  A crucial problem in quantum cosmology is a careful analysis of the one-loop
semiclassical approximation for the wave function of the universe, after an
appropriate choice of mixed boundary conditions. The results for Euclidean
quantum gravity in four dimensions are here presented, when linear covariant
gauges are implemented by means of the Faddeev-Popov formalism. On using
zeta-function regularization and a mode-by-mode analysis, one finds a result
for the one-loop divergence which agrees with the Schwinger-DeWitt method only
after taking into account the non-trivial effect of gauge and ghost modes. For
the gravitational field, however, the geometric form of heat-kernel asymptotics
with boundary conditions involving tangential derivatives of metric
perturbations is still unknown. Moreover, boundary effects are found to be
responsible for the lack of one-loop finiteness of simple supergravity, when
only one bounding three-surface occurs. This work raises deep interpretative
issues about the admissible backgrounds and about quantization techniques in
quantum cosmology.
