Functional truncations in asymptotic safety for quantum gravity
  Finite dimensional truncations and the single field approximation have thus far played dominant roles in investigations of asymptotic safety for quantum gravity. This thesis is devoted to exploring asymptotic safety in infinite dimensional, or functional, truncations of the effective action as well as the effects that can be caused by the single field approximation in this context. It begins with a comprehensive analysis of the three existing flow equations of the single field f(R) truncation by determining their spaces of global fixed point solutions and, where applicable, of corresponding eigenoperator solutions. As a second result, it is then shown that one incarnation of the single field f(R) approximation actually breaks down in the sense that there is no physical content left to explore. In order to clarify whether such drastic findings can be caused by the approximations used in setting up the renormalisation group flow, we identify the single field approximation as a prime candidate and show in the more familiar context of scalar field theory that it can indeed lead to many types of non-physical results. As a way to avoid such non-physical behaviour we highlight the importance of the previously known split Ward identity and exemplify its usefulness by fully restoring the correct physical picture in scalar field theory. Taking this result as evidence that the split Ward identity may lead to well behaved functional truncations also in gravity, we derive the flow equations of conformal gravity in a bi-field truncation of the effective action that goes beyond the local potential approximation in the fluctuation field. It is found that the split Ward identity leads to a simplified set of renormalisation group equations for the conformal factor that, while differing at crucial points, bear close resemblance to flow equations obtained in scalar field theory. 
Juergen Dietz
hep-th