# A Averaging holonomies

1. Sep 5, 2016

### naima

In this paper the author shows that holonomies on Wilson loops are useful tools in GR. I have no problem with the gauge invariance on these loops which comes from the cyclicity of thr trace.
Bonzom writes then:
We are moreover interested in SU(2) gauge invariant states. Gauge transformations act on holonomies only on their endpoints. If h is a map from Σ to SU(2), then the holonomy transforms as
$U_e (A^h ) = h(t(e))U_e (A)h(s(e))^{−1}$, with t(e), s(e) being respectively the source and target points of the path e. When focusing on a single graph Γ, this reduces gauge transformations
to an action of SU(2)^V on the set of cylindrical functions over Γ. So from any function f over SU(2)^E , one gets an invariant function by averaging over the SU(2)^V action.

Here we have edges and vertices.What is this averaging over the action ar the nodes?

2. Sep 5, 2016

### Ben Niehoff

This is not really a GR paper, I'm not sure why you would expect classical GR people to be able to answer a detailed question about LQG.

If the authors haven't made it clear what they mean, then you could ask them, or perhaps try to dig in the literature.

3. Sep 5, 2016

### naima

Could this question be sent to a better place?
thanks

4. Sep 5, 2016

### Staff: Mentor

Moved to Quantum Physics.

5. Sep 10, 2016

### naima

I can describe now why i had a problem with this averaging.
In LQG we have oriented edges which are colored by a representation if SU(2).
Each edge may be considered as an open path with a starting point and a target point. we call them the nodes.
let us consider the simpler spin network with only one edge. it may be a sub network of a bigger one. suppose that it is colored by h a given matrix of SU(2). A gauge transformation acts like this: a pair of elements of the group g1 and g2 transforms h
in $h' = g1.h.(g2)^{-1}$
We will say that h and h' belong to the same class of equivalence if one can find such a pair to map h to h'. it is obvious to see that the identity is equivalent to any element of SU(2). If you have a function f on the set of the representation you do not need averaging or something else to get a gauge invariant function. the constant function f(id) works!
This is not so easy with more complex cases. take the Mercedes Benz network. It has
6 edges and 4 nodes. and there are several classes of equivalence.
if each class "i" was equipped with a pecular element $m_i$ we could build a gauge invariant function f' from any f by taking for any matrix m f'(m) = #f(m_i)## .
The author gives a more elegant way to build such a gauge invariant function. it is also constant on each equivalence class but its value is the mean value of f on the class.
This is why they are talking about the average of the holonomies.

Last edited: Sep 10, 2016