  Perturbative canonical quantum gravity is considered, when coupled to a
renormalizable model for matter fields. It is proposed that the functional
integral over the dilaton field should be disentangled from the other
integrations over the metric fields. This should generate a conformally
invariant theory as an intermediate result, where the conformal anomalies must
be constrained to cancel out. When the residual metric is treated as a
background, and if this background is taken to be flat, this leads to a novel
constraint: in combination with the dilaton contributions, the matter
lagrangian should have a vanishing beta function. The zeros of this beta
function are isolated points in the landscape of quantum field theories, and so
we arrive at a denumerable, or perhaps even finite, set of quantum theories for
matter, where not only the coupling constants, but also the masses and the
cosmological constant are all fixed, and computable, in terms of the Planck
units.
