Holographic / entropic approaches - why SU(2)?

[tom.stoer]

Let's forget about the history of LQG, that means let's forget about that it has been derived from GR in an appropriate formulation (Ashtekar variabes), let's forget about the constraints that had to be implemented; then we can simply use SU(2) spin networks as the starting point for a theory of quantum gravity.

The new discussion regarding entropic and holographic approaches (Verlinde, Smolin) are pointing into the same direction. One does not need the GR roots, the theory could be viable w/o its own history. Instead one has to turn this around and derive 4-dim spacetime, GR, dynamics etc.

Then there is one central question: why SU(2)

SU(2) is the unique remainig structure; it knows something about the 4-dim. spacetime as it is essentially one-half of SO(3,1) ~ SU(2)*SU(2). But if we forget about the history we must forget about SO(3,1), too.

So I can rephrase my question: why not SU(N), or SO(N), or some exceptional group?

 

[arivero]

Of course it could be argued that it is the smallest non abelian (and why? No idea).

If we are about rotations, and covering of rotations, then we need to look not SO(N) but the covers: http://en.wikipedia.org/wiki/Spin_group

There are some concepts which could be helpful here: the spinorial chessboard (Bott periodicity), and the https://www.physicsforums.com/showthread.php?t=181194".

 

[tom.stoer]

The main point is that we cannot answer the question "why SU(2)?" w/o having the result already in mind.

 

[Fra]

I sure don't have any answer but I do expect a satisfactory argument would go along the lines that the symmetries we see in nature are the simplest possible distinguishable symmetry, BUT where simplicity has a specific meaning of minimally speculative, and where the set of possible distinguishable symmetries are CONSTRAINED by the complexit of the observer. And if we start from the low complexity limit, then the complexity is so low that there choices are finite. Then question is then how these preferencs SCALE with complexity, up to the point where continuum like structures start to appear. The continuum does not exist at the low complexity limit IMO.

This is why I think that models like ST, are IGNORING part of the construction when STARTING with continuum structures. I think this may even be the seed to the landscape problems - or in particular the lack of selection principle.

I think this is an interesting question and I think there ought to be a good answer to this out there.

/Fredrik

 

[arivero]

Hmm are we sure that this SU(2) is defined as coming from SO(3,1)?

The fundamental object flowing across spin networks is $$\hbar$$. Perhaps SU(2) is imposed because of it.

 

[arivero]

In the original paper (which, incidentally, is titled "Angular momentum: an approach to combinatorial space-time") Penrose insinuates that SU(2) would be fixed by some kind of self-consistency: there should be rules to build a space from a group, and a group from a space, and you should get the original space (or the original group) after going across all the process and back. He seems to insinuate that only the peculiar properties of SU(2) allow for the game he is playing.

 

[Fra]

In the original paper (which, incidentally, is titled "Angular momentum: an approach to combinatorial space-time") Penrose insinuates that SU(2) would be fixed by some kind of self-consistency: there should be rules to build a space from a group, and a group from a space, and you should get the original space (or the original group) after going across all the process and back. He seems to insinuate that only the peculiar properties of SU(2) allow for the game he is playing.

Thanks for reminding of that paper, I've probably seen that paper before but I'll read it again.

What you describe in principle makes sense, as it describes an evolving almost circular process, where the so called "self-consistency" condition sound like it really is a simply some assumption of the existence of a steady state? (equilibrium) in this game, and this equiblirium state would then be characterised by this symmetry?

This could be an alternative view of things, that would be more to my taste than the way rovelli describes it.

Wether his reasoning is right is anothre story, but the general reasoning seems rational enough to me,. I'll get back when I read that

/Fredrik

 

[tom.stoer]

I know that Penrose introduced spin networks quite early, but I have never seen this paper before; is there an online version available somewhere?

I would appreciate hints regarding self consistency for the "space-time symmetry structure".

@arivero: yes, I would say we can be sure; where else should this symmetry come from? We start with SO(3,1), reformulate it according to Ashtekar's approach and end up with SU(2) connections. The local gauge invariance formulated in the tangent space is just this SU(2) symmetry. Constructiing the Hilbert space uses SU(2) holonomies and cylindrical functions which eventually result in SU(2) spin networks. As far as I can see there is no choice.

 

[arivero]

@arivero: yes, I would say we can be sure; where else should this symmetry come from? We start with SO(3,1), reformulate it according to Ashtekar's approach and end up with SU(2) connections. The local gauge invariance formulated in the tangent space is just this SU(2) symmetry. Constructiing the Hilbert space uses SU(2) holonomies and cylindrical functions which eventually result in SU(2) spin networks. As far as I can see there is no choice.

