  We carry out to completion the quantization of a Friedmann-Robertson-Walker
model provided with a conformal scalar field, and of a Kantowski-Sachs
spacetime minimally coupled to a massless scalar field. We prove that the
Hilbert space determined by the reality conditions that correspond to
Lorentzian gravity admits a basis of wormhole wave functions. This result
implies that the vector space spanned by the quantum wormholes can be equipped
with an unique inner product by demanding an adequate set of Lorentzian reality
conditions, and that the Hilbert space of wormholes obtained in this way can be
identified with the whole Hilbert space of physical states for Lorentzian
gravity. In particular, all the normalizable quantum states can then be
interpreted as superpositions of wormholes. For each of the models considered
here, we finally show that the physical Hilbert space is separable by
constructing a discrete orthonormal basis of wormhole solutions.
