Dimensional reduction in quantum gravity

The phenomenon of dimensional reduction in quantum theories of gravity is described. The five‐dimensional Kaluza‐Klein model is studied and the one‐loop effective potential, as a function of the five‐five component of the metric, is computed. For values of this field such that the distance around the fifth dimension is larger than the Planck length, the loop expansion for the effective potential is reliable. The form of the potential then implies the existence of a force tending to make the fifth dimension contract to a size on the order of the Planck length. This result can be interpreted as a gravitational version of the Casimir effect in quantum electrodynamics.