  In canonical quantum gravity, the formal functional integral includes an
integration over the local conformal factor, and we propose to perform the
functional integral over this factor before doing any of the other functional
integrals. By construction, the resulting effective theory would be expected to
be conformally invariant and therefore finite. However, also the conformal
integral itself diverges, and the effects of a renormalization counter term are
considered. It generates problems such as unitarity violation, due to a
Landau-like ghost, and conformal anomalies. Adding (massive or massless) matter
fields does not change the picture. Various alternative ideas are offered,
including a more daring speculation, which is that no counter term should be
allowed for at all. This has far-reaching and important consequences, which we
discuss. A surprising picture emerges of quantized elementary particles
interacting with a gravitational field, in particular gravitons, which are
"partly classical". This approach was inspired by a search towards the
reconciliation of Hawking radiation with unitarity and locality, and it offers
basic new insights there.
