The Φ quantum field in a scale invariant random metric

We discuss 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 Φ theory in four dimensions we obtain explicit lower and upper bounds for each term of the perturbation series. It comes out that there is no charge renormalization in Φ model in four dimensions. We show that in a particular range of the scale dimension there are models in D = 4 without any divergencies.