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 spacetime sector nor the extra dimensional sector of this model, but two spectrums of quantum wormholes as the solutions of Wheeler-DeWitt equation exist which 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.