Kaluza-Klein tower ($m_n^2=M^2+n^2/R^2$)

For a field periodic on a circle of radius $R$, a Fourier series gives modes with quantized internal momentum $n/R$. Each mode is a lower-dimensional field with the displayed squared mass. The entire sequence is the tower; retaining only modes below a chosen energy gives an effective field theory. The tower is distinct from any individual Kaluza-Klein mode.

The compactification scale is the first nonzero internal-mode mass. On a circle it equals $1/R$ for a higher-dimensional massless field. A zero-mode dimensional reduction applies at energies well below it. Warping, nontrivial geometry and boundary conditions can change the relation between the first mass and a geometric radius.

A pointlike pulse can contain frequencies above the compactification gap even when its travel time and distance greatly exceed the circle size. Its retarded circle-compactified massless propagator can resolve winding images and massive-mode oscillations. Suppressing those modes requires low-frequency or long-wavelength sources/measurements, or suitable smearing; mere coordinate separation is not that restriction. The static Green function across a circle compactification gives a precise long-distance zero-mode limit without this ambiguity.