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.
In our article we address this issue, and explore a representation based on generalized spin-network states which carry a label corresponding to smooth embedding spacetimes in addition to the discrete group-representation labels. Such states were originally introduced by Koslowski-Sahlmann (K-S) in the context of LQG in order to capture the smoothness of classical spacetime in an effective sense. As a preparation for an analysis of asymptotically flat gravity using the generalized kinematical representation, we construct a quantization of two dimensional theory of a parametrized scalar field (PFT) on noncompact spatial slices. Since PFT is a model where for compact slices the Hamiltonian constraint can be solved unlike gravity, it is a good testing ground to see if K-S representation in noncompact case also can be used to solve dynamics.
Here we demonstrate that the asymptotic conditions on the canonical fields at spatial infinity can be realized consistently using the generalized states. The corresponding quantum spacetime, as obtained after solving the constraints, is discrete at the interior, and smooth at asymptotia. Such a construction signifies an important step, since these features do not emerge within the standard kinematical formulation. We also demonstrate that although the Lorentz symmetry is broken in the quantum theory, the symmetry can be recovered in an approximate sense at large scales. The framework set up here is expected to be relevant for a quantization of asymptotically flat gravity, and also for exploring questions regarding Lorentz violation in quantum gravity.
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