  We carry out the nonperturbative canonical quantization of several types of
cosmological models that have already been studied in the geometrodynamic
formulation using the complex path-integral approach. We establish a relation
between the choices of complex contours in the path integral and the sets of
reality conditions for which the metric representation is well defined, proving
that the ambiguity in the selection of complex contours disappears when one
imposes suitable reality conditions. In most of the cases, the wave functions
defined by means of the path integral turn out to be non-normalizable and
cannot be accepted as proper quantum states. Moreover, the wave functions of
the Universe picked out in quantum cosmology by the no-boundary condition and
the tunneling proposals do not belong, in general, to the Hilbert space of
quantum states. Finally, we show that different sets of reality conditions can
lead to equivalent quantum theories. This fact enables us to extract physical
predictions corresponding to Lorentzian gravity from quantum theories
constructed with other than Lorentzian reality conditions.