There is a hidden choice: that we have chosen to preserve angular momentum. This is a logical choice in the origin of mechanics; it is needed to formulate an action (which has units of angular momentum), or to formulate a contact transformation (which is the origin of Feynman method, following a idea of Dirac), or to grant Kepler's second law. Probably it is also the cause for the use of Riemaniann Curvature.

Now, it is true that we can preserve angular momentum in an arbitrary n-dimensional space time, so we are not limited to SO(3,1). On other hand, now I am not immediately sure about if it is possible, for any dimensionality, to represent and angular momentum J as a product of linear momentum p cross position x.

 

[Fra]

I know that Penrose introduced spin networks quite early, but I have never seen this paper before; is there an online version available somewhere?

I would appreciate hints regarding self consistency for the "space-time symmetry structure".

http://math.ucr.edu/home/baez/penrose/Penrose-AngularMomentum.pdf

I haven't had time to read it again yet, maybe tomorrow.

/Fredrik

 

[Fra]

I would appreciate hints regarding self consistency for the "space-time symmetry structure".

Given what I know of penrose general reasoning since before, I'm quite sure I would interpret his own reasoning at right angle, but still share the main conclusion. But that has happened before regarding his gravitationally induced wave collapse ideas. He tries to use gravity to construct a collective collapse; I think the better way is to use the subjective collapses to EXPLAIN the emergence of gravity.

My hunch is that I am similarly inverting his "consistency requirement" regarding the symmetry. I'll try to catually read the paper before adding more ramblings.

/Fredrik

 

[Prathyush]

Hi,
Can some one explain the question to me. I am familiar with the recent interest in holography and connection to gravity . I will be very helpful if some one can point me to resources for further reading on what spin networks are and their importance in quantum gravity and holography.

 

[tom.stoer]

There is a hidden choice: that we have chosen to preserve angular momentum. ...

This is not a hidden choice; if you chose n-dim. Minkowski space you have SO(n-1, 1) as symmetry group. The generators T^a with (T^a)₀₀ = 1 generate angular momentum.

Now, it is true that we can preserve angular momentum in an arbitrary n-dimensional space time, so we are not limited to SO(3,1). On other hand, now I am not immediately sure about if it is possible, for any dimensionality, to represent and angular momentum J as a product of linear momentum p cross position x.

It is rather simple; in 3 dim. angular momentum is defined as

J^a = ε^abc x_b p_c; a=1..3

Now you can identify ε^abc with the bc-component of the generator ε^a; that means you write it as

J^a = (ε^a)_bc x_b p_c

Going from SO(3,1) to SO(n-1,1) means that you replace (ε^a)_bc with (t^a)_bc where t^a is the generator a of the symmetry group of n-dim. spacetime.

Be careful: in 4 dim. spacetime you have SO(3,1) ~ SU(2)*SU(2); this is a special situation and does not hold in arbitrary dimensions. In addition the dimension of the rotation group is no longer identical to the dimension of space. In n-dim spacetime i.e. in n-1 dim. space you have

dim SO(n) = dim SO(n-1,1) = n(n-1)/2

The "spacelike" subgroup SO(n-1) generates angular momentum.

Summary: chosing n-dim. Minkowski space automatically fixes the structure group. So my question "why SU(2)" is related to the question "why 4-dim. spacetime". Now if we forget Minkowski spacetime we need longer longer answer the question "why 4-dim. spacetime", but instead we have to answer the question "why SU(2)". Perhaps the latter question is easier ...

 

[arivero]

J^a = ε^abc x_b p_c; a=1..3

Now you can identify ε^abc with the bc-component of the generator ε^a; that means you write it as

J^a = (ε^a)_bc x_b p_c

Going from SO(3,1) to SO(n-1,1) means that you replace (ε^a)_bc with (t^a)_bc where t^a is the generator a of the symmetry group of n-dim. spacetime.

Be careful: in 4 dim. spacetime you have SO(3,1) ~ SU(2)*SU(2); this is a special situation and does not hold in arbitrary dimensions. In addition the dimension of the rotation group is no longer identical to the dimension of space. In n-dim spacetime i.e. in n-1 dim. space you have

Yes it seems that for 3 dimensions the interesting object is J^a while for generic dimensions the object to look is (x_b p_c-x_c p_b).

Were we to need really a vector J, is should justify three dimensions, should it?

