On signature transition and compactification in Kaluza-Klein cosmology

We consider an empty (4 + 1)-dimensional Kaluza-Klein universe with a negat ive cosmological constant and a Robertson-Walker-type metric. It is shown t hat the solutions to the Einstein field equations have a degenerate metric and exhibit transitions from a Euclidean to a Lorentzian domain. We then su ggest a mechanism, based on signature transition which leads to compactific ation of the internal space in the Lorentzian region as a similar to \Lambd a\(1/2). With the assumption of a very small value for the cosmological con stant we find that the size of the universe R and the internal scale factor a would be related according to Ra similar to 1 in the Lorentzian region. The corresponding Wheeler-DeWitt equation has an exact solution in the mini -superspace giving rise to a quantum state which peaks in the vicinity of t he classical solutions undergoing signature transition.