​Super-Renormalizable Quantum Gravity

It can be proved that the class of finite quantum gravitational theories (in odd as well as in even dimensions) is actually a range of anomaly-free conformally invariant quantum theories in the spontaneously broken phase of the conformal Weyl symmetry. The fact that conformal symmetry present at classical level is preserved at quantum level allows us to address the singularity issue on the base of a fundamental symmetry principle. Indeed, the Weyl conformal invariance is likely able to tame the spacetime singularities that plague not only Einstein gravity, but also local and weakly non-local higher derivative theories. This latter statement can be rigorously proved by a simple singularity theorem that applies to a large class of weakly non-local theories. Therefore, contrary to the claim that we can find in literature the non locality is not sufficient to tame the singularities' issue of Einstein's gravity. Now following the seminal paper by Narlikar and Kembhavi, we can explicit construct singularity-free exact black hole solutions conformally equivalent to the Schwarzschild metric. In conformal gravity it is also trivial and tautological to show that the FLRW cosmological solutions are singularity free. Finally, the Belinskii, Khalatnikov, Lifshitz (BKL) spacetimes, which exactly solve the classical equations of motion, are conformally equivalent to regular spacetimes.

So far we have many examples of spacetime in which conformal invariance places a crucial role in removing the singularities. We are so temped to believe that there are no singularity on a conformal manifold. Nevertheless, we do not have yet a general prove of the latter statement.  

 

Another related issue is about the closed time-like curves (CTC) in a conformal invariant theory. Can the Weyl rescaling makes harmless the (CTC) with a proper choice of the overall conformal factor?

We will attempt to generalize the found results and answer the new question in future works.