The results of this paper came as a surprise to me. It started out from a challenge I gave to myself: find a suitable model for a local discretisation of the hypersurface deformation algebra in a single simplicial cell. And do it without creating any unwanted anomalies. Only in a second step, would I want to ask how to glue adjacent cells to build a discretisation of a realistic geometry. To my surprise, this was easier than expected. There is a straight-forward way to discretise the constraints using holonomy-flux variables for Ashtekar's self-dual variables. How far the results generalize beyond a single building block, I do not know, but I lay out a strategy in the paper how to answer this question in a systematic way. Only future research can tell. Possible applications range from group field theory, loop qantum gravity, spinfoams and loop quantum cosmology.
This is a review article on quasi-local holography and bulk and boundary modes in general relativity and quantum gravity.
An analysis is given of the local phase space of gravity coupled to matter to second order in perturbation theory. Working in local regions with boundaries at finite distance, we identify matter, Coulomb, and additional boundary modes. The boundary modes take the role of reference frames for both diffeomorphisms and internal Lorentz rotations. Passing to the quantum level, we identify the constraints that link the bulk and boundary modes. The constraints take the form of a multi-fingered Schrödinger equation, which determines the relational evolution of the quantum states in the bulk with respect to the quantum reference fields for symmetries at the boundary. Taking the boundary to infinity, we obtain quantum reference frames for asymptotic symmetries.