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That's already kind of interesting because graphs (and higher analogs such as foams) are used to describe geometry. All we can know about a geometry (based on a finite number of observations, which incidentally are of discrete-spectrum type) is naturally described by such objects. So they are the natural way to represent geometries.

On the other hand quantum

**field**theory is naturally represented in terms of C* algebras---the axiomatic generalization of von Neumann algebras of observables. So an obvious way to go if you want a

**general covariant**QFT, in other words a GCQFT is to inject the underlying geometry ALSO into the C* algebra, as well as the fields.

This suggests we should be looking at a mathematical object which is simultaneously a spin network, or spin foam, AND a pair (M,ω) where M is a C* algebra and ω is a state defined on M. Essentially M is all the measurements LIVING on the particular graph/complex and ω is what we think their expectation/correlation values are.

It means that it might be rewarding to look at the conventional way people have worked out to build a C* algebra corresponding to any given directed graph.

I'll assemble the links. Anyone who has already studied this area please contribute links to any sources you found helpful.