Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 59 / 6 / c

Each compact Fourier mode obeys the noncompact equation with dispersion relation ${\omega }^2={\mathbf{k}}^2+m_n^2$, whereThus momentum in the extra dimension appears as rest mass to a four-dimensional observer. There is one massless zero mode and equally massive $n,-n$ modes. For a real scalar they can be organized into sine and cosine components rather than independent complex fields.

At equal compact positions all phase factors are one, and the propagator becomes the Kaluza-Klein towerThe factor $1/R$ follows from the normalized compact eigenfunctions. It cannot be discarded if the five-dimensional field and source normalizations are retained.