  We show that if the space of physical states spanned by the wormhole wave
functions can be equipped with a Hilbert structure, such a Hilbert space must
coincide with that of the Lorentzian gravitational system under consideration.
The physical inner product can then be determined by imposing a set of
Lorentzian reality conditions. The Hilbert space of the gravitational model
admits in this case a basis of wormhole solutions, and every proper quantum
state can be interpreted as a superposition of wormholes. We also argue that
the wave functions that form the basis of wormholes must be eigenfunctions of a
complete set of compatible observables. The associated eigenvalues provide a
set of well-defined wormhole parameters, in the sense that they can be employed
to designate the different elements of the basis of wormholes. We analyse in
detail the case of a Friedmann-Robertson-Walker spacetime minimally coupled to
a massless scalar field. For this minisuperspace, we prove the validity of all
the above statements and discuss various admissible choices of bases of
wormhole wave functions.

General Relativity and Quantum Cosmology