Extra Dimensions and Kaluza–Klein

Abstract

In this chapter we will study the idea of extra dimensions, as proposed by Kaluza and Klein. The idea of having extra dimensions, besides our usual 3 $+$ 1, is an old one. Already Theodor Kaluza (in 1921) and Oskar Klein (in 1926) had considered it. The general idea of Klein was to consider that our space is a product of our four dimensional space $M_4$ and a small compact space $K_n$, i.e., $K_D=M_4\times K_n$, with $D=4+n$. Then if the volume (or rather, the radii) of the extra dimensional space is really small, they would be unobservable. Perhaps the size is comparable to the Planck length $l_\mathrm{Pl}$, which would be natural in a quantum gravity theory, or maybe otherwise larger, yet still small.