Super-renormalizable Multidimensional Quantum Gravity

In this paper we introduce a perturbatively super-renormalizable and unitary theory of quantum gravity in any dimension D. The theory presents two entire functions, a.k.a. "form factors", and a finite number of local operators required by the quantum consistency of the theory itself. The main reason to introduce the entire functions is to avoid ghosts (states of negative norm) like the one in the four-dimensional Stelle's theory. The new theory is indeed ghost-free since the two entire functions have the property to generalize the Einstein-Hilbert action without introducing new poles in the propagator. The theory is renormalizable at one loop and finite from two loops upward. In this paper we essentially study three classes of form factors, systematically showing the tree-level unitarity. We prove that the gravitation potential is regular in r = 0 for all the choices of form factors compatible with renormalizability and unitarity. We also include Black hole spherical symmetric solutions omitting higher curvature corrections to the equation of motions. For two out of three form factors the solutions are regular and the classical singularity is replaced by a "de Sitter-like core" in r=0. For one particular choice of the form factors, we prove that the D-dimensional "Newtonian cosmology" is singularity-free and the Universe spontaneously follows a de Sitter evolution at the "Planck scale" for any matter content. We conclude the article stating that, in the ultraviolet regime, the spectral dimension takes on different values for the three cases: less than or equal to "1" for the first case, "0" for the second one and "2" for the third one. Once the class of theories compatible with renormalizability and unitarity is defined, the spectral dimension has the same short-distance "critical value" or "accumulation point" for any value of the topological dimension D.