Before plunging into the group representation mess, I wanted to make up my mind about why "spin networks" are called "spin" networks. The conclusion I came to is quite different from those I’ve (rather superficially) taken a look at, but it is well suited for NESM (Non-Equilibrium Statistical Mechanics) since it deals with oriented graphs and with bits of information spreading through a thermodynamical system (i.e., a system where information diffuses).

Oriented spin networks

Spin networks (SN) are in the first place graphs (lines connecing dots in space) with an integral weight attached to each edge, which bears a microscopic interpretation in terms of collections of strands (think of a rope made up of several filaments):

 The black transversal box is the permutator, meaning that each straight edge is actually identified with the ensemble of strands in all their possible permutations, e.g.

SNs, when closed (no strands ending up in the middle of nowhere), can be algebraically manipulated so to yeald a number. Open SNs can be identified with tensors, permutators with determinants etc. At the moment I have no idea what the physical meaning of that number is and how it relates to other numbers in graph theory, such as cofactors of the laplacian matrix and so on (I have a vague hypothesis that such evaluations could find a place in NESM when restricting variational principles to given domains in the space of systems which share similar response properties (= which have similar inertias to external stimuli). Sort of "universality class index"… could be, but could very well not…). Let’s skip that.

I would rather deal with oriented graphs. So let’s put an arrow on each strand:

Now the permutator can only be taken among strands with the same orientation. We have also assigned an arbitrary orientation to the edges of the graph. We now have two quantum numbers for each edge: the total number of strands 2j (factor 2 not casual) and the difference 2m between forward and backward strands in the direction of the orientation of the corresponding edge; each time the direction of a strand is reversed there is an even jump in 2m: m ranges in -j/2,-j/2+1…j/2. So it is not hard to see that j and m behave respectively as the label of the SU(2) representation and as the eigenvalue of the third component of the angular momentum (that along which measurement is carried on).

So we can define angular momentum eigenvectors |j,m> this way:

Directed or signed SNs?

Before moving on, a note on directedness. Surfing the SN literature I can find mention of directed spin networks, but in a very different taste (and in fact, I can’t find oriented strands anywhere). Since when evaluating a SN one antisymmetrizes permutations according to their parity, as Rovelli and Smolin put it in Spin Networks and Quantum Gravity (here)

It follows that the (trivalent) states that we have obtained by antisymmetrizing the ropes are fully determined solely by their support, the order of each rope and an overall sign. Equivalently, they are determined by a trivalent graph (the support), with integers assigned to each link (the order of the rope), plus an orientation of the graph.

Angular momentum composition

Up to now it is only a very feeble correspondence. Things get more interesting when one tries to compose angular momenta. My first aim has been that of deriving the decomposition into irreducible representations (bold j’s label representations)

         (1)

We have to define the tensor product. In the first place, let’s combine trivial things such as

Graphically speaking, this looks like this:

So we conclude that the tensor product is a sort of multinomial permutator which permutes objects belonging to different sets without permuting objects in the sets. Unfortunately, this is far from being a good definition and we will encounter problems: one has to choose representative permutations somehow. Things depend heavily on this choice.

To generalize, we need a rule to estabilish what happens when connecting strands with opposite directions. The rule is: they annihilate! For example, consider the simplest example |1/2,1/2> X |1/2,-1/2>:

(2)

As you can see, we have connected "inverted" strands in the only feasible way, obtaining a well-known relation resonating with knot theory (the definition of Kauffman bracket), graph theory (the definition of Tutte polynomial in terms of deletion/contraction), statistical mechanics (Potts model), electric circuits (parallel vs. series connection) and most prominently QFT’s Feynmann diagrams

Now with the aid of some knot theory one can just "pull the string" in (2) and ambient-isotopically obtain a solitary strand directed rightwards, that is, a spin-0 particle (empty relationships are spin-0 particles?). One can easily check that (1) is satisfied with whichever notion of tensor product as something permutating strings among them, with the rule that opposite threads annihilate.

The real problem: probabilities

OK, it is still quite a stupid trick. But things can get worse. Since quantum states are ensembles of microscopic states (Bell would be happy to hear this, von Neumann would strongly argue), one can, statistically speaking, associate a probability to the ensemble by assigning microcanonical equal-a-priori probabilities to microstates, according to Liebniz principle of identity of indescernibles. In his original paper Angular momentum: an approach to combinatorial space-time, Roger Penrose writes

The idea, then, is that any ‘pure probability’ (if such exists) ought to be something arising ultimately out of a choice between equally probable alternatives.

So the idea is that we count the number of realizations which combine to give a quantum state and we interpret that (normalized) number as its probability. We are thus looking for the Clebsh-Gordan coefficients. This is not yet done, mainly because the CG explicit formula is monstruous

and is better found with creation-destruction operators rather than combinatorially testing it. How should it work? When m_1 and m_2 are extremal things work nicely, since taking the tensor product returns all permutations, e.g.

One can easily chek that this generalizes to any such cases. When initial states have an internal structure instead, one has to estabilish what it means to take the tensor product of a set (A,B) with C: it could be (A,B,C),(A,C,B),(C,A,B), or it could be (A,B,C),(A,C,B),(C,B,A), or anything else. Note that permuting A and B yealds all permutations in both cases. It turns out that when one combines any angular momentum with a spin-1/2 particle, one has to independently permute all of the elements of the first set with the solitary spin-1/2 strand:

It remains utterly misterious why we had to choose this particular form for the multinomial permutator, and how to generalize these particular cases to the general case. Special cases already become quite discouraging with 1 X 1, so it is hard to elaborate a precise conjecture and test it, before trying to give a proof. But we’re almost there…

All of this could well exist somewhere in the literature, and I hope someone will alert me before I spend too much precious time doing stupid things.

Anyway, once one gives a full derivation of Clebsh-Gordan coefficients, one can try to interpret all of the QM machinery, such as ladder operators and so on. It might be interesting to work out the SU(3) case; possibly, not with directed graphs but RGB-coloured graphs.

From a pseudo-phylosophical point of view (hold tight!), we have a microscopic statistical interpretation of QM in a very different fashion than that of Bohm & Co. Relevant for NESM is the fact that information is thought to be made up of a discrete number of bits, and ultimately bits are SU(2) spins. Note that spin-0 particles carry no information, so they are physically unobservables, yet statistically relevant for the counting: they account for symmetries (so Higgs boson does not exist, but it does affect other particles). I leave you with the question: could the theory of information (flows) emerge in a natural way from Quantum Gravity?