The quest for quantum gravity: from weyl invariance to transverse theories

Abstract

Tesis Doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de Lectura: 16-07-2021This thesis, written as a compendium of articles, addresses some of the fundamental problems encountered when trying to build a theory of Quantum Gravity. Taking General Relativity (GR) as the starting point, its well-known non-renormalizable character leads to the need for an ultraviolet completion. Besides, the Cosmological Constant (CC) problem is still one of the cornerstones of Theoretical Physics. The following articles explore possible insights into these problems within the context of gravitational Effective Field Theories. The first article is devoted to the study of quadratic (in curvature) theories of gravity when treated in the First Order formalism, where the metric and the connection are considered as independent fields. These renormalizable theories are quadratic in the derivatives of the connection and do not contain quartic propagators, leaving a priori some room for unitarity. Nevertheless, it is not clear whether these theories include a graviton or whether they are free of ghosts, as all the dynamics is now encoded in the connection field. A complete study of the propagating degrees of freedom is then needed. In this work, we analyze the spin content of a generic torsion-free connection by constructing a complete basis of 22 six index spin projectors. We find that these theories generically propagate a spin three piece together with several lower spin components. One of the classical solutions to the CC problem is to consider Weyl invariant theories, as they forbid a CC term in the action. This symmetry is a generalization of the usual conformal invariance to cases where gravity is present. In the second article of the thesis, we carry out an analysis of the (in)equivalence of conformal and Weyl invariant theories for the gravitational field. The most general Lagrangian for spin two particles up to dimension six operators is explored, corresponding to the low-energy expansion of linear and quadratic (in curvature) theories of gravity. We carry out a full classification of the theories invariant under linearized (transverse) diffeomorphism, linearized Weyl transformations, and the usual conformal and scale symmetries. In the last part of the thesis, the theory of Unimodular Gravity (UG) is examined. This theory is an alternative low-energy description of gravity defined as the truncation of GR to unit determinant metrics. In UG the CC does not couple directly to gravity due to the unimodular constraint, and thus, it possesses a completely different nature. In particular, it does not receive radiative corrections, partially solving the CC problem. Apart from the character of the CC, UG is found to be classically equivalent to GR, and the question of the full (in)equivalence of both theories is still an open debate when quantum corrections are considered. The potential differences arising when studying the coupling to matter are investigated, via the introduction of a non-minimally coupled scalar field. We compute all the one-loop divergences in both theories and find a physical combination of couplings whose running differs for intermediate values of the non-minimal couplin