Exact Results in Quantum Theory

I will present a framework of a non-perturbative quantization of general relativity coupled to dust and other matter fields. The irrotational dust provides a natural time variable, leading to a physical Hamiltonian with spatial diffeomorphism symmetry. The surprising feature is that the Hamiltonian is not a square root. This property, together with the kinematical structure of loop quantum gravity, provides a complete theory of quantum gravity with strict control over the physical Hilbert space, action of the Hamiltonian on it and the construction of physical observables. This puts in technical reach applications to cosmology, quantum gravitational collapse and Hawking radiation.