  We consider the Hartle-Hawking wavefunction of the universe defined as a
Euclidean path integral that satisfies the "no-boundary proposal." We focus on
the simplest minisuperspace model that comprises a single scale factor degree
of freedom and a positive cosmological constant. The model can be seen as a
non-linear $\sigma$-model with a line-segment base. We reduce the path integral
over the lapse function to an integral over the proper length of the base and
use diffeomorphism-invariant measures for the ghosts and the scale factor. As a
result, the gauge-fixed path integral is independent of the gauge. However, we
point out that all field redefinitions of the scale factor degree of freedom
yield different choices of gauge-invariant path-integral measures. For each
prescription, we compute the wavefunction at the semi-classical level and find
a different result. We resolve in each case the ambiguity in the form of the
Wheeler-DeWitt equation at this level of approximation. By imposing that the
Hamiltonians associated with these possibly distinct quantum theories are
Hermitian, we determine the inner products of the corresponding Hilbert spaces
and find that they lead to a universal norm, at least semi-classically. Quantum
predictions are thus independent of the prescription at this level of
approximation. Finally, all wavefunctions of the Hilbert spaces of the
minisuperspace model we consider turn out to be non-normalizable, including the
no-boundary states.
