Final Report Summary - OBSERVABLES IN QG (Physical observables in quantum gravity) The Intra-European Fellowship 'Physical observables and states in quantum gravity' was concerned with approaches to construct a quantum theory of gravity and strategies to extract predictions out of such a theory. In recent years several approaches to quantum gravity, such as loop quantum gravity, spin foam models and causal dynamical triangulations showed promising developments. Loop quantum gravity is a canonical approach to quantum gravity that succeeded in constructing geometrical operators, such as areas and volumes. It turned out that these operators have a discrete spectrum (at least on the kinematical level), realising rigorously the expectation of a Planck size discreteness of space. Spin foam models are background independent path integrals motivated by loop quantum gravity, with interesting connections to lattice gauge theory and topological field theories. Causal dynamical triangulations is another path integral approach to quantum gravity based on discretisation which has recently produced evidence that four-dimensional space-times emerge in the continuum limit of the theory.All these approaches have in common that they use some kind of discreteness either as an auxilary structure or as a derived property. This opens the question of how this discreteness interacts with the symmetry at the heart of general relativity, which is the so-called diffeomorphism symmetry. In the classical and continuum theory this symmetry is deeply intertwined with the dynamics of the theory, at the same time defining its observables. Every theory of quantum gravity has to recover a representation of diffeomorphism symmetry in its classical and/or continuum limit. Otherwise one cannot hope that the theory displays the correct low energy limit.Moreover the question whether a quantum representation of the diffeomorphism symmetry is possible at all or not is still open. However that would also answer phenomenological questions, for instance whether we should expect a theory of quantum gravity to be Lorentz-violating or not.During the seven month of the fellowship the fellow undertook first steps to discuss representations of the diffeomorphism group in discretised theories of (quantum) gravity. After providing a clear definition of diffeomorphisms in discretised theories, it was shown that in general diffeomorphism symmetry is broken on the discretised level. In collaboration with another author implications were derived and discussed for approaches to obtain a consistent dynamics for quantum theories of gravity.As mentioned diffeomorphism symmetry is deeply intertwined with diffeomorphisms, hence a breaking of these symmetries has indeed a severe impact on the dynamics. In particular the proposal to define the dynamics via a so-called constraint algebra in discretised theories seems not to be feasible and other strategies may need to be found.Although diffeomorphism symmetry is in general broken for the discretised theories one might re-obtain these in a continuum limit. An alternative approach is that of the so-called perfect action program, which attempts to mirror the exact continuum dynamics on the discrete level and which has relations with the renormalisation group program. This is however only possible if one allows for non-local discrete theories, a notion which so far has not been discussed much in the quantum gravity community.These considerations open up a new line of research, leading to the incorporation of methods from statistical field theory via renormalisation techniques into quantum gravity. In particular the question arises, how an improvement of diffeomorphism symmetry can guide the renormalisation flow in the parameter space describing (quantum) gravity theories. A very first step is taken in a paper by the fellow and another author (to appear), discussing alternative discretisations of general relativity, which have an improved notion of diffeomorphism symmetry.