I think that a way to understand why the dimension of the rotation group must be greater than the dimension of space is that the rotation group actually needs to label "areas" where the cosines and sines will act.

 

[arivero]

Summary: chosing n-dim. Minkowski space automatically fixes the structure group. So my question "why SU(2)" is related to the question "why 4-dim. spacetime". Now if we forget Minkowski spacetime we need longer longer answer the question "why 4-dim. spacetime", but instead we have to answer the question "why SU(2)". Perhaps the latter question is easier ...

You can guess the answer. Choose between:

- It is because of the coincidences between Lie algebras in low dimensions.

- It is because of R C H O.

Usually it is one or another, at the end :-D

 

[tom.stoer]

Yes, for generic dimensions the object is slightly more complicated. Three dimensions are singled ouf if you insist on the pseudo-vector structure. You can trace it back similar to the tensor structure for (2,0) tensors like F for the electromagnetic field; only in four dimensions the dual *F has the same number of indices: let F be 2-form in D space; then its dual *F is an D-2 form, which means that 2 = D-2 is solved by D=4.

The dimension of the rotation group is fixed by the requirement that is generated by real, hermitean matrices. Geometrically the number of Euler angles grows faster than the number of dimensions.

 

[atyy]

Lubos Motl has an interesting post about spin networks with a different group.
http://motls.blogspot.com/2008/06/nathan-berkovits-proof-of-adscft.html

 

[Fra]

I just got half way through the paper last night, but I think Penrose isn't radical enough. I also recognise this so I read this sometime in that past.

These traits of his, are the ones I fully share in my own thinking:

+) "get rid of the continuum and build up physical theory from discreteness"
+) "build up space-time and QM simultaneously from combinatorical principles"
+) "continous concepcts emerget in a limit when we take more complex systems"
+) His search for so called "pure probabilities" which are rational numbers, rather than real numbers.

This SOUNDS ambitious, but then comes these things

-) "build up space-time and QM simultaneously from combinatorical principles - but not (at least in the first instance) to try and change physical theory"

This means he does not quite aim to exaplain things properly from first principles, he is - just like rovelli - just doing a REformulation and an alternative REconstruction, to arrive at an already set result.

-) "the most obvious physical concept that one has to start with, where QM says something is discrete and which is conncted to the structure of spacetime in a very intimate way is angular momentum. The idea here is to start with the concept of angular momentum - here one has a discrete spectrum - and use the rules for combining angular momentum tp see of one cn construct space from this"

This is not the most obvious choice of starting point to me. It also seems that far too much baggge is taken in, motivated by what we are looking for.

I am trying to connect to my own reasoning and find IMO a more sensible interpretation of Penrose networks, as more general action networks or state history networks, but I need to keep thinking of this. It's at least sure that while I share a number of his inspirational starting points, his ideas are not "first principle" enough for me.

I do not think we should just REconstruct, using what we know as constraints, I think we should REconstruct, without cheating, and if the idea is right the right physics should come out by itself, if not, the idea is wrong. Otherwise, what have we proven? Nothing as far as I can see.

/Fredrik

 

[tom.stoer]

I started to read the paper. Because now we know that there is a derivation of spin networks in LQG, it is more interesting to learn, why Penrose - w/o knowing this derivation - was interested in these mathematical objects.

In addition I would like to see if there is some principle of uniqueness forcing him to study SU(2). His spinor program was somehow specific for 4-dim spacetime, even though I was never able to find out why not something similar should by possible in other dimensions (there seems to be no no-go theorem).

Regarding R, C, H, O: could be that it's related to these structures, but it is not clear what the argument for uniqueness should be. Simplicity singles out R, largest commutative divison algebra singles out C, largest divison algebra singles out O, ...

John Baez was very interested in these topics for years, but I could not find a single hint for C, he seems to be more interested in O. The unique structure from Lie algebras is ceratinly E(8) as it is defined entirely in terms of e(8). But there's a long way down from E(8) to SU(2).

 

[Fra]

In addition I would like to see if there is some principle of uniqueness forcing him to study SU(2). His spinor program was somehow specific for 4-dim spacetime, even though I was never able to find out why not something similar should by possible in other dimensions (there seems to be no no-go theorem).

Regarding R, C, H, O: could be that it's related to these structures, but it is not clear what the argument for uniqueness should be. Simplicity singles out R, largest commutative divison algebra singles out C, largest divison algebra singles out O, ...

