at a $d$ dimensional Lagrangian $L_d=\sqrt{-g}(R - \frac{1}{2} (\partial \phi)^2 - \frac{1}{4} e^{-2(d-1) \alpha \phi} F^2)$.
We can see that the Kaluza Klein vector in the $(d+1)$ dimensional metric manifests itself as a $d$ dimensional gauge field in the lower dimensional system. This gauge field has some associated electric charge and I would like to know how, and why, this gets quantized as a result of the identification $z \sim z + 2 \pi r$.
Thanks very much.
 A: The Kaluza-Klein equations of motion (the geodesic equations) for a particle moving in the 5D spacetime contain the equations of motion of a particle in 4D spacetime under influence of electromagnetism if and only if one identifies $p^5 = mU^5 = \frac{1}{\sqrt{G}}cq$, i.e. relates the momentum in the fifth dimension $p^5$ to electric charge $q$. (And yes, the momentum in the fifth dimension is a constant of motion, so this is allowed.)
Now, for the Kaluza-Klein theory on a 5D cylinder $\mathbb{R}^{3,1}\times S^1$, the fifth coordinate is that of a circle, and since position and momentum are Fourier transforms of each other in the quantum theory, this means the allowed momenta $p^5$ are discrete, yielding discretization of the electric charge.
A: You can read lecture notes by Malcolm Perry on the Applications of Differential Geometry to Physics. A detailed answer to your question is given in pages 37 & 38. I am providing the link for the (unofficial) lecture notes 
http://www.aei.mpg.de/~gielen/diffgeo.pdf
