Why do representations label the edges in LQG?

[Heidi]

TL;DR Summary

from Penrose diagrams to spin networks

Hi Pfs
Marcus wrote a huge bibliography during many years about LQG.
but i do not see where to find an answer to a question. Penrose draw diagrams with edges labelled by numbers. What is the reason why later number were replaced by SU(2) representations?tÿ

 

[Heidi]

There is another difference. the nodes in the Penrose diagrams are not labelled. This comes from the fact that they are trivalent. there is a geometrical analogy. When you know the lengths of a triangle there is no freedom for its angles. In the case of a 4valent node we can see it like 2 close trivalent nodes linked by a virtual edge. It has 1 d.o.f. Has the intertwiner of a nvalent node n-3 degrees of freedom?

 

[Milsomonk]

I believe the edges use representations of su(2) to encode curvature, the representations are the holonomy of the connection, this provides the link from curvature on a Riemmanian manifold to the discretized space of LQG. If we take a number of infinitesimal rotations in the form of su(2) representations sufficient to complete a full rotation or "loop" and we get the identity then there is no curvature, if there is a deviation from the identity then there is curvature present. A great book on this is Carlo Rovelli's Covariant Loop Quantum Gravity.

 

[Heidi]

I agree but why are the intertwiners necessary? is it for equivalence classes reasons?

 

[Heidi]

I read that a spin-network is a functional on connections. Say S[A]. I am not sure but i think we have to contract the intertwiners of the nodes with the SU(2) of the edges. but how to do that with the connection A? The wiki article about spin-networks says to take the matrices associated to edges amd the nodes, to multiply them (tensor produc?) and then to contract the result according to a rule that is not given...

 

[Heidi]

When i have a spin-network, i see nodes edges and matrices M(g) g belonging to SU(2). I read that it can become a functinal of the connections A. have i to replace all the matrices by A? in the edges? in the noder? and then to contract?

 

[Heidi]

I found answers in a paper by John Baez:
Spin networks in nonperturbative quantum gravity.
When a spin network in the curved 3d space it discretizes it on a lattice. giving a SU(2) matrix on each edge is the discretized way to give the connection. on the nodes we only have linear combination of tensor products of Id2 and epsilon. (it is the 2+2 antisymmetric matrix). Why can we higher dimension matrices on the edges?

 

[Heidi]

I found why the intertwiners are useful. The name LQG contains the word loop. In a first step there were multiple Wilson loop states of functionals of connexions A. They still exist with spin networks but they are hidden. to see them we have to consider close circuits along the edges. intertwines give them.

 

[Heidi]

Closed circuits (not close). an intertwiner is a linear combination of tensors invariant by SU(2). when you follow an edge each one is pointing to the next edge to get a loop.