The big breakthroughs of general relativity and quantum theory of the past century have taught us two important lessons: firstly, space and time are not a static background for matter, but are themselves dynamical and depend on the distribution of matter, and secondly, this matter does not have a fixed position and velocity at small scales. Such properties should instead be described probabilistically.
Together, these results of general relativity and quantum theory lead to a radical conclusion for what is called quantum gravity. Space and time could be described as an ensemble of many versions of itself of different shapes, in the form of a path integral. Existing concepts such as position and a moment in time are no longer meaningful in this view of space and time and new measurable quantities, or observables, must be found to be able to say something about the geometric properties of space and time at the smallest scales of nature.
In my doctoral thesis, I describe three proposals for new observables in non-perturbative quantum gravity and illustrate the difficulties that are encountered. These include an observable based on the gravitational Wilson loop, an observable related to diffusion of m-form fields and an observable based on approximate Killing symmetries. I also give an outlook on possible interesting new phenomena which could be discovered with these observables, such as an effective dimension for propagating matter at the smallest scales of the theory that depends on the type of matter.