  We discuss a D-dimensional Euclidean scalar field interacting with a scale
invariant quantized metric. We assume that the metric depends on d-dimensional
coordinates where d<D. We show that the interacting quantum fields have more
regular short distance behaviour than the free fields. A model of a Gaussian
metric is discussed in detail. In particular, in the \Phi^4 theory in four
dimensions we obtain explicit lower and upper bounds for each term of the
perturbation series. It turns out that there is no coupling constant
renormalization in the \Phi^4 model in four dimensions. We show that in a
particular range of the scale dimension there are models in D=4 without any
divergencies.
