Dynamical refinement in loop quantum gravity
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In Loop Quantum Gravity, a quantum state of the gravitational field has a semiclassical interpretation as a three-dimensional lattice discretization of space. We explore the possibility that the scale of the lattice is only as fine as it needs to be in order to carry the dominant frequency excitations of the auxiliary fields living on the lattice, by considering graph-changing transition amplitudes in the context of a pure gravity quantum theory. We define regular graphs that correspond to closed spatial slices of FLRW spacetime in a novel way, with coherent state labels that correspond to physical observables. This correspondence is obtained using the novel concept of a pseudoregular polyhedron which affords a dimensionless volume to surface area ratio in terms of the number of faces of the polyhedron. We normalize these regular graph states using a new method, employing a saddle point approximation based on the valence of the nodes rather than the large-scale semiclassical limit to obtain a result that holds in the quantum limit. Finally we employ the EPRL spin foam model to obtain a transition amplitude between single-node graphs of arbitrary valence that is valid in both the semiclassical and quantum regimes, using an improved method of normalizing the amplitude. We find that if we fix the scale factor and the fiducial volume of space the amplitude favors final states with infinitely large valence.