Evaluating Spin Network States

[Michael McBreen]

Hello. I'm finishing an undergraduate research project on lattice
gauge theory, and I have a question about spin network states: in
short, is there any efficient method for evaluating them when some
random group elements g_1, g_2, g_3... are assigned to the edges,
beyond doing the whole contraction ``by hand``? My gauge groups are
SU(2) and SU(3).

To avoid confusion, here's my understanding of spin network states:
1) Take some lattice (I'm interested in large cubic lattices) with
nodes n_i and edges e_ij.
2) For each edge e_ij, choose a gauge group variable g_ij and
irreducible representation rho_ij.
3) For each node, choose an intertwiner t_i from the tensor product
of the representations of the in-going edges and dual reps of the
outgoing edges to the trivial representation.
4) Contract all the rho_ij(g_ij) with the corresponding t_i and t_j,
so that the whole thing is contracted to a number.
5) The spin network state is the function of the g_ij that returns this number.

I'm interested in the cases with either arbitrary g_ij or, less
ambitiously, with all the g_ij = 1 (i.e. a pure contraction of
intertwiners). The number I want is the sum of the results for each
specific choice of intertwiners, i.e. I sum the result over an
orthonormal basis of intertwiners at each node.

At any rate, that's the function I want to evaluate, whether or not
it's called a spin network state. For a large lattice, I figure it
would take a computer a good while to assign all the rho_ij(g_ij)
(Especially in SU(3), where the explicit parametrizations of
arbitrary representations are rather complicated) , and another good
while to contract all the rho_ij(g_ij) with the intertwiners.

I'd hugely appreciate any references to articles or textbooks, I've
searched and searched but I've found only scattered hints. Thanks a
lot for your time,
Michael McBreen

[Igor Khavkine]

On 2006-08-19, Michael McBreen <michael.mcbreen@mail.mcgill.ca> wrote:
> Hello. I'm finishing an undergraduate research project on lattice
> gauge theory, and I have a question about spin network states: in
> short, is there any efficient method for evaluating them when some
> random group elements g_1, g_2, g_3... are assigned to the edges,
> beyond doing the whole contraction ``by hand``? My gauge groups are
> SU(2) and SU(3).
>
> To avoid confusion, here's my understanding of spin network states:
> 1) Take some lattice (I'm interested in large cubic lattices) with
> nodes n_i and edges e_ij.
> 2) For each edge e_ij, choose a gauge group variable g_ij and
> irreducible representation rho_ij.
> 3) For each node, choose an intertwiner t_i from the tensor product
> of the representations of the in-going edges and dual reps of the
> outgoing edges to the trivial representation.
> 4) Contract all the rho_ij(g_ij) with the corresponding t_i and t_j,
> so that the whole thing is contracted to a number.
> 5) The spin network state is the function of the g_ij that returns this number.
>
> I'm interested in the cases with either arbitrary g_ij or, less
> ambitiously, with all the g_ij = 1 (i.e. a pure contraction of
> intertwiners). The number I want is the sum of the results for each
> specific choice of intertwiners, i.e. I sum the result over an
> orthonormal basis of intertwiners at each node.

Out of curiosity, why do you want your result to depend on the group
element g_ij? If your lattice has L links and the gauge group is N
dimensional, then you are essentially calculating a function of L*N
parameters. As a rule of thumb, the more parameters you have the harder
your function is to evaluate.

In the context of lattice gauge theories, we are usually interested in
integrating an amplitude calculated in the way you outline over all
possible assignments for the group elements g_ij. This integral factors
into integrals over individual g_ij which can be done knowing a little
bit about representation theory (you need to know how to integrate
matrix elements of a representation over the group). Henceforth, the
dependence on g_ij is eliminated and instead of an integral over over
many copies of the gauge group, you'll have a sum over irreps and
intertwiners labeling vertices and edges of the lattice (details depend
on the situation).

Depending on what you get after group integration, you may need only to
evaluate small closed spin networks. For those you might find tricks
similar to the Christensen-Egan algorithm that Greg Egan already
mentioned.

Hope this helps.

Igor

[Michael McBreen]

Well, I start out with a non gauge invariant state that assigns a
specific group element (link variable) to each link on the lattice,
i.e. a Bargman state.

To get rid of the gauge-variant part of the Bargman state, I project
it onto a basis of spin network states. The inner product of the
Bargman state with a spin network state gives the spin network state
evaluation I described in my first post.

Of course, the inner product is gauge invariant and doesn't depend on
the specific Bargman state but rather on its "gauge equivalence
class", but I haven't extracted any major simplifications from this.