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