Holographic / entropic approaches - why SU(2)?

[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.

 

[tom.stoer]

do you know where explicitly something else but SU(2) has been used for a physical system (not simply mathematics)

 

[Physics Monkey]

The material \text{Zn}\text{Cu}_3\text{(OH)}_6\text{Cl}_2 also known as Herbertsmithite may support a string net ground state (big emphasis on may). If so, then the ground state would be a superpositon of networks of strings each of which basically form a U(1) spin network.

Fractional quantum hall systems definitely realize certain Chern-Simons theories including those based on U(1), SU(N), and Sp(N). These theories may be reformulated in terms of spin networks as Turaev and Viro and others have done.

 

[MTd2]

Just a retarded question:

does anyone know of an anyon spin network?

 

[tom.stoer]

thanks! I found this

http://arxiv.org/abs/gr-qc/9408013
Spin Networks, Turaev-Viro Theory and the Loop RepresentationTimothy J. Foxon
(Submitted on 10 Aug 1994)
Abstract: We investigate the Ponzano-Regge and Turaev-Viro topological field theories using spin networks and their $q$-deformed analogues. I propose a new description of the state space for the Turaev-Viro theory in terms of skein space, to which $q$-spin networks belong, and give a similar description of the Ponzano-Regge state space using spin networks.
I give a definition of the inner product on the skein space and show that this corresponds to the topological inner product, defined as the manifold invariant for the union of two 3-manifolds.
Finally, we look at the relation with the loop representation of quantum general relativity, due to Rovelli and Smolin, and suggest that the above inner product may define an inner product on the loop state space.