Asymptotic flatness and quantum geometry

From the perspective of quantum gravity, the spacetime is smooth only in an effective sense, and is expected to exhibit a discrete structure at suitably small length scales. Within the gauge theoretic formulation of gravity, there are certain kinematical states which provide an elegant realization of such a scenario. These are known as the spin-network states, and are used extensively in certain quantization approaches, e.g. Loop Quantum Gravity (LQG). However, since these states correspond to a spatially discrete quantum geometry, they cannnot be used to capture the notion of a classical spacetime continuum. This leads to a serious obstacle towards a quantization of asymptotically flat gravity, where the asymptotic geometry approaches the flat continuum. Here we address this issue, and explore a representation based on generalized spin-network states in order to implement asymptotic flatness in the quantum theory. As a preparation for an analysis of asymptotically flat gravity based on such states, we construct a quantization of two dimensional theory of a parametrized scalar field (PFT) on noncompact spatial slices. This exercise is particularly motivated by the fact that PFT is an exactly solvable toy model (field theory) of quantum gravity. We demonstrate that the asymptotic conditions on the canonical fields at spatial infinity can be realized consistently using the generalized states, and discuss the fate of the Lorentz symmetries in this context. The framework set up here is expected to be relevant for a quantization of asymptotically flat gravity using spin-network states, and also for exploring questions regarding Lorentz violation in quantum gravity.