Matter fields in asymptotically safe quantum field theories of gravity

In this doctoral thesis we present the research of Refs. [1–4] concerning the asymptotic safety scenario in quantum gravity. We motivate the theoretical and conceptual need for the existence of a theory of quantum gravity, and explain how such a theory might be realised through the construction of an ultraviolet complete quantum field theory defined via a high-energy interacting fixed point. In particular, we give a pedagogical introduction to the functional renormalisation group, as well as fixed points and critical phenomena. Thereafter we investigate the issue of background independence, which is introduced when constructing a scale-dependent effective action of a gauge theory using the background field formalism through the introduction of an infrared cutoff operator and the gauge fixing procedure. To this end we study simultaneous solutions of the flow equation combined with so-called modified split Ward identities in a conformally truncated theory in the derivative expansion. In the main part of this thesis we study the dynamics of a gravitational system coupled to a number of scalar fields. In doing so, we find scaling solutions in a fully functional truncation using the derivative expansion and background field approximation on a d-dimensional sphere. We then study a similar system using the vertex expansion in a fully dynamical fluctuation field calculation. This is done in the exponential as well as the linear parametrisation of the metric, using different gauge fixing procedures. We find that the overall behaviour is very similar in both cases. We furthermore find however that the contribution of the scalar fields to the gravitational coupling differs in its sign compared to older results obtained in a background field approximation. To correct this behaviour, we supplement the background field equations with modified split Ward identities, with the aim to obtain the fluctuation field behaviour from a background approximation. This seems to give satisfactory results in the case of the cosmological constant, whilst the running of Newton’s constant still differs significantly.