Asymptotic Flatness and Quantum Geometry

We construct a canonical quantization of the two dimensional theory of a parametrized scalar field on noncompact spatial slices. The kinematics is built upon generalized charge-network states which are labelled by smooth embedding spacetimes, unlike the standard basis states carrying only discrete labels. The resulting quantum geometry corresponds to a nondegenerate vacuum metric, which allows a consistent realization of the asymptotic conditions on the canonical fields. Although the quantum counterpart of the classical symmetry group of conformal isometries consists only of continuous global translations, Lorentz invariance can still be recovered in an effective sense. The quantum spacetime as characterized by a gauge invariant state is shown to be made up of discrete strips at the interior, and smooth at asymptotia. The analysis here is expected to be particularly relevant for a canonical quantization of asymptotically flat gravity based on generalized spin-network states labelled by smooth geometries.