A What does Rovelli mean with "oriented and ordered graph"?

[Heidi]

Hi Pfs
Rovelli writes this in his book (Qunatum Gravity) about spin networks:
Given an oriented and ordered graph there is a finite disgrete group of maps that change its order or orientation and that can be obtained as a diffeomorphism.
A link is equipped the source and target functions. this give the orientation.
But what is the order he is talking about.
the paragraph is the 6.4 (Diff invariance)
thanks

 

[julian]

You have to label the links first, as $l_1, l_2, \dots$ say, before you can assign a colouring $j_1 , j_2 , \dots$. It is this labelling of the links that is the ordering, I think. With some graphs there are diffeomorphisms that simply swap the links around and hence change the ordering.

 

[Heidi]

yes we can do like that to color the links but it is not necessary to have oriented ordered links.
The Penrose's diagrams were the ancestors of the spin networks.
they were trivalent with no explicit intertwiners. the links were not oriented and each link was coloured by a number (not a representation)
So we had a numerical function on a graph without necessary ordering:
to each pair of connected node we assign a number.

 

[Heidi]

Rovelli writes later that if changing the order correspond to swap the variables, changing the orientation leads to replace a variable by its inverse.
If in an oriented loop the holonomy gives a matrix, changing the orientation gives the inverse matrix.

 

[Fractal matter]

I think, ordering means specifying the predecessor and successor to each node of the graph.