# Dimensional reduction

Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero. For example, consider a periodic compact dimension with period L. Let x be the coordinate along this dimension. Any field $\phi$ can be described as a sum of the following terms:
$\phi_n = A_n \cos \left( \frac{2\pi n x}{L}\right)$
with An a constant. According to quantum mechanics, such a term has momentum nh/L along x, where h is Planck's constant. Therefore as L goes to zero, the momentum goes to infinity, and so does the energy, unless n = 0. However n = 0 gives a field which is constant with respect to x. So at this limit, and at finite energy, $\phi$ will not depend on x.