Universal regular short distance behavior from an interaction with a scale invariant gravity

We assume that the fourdimensional quantum gravity is scale invariant at short distances. We show through a simple scaling argument that correlation functions of quantum fields interacting with gravity have a universal (more regular) short distance behavior. We consider the tetrads ea(x) as the basic variables for quantum gravity. We assume that correlation functions of ea(x) are scale invariant. This means that (with a certain real γ) λea(λx) and e μ a(x) have the same correlation functions for all real λ,i.e., these are equivalent random variables. Moreover, we assume that at short distances < ea(x)e ν c (x ) > depends only on x − x. The scale invariance cannot apply to intermediate distances where the correspondence with the classical gravity is required. Then, we expect that the mean value of ea(x) describes the classical gravity. We could apply our arguments if the mean value of the tetrad was different from zero and the correlation functions were scale invariant only for short distances. However, for simplicity of the presentation we assume the exact scale invariance. We show that quantum fields interacting with gravity have a more regular behavior than the canonical one. It is determined solely by the scaling index γ. In reality quantum gravity is not expected to be exactly scale invariant. There will be a distance scale (presumably the Planck scale) which determines the region where the scaling applies. There are renormalizable models of gravity (with the square of the curvature tensor) [1][2] [3][4] which may be scale invariant at short distances. Experiments support the canonical behavior of QED at short distances (modified eventually by logarithmic corrections). We suggest that the regular (superrenormalizable) behavior takes over at distances below the Planck scale ( there were earlier papers on this topic reviewed by S. Deser in [5], see also [6][7]). We consider first the free scalar field in four dimensions interacting with gravity. The two-point function in an external gravitational field is (in the