On Signature Transition and Compactification in Kaluza-Klein Cosmology

We consider an empty (4+1) dimensional Kaluza-Klein universe with a negative cosmological constant and a Robertson-Walker type metric. It is shown that the solutions to Einstein field equations have degenerate metric and exhibit transitioins from a Euclidean to a Lorentzian domain. We then suggest a mechanism, based on signature transition which leads to compactification of the internal space in the Lorentzian region as $a \sim |\Lambda|^{1/2}$. With the assumption of a very small value for the cosmological constant we find that the size of the universe $R$ and the internal scale factor $a$ would be related according to $Ra\sim 1$ in the Lorentzian region. The corresponding Wheeler-DeWitt equation has exact solution in the mini-superspace giving rise to a quantum state which peaks in the vicinity of the classical solutions undergoing signature transition.