John Baez was very interested in these topics for years, but I could not find a single hint for C, he seems to be more interested in O. The unique structure from Lie algebras is ceratinly E(8) as it is defined entirely in terms of e(8). But there's a long way down from E(8) to SU(2).

I'm not sure how you reason, but to make any progress and be able to find something that is more properly an explanation, rather just a constructed explanation, I think we need to step back alot.

To me, starting from repetitive occurences of distinguishable events, one of the most fundamental symmetry is permutation symmetry (operating on historical sequences). And this symmetry probably somehow breaks into sub symmetries as the complexity increases, because the permutation symmetry are likely to be be broken in the sense that the breakaing becomes distinguishable only at a certain complexity level, since the breaking is yet another measures, requiring represenation cmoplexity.

Each time I've went this road, it seems to me the simplest possible continuum-like structure, appearing after permutation symmetry is broken is like a "string", but one with unknown embedding, and the string index is defined as the limit of the rational probability values, with in the limit is the continuum ]0,1]. Next one would ponder what happens when this grows more complexity, new dimensions are bound to grow.

I have not yet come as far as to infer symmetries from this, since I was stalled by trying to understand the process by which complexity is in fact increasing (generation of complexity).

Penrose also ponders the "large N" limit of what he called "directions". He also agreed that to define aconnection these must further be connected to a common network, which ha labeled K. This is interesting except I don't quite see it the same way.

But the exact "scaling" of the N->inf is important, and I think the reason for this scaling is too. How interacting networks compete for complexity.

I have to admit that I do nto yet understand penrose endavours in any other way that he is actually using a designed construction to arrive at something he wants, rather than using real first principles?

Does anyone agree or is it just me?

/Fredrik

 

[arivero]

Lubos Motl has an interesting post about spin networks with a different group.
http://motls.blogspot.com/2008/06/nathan-berkovits-proof-of-adscft.html

Dunno. This comparison between spin networks and Wilson lines seems to miss the point of angular momentum, which was the main motivation of Penrose, as it has been told. A spin network link is not (only) a kind of particular "gauge current line".

 

[tom.stoer]

Angular momentum emerges from spin networks b/c one starts with SU(2) or SO(3) which means one starts with angular momentum ...

Does group field theory tell us something regarding spin networks based on different groups?

 

[atyy]

Does group field theory tell us something regarding spin networks based on different groups?

GFT links to spin foams. I don't know if spin foams in general link to spin networks.

Another place that spin networks are popping up is condensed matter! http://arxiv.org/abs/0907.2994

 

[tom.stoer]

I do not care if spin networks or spin foams; what I would like to understand is whether different groups can be used and what the results are. I googled "SU(N) spin networks" and "SU(N) spin foams" and found nothing!

 

[Physics Monkey]

I can tell you one situation where the answer is reasonably well understood. I have in mind the situation of topological quantum field theory in 2+1 dimensions. This is clearly not quantum gravity in 3+1, but at least quantum gravity in 2+1 seems quite like one of these topological theories. No background metric is required, for example.

In this setting there is no sense in which SU(2) is unique. One may in general have various kinds of so called Chern-Simons theories which can be reformulated in a spin network language. One of the simplest is the "quantum group" called SU(2) level k (a truncated version of SU(2) ) that can be used to define a regularized version of Regge quantum gravity. But you can have many different groups. Of course, demanding that some kind of semiclassical limit gives 2+1 GR may fix the group, but I think the group is not fixed by very general considerations like background independence, etc.

 

[tom.stoer]

Where does the level k in general come from in 2+1? I know that in LQG in 3+1 there's a relation to the cosmo const.

Do you think that we are asking the wrong question? I mean in three-space SU(2) is somehow related to the structure group, but one has to "factorize" SO(3,1) and it is not automatically clear why this should be a fundamental step. If one uses the holographic principle SU(2) with some k-deformation emerges quite naturally from two-space.

The holographic principle can be used for any dimension n; the boundary is n-1 dim. and again we face the question "why this specific n?"

 

[Fra]

The holographic principle can be used for any dimension n; the boundary is n-1 dim. and again we face the question "why this specific n?"

If you find out, don't forget to tell me too. I agree with your insisting on a deeper answer :)

I think the answer lies in reconstructing the contiuum. In the low complexity limit, there is probably not even a thing such as dimension in the continuum sense, just cardinality.

It seems plausible that a minimal cardinality is required to distinguish the first dimension to a given level of confidence by an inference process etc. The question may rather be, why would it stop, or reach an equilibrium at a particular dimension? Apparently the structure would have to be an extremely fit compresson? That neither a lower nor a higher dimensional model could beat?

