  We present a non-perturbative quantization of general relativity coupled to
dust and other matter fields. The 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, and puts in technical reach applications to
cosmology, quantum gravitational collapse and Hawking radiation.
