Super-Renormalizable Multidimensional Gravity: Theory and Applications

Abstract

In this paper we introduce a perturbatively super-renormalizable and unitary theory of quantum gravity in any dimension D. In four dimensions the theory is an extension of the Stelle higher derivative gravity that involves an infinite number of derivative terms characterized by two entire functions, a.k .a. “form factors”. In dimension D we preserve two entire functions and we implement a finite number of local operators required by the quantum consistency of the theory. 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. By expanding the form factors to the lowest order in a mass scale we introduce, the local high derivative theory is recovered. Any truncation of the entire functions gives rise to the unitarity violation and it is only by keeping all the infinite series that we overcome similar issues. The theory is renormalizable at one loop and finite from two loops upward. More precisely, the theory turns out to be super-renormalizable because the covariant counter-terms have less derivatives then the classical action and the coefficients of the terms with more derivatives do not need any kind of infinity renormalization. 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 (either dust or radiation). We conclude the article providing an extensive analysis of the spectral dimension for any D and for the three classes of theories. 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 t heories 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.