/Fredrik

 

[Physics Monkey]

The level may be fixed by the cosmological constant if you're trying to do 2+1 GR in AdS, however, vaguely GR-like topological theories make sense for any value of the level and for different quantum groups. So I think we may indeed by asking the wrong question in some sense. At least in 2+1 I think we know that very general considerations alone do not fix the theory.

I also feel that holography points to the same conclusion The existence of perfectly sensible quantum theories of gravity in various dimensions suggests to me that consistency alone or background independence alone or whatever are insufficient to pick out our universe. Said more directly, there are consistent background independent theories in different macroscopic dimensions, so neither of these principles by themselves can logically lead to 4 dimensions. One must at least require information about the semiclassical limit, and even then it's quite hard to imagine that consistency plus the low energy limit can uniquely fix the theory.

Of course, this is also what the string landscape is telling us. Here one may try to answer the "why four dimensions" question anthropically. For example, a pretty famous consideration along these lines concerns the stability of orbits in more than three dimensions.

 

[marcus]

Not to interrupt the discussion but as a side comment, you call the thread
Holographic / entropic approaches - why SU(2)?
and if that is the topic, then it should be mentioned that Kowalski-Glikman's
holographic/entropic approach does not use LQG, but something closely related,
and it does not use SU(2).

It uses SO(4,1) BF theory.

Here's a thread about Kowalski-Glikman's February 2010 paper:
https://www.physicsforums.com/showthread.php?t=377015

A note on gravity, entropy, and BF topological field theory
Jerzy Kowalski-Glikman
(Submitted on 4 Feb 2010)
"In this note I argue that the expression for entropic force, used as a starting point in Verlinde's derivation of Newton's law, can be deduced from first principles if one assumes that that the microscopic theory behind his construction is the topological SO(4,1) BF theory coupled to particles."

What I mean to suggest is that while you are here asking "Why SU(2)?" it could be that the right way is not to concentrate on SU(2) but to use SO(4,1) beef.

Eventually SU(2) might turn up, as gaugefixing the cosmological constant breaks it down to SO(3,1) and then SU(2) appears as half or the square root of that. But one wouldn't be starting with SU(2). Correct me if I'm way wrong, please.

==quote Jerzy K-G==
The plan of this note is as follows. In the next section I will recall the formulation of gravity as a constrained SO(4, 1) BF theory and its coupling to particles.

These technical results will be needed for the derivation
of Verlinde’s entropic force... (page 2)
...

III. ENTROPY AND GRAVITY FROM TOPOLOGICAL FIELD THEORY
In the previous section I argued that if one couples the SO(4,1) topological BF theory (which after gauge breaking down to SO(3,1) is equivalent to General Relativity) to point particles, then the theory forces the particles to be accompanied by semi-infinite Misner strings... (page 3)

...
Knowing this let us turn to deducing the form of entropic force acting on the particle. Suppose the test particle of mass m is at distance R from the mass M, which we can assume to be also point-like. Consider now, as in Verlinde’s argument, ...(page 4)

Let me now turn to the main argument of this paper. It is well known that there is entropy associated with
Misner string, see [19], [20], [21], and [22] where it is argued that the entropy of Misner string ... (page 4)...The entropy (3.2) adds to the original entropy of the screen, and since it is proportional to the test particle
displacement it leads to the emergence of the entropic force. Notice that since entropy increases when the test
particle moves towards the mass M this entropic force is attractive. Also when the test particle which was initially inside the screen moves outside, the entropy decreases...

Having (3.2) it is possible now to run the remaining part of the Verlinde’s argument essentially without modifications...(page 4)

IV. CONCLUSIONS AND OUTLOOK
In this note I argued that the form of entropic force being the starting form of the recent proposal of Verlinde
[6] to seek the origin of gravity in thermodynamics can be understood if one assumes that the fundamental degrees of freedom behind it are described by the topological BF theory coupled to particle(s). The reason for this is that, as shown in [16] and discussed in [17], a particle carrying the charge...

==endquote==

 

[tom.stoer]

Again I googled "SU(N) spin networks"; the first hit shows a link to this discussion :-) I will continue to investigate if one can generalize the spin network approach

 

[arivero]

"G spin networks", with quotes, gives another couple of entries. ND Dass C Florentino and V Husain are listed as authors in some searches. Also H Grosse

 

[tom.stoer]

In the original paper (which, incidentally, is titled "Angular momentum: an approach to combinatorial space-time") Penrose insinuates that SU(2) would be fixed by some kind of self-consistency: there should be rules to build a space from a group, and a group from a space, and you should get the original space (or the original group) after going across all the process and back. He seems to insinuate that only the peculiar properties of SU(2) allow for the game he is playing.

