Area propagator and boosted spin networks in loop quantum gravity
Quantum states of geometry in loop quantum gravity are defined as spin networks, which are graph dressed with 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 holonomy along the edge with a Lorentz boost into 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.