Feynman diagrams are a good way of calculating effects in quantum electrodynamics on a constant background space-time.
Spin foams (LQG) are conjectured to be a good way of calculating quantum effects with gravity.
Is there any recent ideas of how the two ideas will fit together?
Are particles thought to be identified with nodes on the spin-network. Or perhaps they are smeared round loops? Or on the lines? Why is this problem so hard to solve?
In other words, how (if it is known) would one embedd a Feynman graph on a spin-foam?
I would imagine if $V$ represents the vertices of a spin foam. Then one should be able to define fields $\phi(V)$ which have values for every vertex. Then one should be able to quantize those fields.