Quantum dynamics of Kaluza-Klein theories

Some of the quantum properties of Kaluza-Klein theories are studied. The classical features of these theories are reviewed, and the quantization of the gravitational field in an arbitrary number of dimensions is described. These results are then applied to a detailed analysis of the five-dimensional Kaluza-Klein model. The fifth dimension is taken to be compact and a quantum effective potential, as a function of the five-five component of the metric, is constructed. It is argued that the one-loop computation is reliable as long as the distance around the fifth dimension is large compared to the Planck length. The effective potential separates into two pieces: an induced cosmological constant, independent of the size of the fifth dimension, and a distance-dependent "Casimir" energy. The cosmological term is subtracted, leaving an attractive Casimir potential which will contract the fifth dimension to a size on the order of the Planck length. Consequences of this result are discussed and some of the ways in which it can be generalized are outlined.