Local cohomology, master equation and renormalization of higher-derivative and nonlocal quantum gravity

In this thesis we have studied the properties of a higher-derivative toy model of quantum gravity and we have used the results to analyze the main features of nonlocal theories of quantum gravity. The renormalizability of higher-derivative quantum gravity have been revisited by using the Batalin-Vilkovisky formalism. The very hearth of this formalism is the master equation which plays a foundamental role in the analysis of a gauge system. In particular, it is very useful to prove the renormalizability of a gauge theory. We have considered two main approaches to prove the renormalizability of higher-derivative quantum gravity, both performed by induction in the number of loops. The “cohomological” approach preserves the validity of the master equation at each inductive step. We assume that the theory is renormalized up to n-loops and then prove that we can cancel out all the divergences at (n + 1)-loops. It turns out that the (n+1)-loops counterterms must satisfy a certain cohomological condition. The cohomological method only works if we can write the solution of this condition as a sum of a gauge invariant functional of the gauge fields plus a trivial term. This is also known as Kluberg-Stern-Zuber conjecture. In the original work on the renormalizability of higher-derivative quantum gravity, Stelle assumes the validity of the conjecture. So far, it has been proved for Yang-Mills theory and Einstein gravity. We have extended the proof to higher-derivative quantum gravity. Our results naturally extend to nonlocal theories of quantum gravity, which turn out to be super-renormalizable. In general, it is not obvious that the Kluberg-Stern-Zuber condition holds. In such a situation, the cohomological approach would fail. A more powerful method which avoid this constraint is a so-called “quadratic” approach. In this case, the inductive procedure absorbs the divergences automatically at each loop. On the other hand, it postpones the solution of the master equation to the end of the inductive procedure, after the theory is renormalized to all orders. We have worked out the exact solution of the master equation at the renormalized level. This result also proves that higher-derivative quantum gravity is renormalizable, and generates the same number of renormalization constants as the cohomological approach. In particular,the structure of the BRST transformations is preserved by renormalization. This last property, extended also to nonlocal gravity, enabled us to conclude that general covariance is the most general gauge symmetry of a renormalizable higher-derivative and nonlocal quantum theory of gravity. Besides the investigation of higher-derivative quantum gravity, we have also considered aspects of nonlocal theories. The interest in studying nonlocal theories is twofold. On the one hand, it brings to light a sector of quantum field theory that is still vastly unexplored and worth of investigation. On the other hand, it is a candidate for a unitary and super-renormalizable quantum theory of gravitation. It is worth to note that these theories are also predictive to a certain extent. Indeed, they have a rather constrained behaviour in the UV limit and provide predictive transition amplitudes at high energies