Classical and quantum wormholes in a 5D compact Kaluza-Klein theory

We study the classical and quantum euclidean wormholes for an empty (4+1) dimensional Kaluza-Klein universe with a positive cosmological constant and a spatially flat Robertson-Walker type metric. It is shown that classical wormholes do not exist neither for the space-time sector nor the extra dimensional sector of this model, but two spectra of quantum wormholes as the solutions of the Wheeler-DeWitt equation exist that are consistent with the Hawking-Page conjecture about the necessary boundary conditions. In the spectrum where the external scale factor R shapes the quantum wormholes the internal scale factor a may play the role of effective matter source, and in the spectrum where the internal scale factor a shapes the quantum wormholes the external scale factor R may play the role of effective matter source.