[I’m switching to english for scientific posting]

So, I am trying to make my way through spin network theory. Spin networks were presumably discovered by Roger Penrose and more recently implemented by Rovelli and Smolin to describe the discretized nature of space-time in Loop Quantum Gravity. How they actually arise from a quantum theory of General Relativity remains utterly mysterious to me, but still one can learn the rules and play the game, since it is widely advertized that spin networks arise in many areas of physics, and I soon hope to take a glance into such fascinating areas such as… such as… Well, in particular I am interested in looking at relationships with diffusion theory on graphs.

A good point to start is Seth A. Major’s A Spin Network Primer (here). Unfortunately, this is far from being a self-contained introduction to the topic and more depressingly it is not "background independent" of the background training of the reader, which had better be from the area. To a poor simple Markov guy like me, it leaves many open questions, most prominently: Why?

I am looking for a physical intuition of what’s going on. I know I can’t argue axioms, as John Baez would say. But still, there is something disturbing in all that. It is like being a classical piano player at a jazz jam-session: everybody is improvising tetrahedra with virtuosistic Redemeister moves while you are staring at those 3j- and 15j-symbols on a sheet someone handled to you, calling it a "standard", but yet there is nothing standard for you in there: too much information and history and practice into a too compressed alphabet.

Here a first physical shot: one is working with undirected weighted graphs, with positive integer weights. Such a weight can be thought to be a collection of strands which make up the rope, and some minor god decided that solitary strands cannot end in the middle of nowhere. So at each crossing one must have an even number of strands going in, and will get a continuity equation. This is pretty much what happens in diffusion theory on graphs, where one has one conservation law at each node (with real weights, though!), and I guess that one could implement orientation so to give this analogy more appeal. Microscopically speaking, this is a theory of knots. Macroscopically, it is a theory of graphs. Now, here come the bad news: those weights are interpreted as labelling representation of the SU(2) Lie group. My main concern is to try to give this fact a physical interpretation purely in terms of diffusion of some form of "structured" information.