Either I read the wrong paper or I cannot find this argument. But I guess I know what Penrose had in mind: Take the Riemann space M⁴ and map it to SL(2,C). Then you get back SU(2):

M⁴ => SL(2,C) = SU(2)*SU(2)

That means that the space is somehow identical with its own structure group. Penrose uses this in his twistor construction quite frequently.

========= EDIT =========

If one maps a vector x to SL(2,C) it is represented by a matrix X on which left- and right rotations from SU(2) act:

X => X' = UXV

This looks similar to the adjoint rep. of SL(2,C).

It is interesting that something similar works for the exceptional group E(8). For E(8) one can define X from the e(8) algebra and define rotations U from E(8); then the adjoint rep. looks like

X => X' = UXU* (* = h.c.)

In E(8) this is identical to the fundamental rep., i.e. the simplest vector space E(8) is acting on is identical to the algebra e(8), so there is no way to define E(8) w/o defining e(8) and vice versa.

 

[arivero]

The statement is about the middle part of the paper, and yes it can not be say that it is an "argument", it is only something Penrose had in mind.

For any group G, if you look at the group as a manifold M, its set of isometries is usually controlled by the group GxG. The most simplest example is SU(2), which is the sphere S3, whose group of isometry is SU(2)xSU(2). I am not sure why it does not work for U(1), perhaps because it is abelian.

Now, I keep thinking that the answer is even more related to Angular Momentum. Perhaps the right question is "why SO(3)?" In a way, a SO(3) matrix keeps the information about cross products; the third column must be, I believe, the crossproduct of the first and second ones, must it? (this same trick is used sometimes to parametrize SU(3)). And in a generic SO(N) the Nth column could be defined probably as a function of the other N-1.

Well, the point is that if you want to carry "Generalised angular momentum" across the lines of the "G network", the only option that let's you codify it not only as a single (pseudo?)vector J but as a pair of vectors x,p is SO(3), or SU(2).

 

[tom.stoer]

For any group G, if you look at the group as a manifold M, its set of isometries is usually controlled by the group GxG. The most simplest example is SU(2), which is the sphere S3, whose group of isometry is SU(2)xSU(2). I am not sure why it does not work for U(1), perhaps because it is abelian.

OK, interesting point; but for 4-dim Minkowski space / SL(2,C) it is the other way round! You do not look at the group as a manifold (which is by definition of a Lie group always possible), but you take the manifold itself and then construct a group from it.

Example: it fails in 3-space: take a 3-vector (x,y,z) and map it to SO(3); this map will not preserve the geometrical properties of 3-space.

Example: it works in 4-dim Minkowski space: you map x^m = (t,x,y,z) to X = x⁰1 + xⁱt^a; then x_m x^m = det X; I think this is unique for 4 dimensions.

 

[Physics Monkey]

Any group can provide a version of a "spin" network. The word spin isn't really appropriate, but I will use it for the time being. For example, the group U(1) is particularly simple. The edges of the spin network are labeled by reps of U(1), that is integers. The branching rules are simply that the sum of the reps at each vertex must be zero which is basically Gauss' law. Another example is given by $$Z_k$$. This group is like U(1) except that charge k is equivalent to charge 0 so there are a finite number of labels for each spin network edge. These groups are for some purposes too simple because they are abelian. One can also use SU(3). Label the edges by reps of SU(3). The branching or "fusion" rule is simply that a vertex is allowed if the trivial rep is contained in the tensor product of the fusing reps. This more general rule reproduces all the rules you know so far. SU(3) is more complicated than SU(2) because you can get multiple copies of the same rep when you "fuse" two other reps. In SU(2) when you "fuse" or add two reps you only get all the intermediate reps once.

This basic structure appears everywhere. You can use it to compute topological invariants of 3-manifolds. It appears in condensed matter physics in the physics of the fractional quantum hall effect, in so-called string net phases, and in discrete gauge theory. It also obviously has relevance to quantum gravity in various dimensions.

Of course, none of what I said necessarily has anything to do with the question of why SU(2) in our world. I have already argued that this question may not be the right one. I simply want to point out that the mathematical concept of non-SU(2) spin networks exists and is useful. Key words to look are Turaev-Viro, quantum groups, topological quantum field theory, Jones polynomial, string net, tensor category theory.