I have some questions regarding the current status of spin networks and their relation to geometry in loop quantum gravity.
The picture I got many years ago is that spin networks in LQG are coloured graphs, based on cylindrical function, constructed as solutions to a subset of the constraints.
The picture that edges (vertices) represent area (volume) is restricted to graphs as duals to polyhedra embedded in space, satisfying shape-matching conditions.
But in full LQG one forgets about these polyhedra and considers the spin networks as independent entities w/o any reference to the 3-space one started with.
Graphs as duals to polyhedra are special. A general graph does not need to be dual to any such collection of polyhedra embedded in 3-space. A simple example is a graph G for which this duality holds, and which we extend by some „non-local“ edges connecting non-adjacent vertices (adjacent in terms of the original graph G). Of course we may also consider „weired“ graphs that may be dual to polytopes in higher dimensional spaces.
Which role do these considerations play in LQG? How does one get rid of „weird“ graphs such that 3-space „emerges“ in some semi-classical limit? Are they excluded by construction? Are they allowed in principle but suppressed due to a dynamical mechanism? Can one calculate effects generated by „non-locality“?