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Quantum states of geometry in loop quantum gravity are defined as spinnetworks, which are graph dressed with SU(2) representations. A spin networkedge carries a half-integer spin, representing basic quanta of area, and thestandard framework imposes an area matching constraint along the edge: itcarries the same spin at its source and target vertices. In the context ofcoarse-graining, or equivalently of the definition of spin networks asprojective limits of graphs, it appears natural to introduce excitations ofcurvature along the edges. An edge is then treated similarly to a propagatorliving on the links of Feynman diagrams in quantum field theory: curvatureexcitations create little loops -- tadpoles -- which renormalize it. Thisrelaxes the area matching condition, with different spins at both ends of theedge. We show that this is equivalent to combining the usual SU(2) holonomyalong the edge with a Lorentz boost into SL(2,C) group elements living on thespin network edges, underlining the fact that the Ashtekar-Barbero connectioncarries extrinsic curvature degrees of freedom. This finally leads us tointroduce a new notion of area waves in loop quantum gravity.

Area Propagator and Boosted Spin Networks in Loop Quantum Gravity