Algebraic approach to quantum gravity IV: applications

We provide a relatively self-contained introduction to the application of quantum spacetime and quantum Riemannian geometry to theoretical physics. Recent successes include calculation of the vacuum energy of spacetime curvature fluctuations in a single-plaquette model of quantum gravity, derivation of the Kaluza-Klein ansatz as a consequence of quantum spacetime, exactly conserved Noether charges from variational calculus on a lattice, and a new theory of classical and quantum geodesics. The latter leads to a theory of generally covariant quantum mechanics applicable in General Relativity with intriguing first results for the case of a black-hole. We discuss several open problems past and present, and how they might be addressed going forward. New results include a phase transition for Euclidean quantum gravity on a 4-pointed star.