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Okay, so we were quantizing this Poisson bracket.

And we introduced for this some new variables. These were parallel transports and fluxes.

And we quantized those variables by introducing a suitable Hilbert space, which was constructed from the space of cylindrical functions.

And in this Hilbert space we found those operators corresponding to those variables by quite exact quantization,

meaning that we just assumed that the action of this PS on a function of connections, that this is just the Poisson bracket,

which is the most direct quantization.

And next we found some useful orthogonal decomposition of Hilbert space, so we defined spin networks.

And we introduced the orthogonal decomposition of our Hilbert space into subspaces,

which in each of those subspaces is labeled by an embedded graph and by two colorings by irreducible representations of our group G.

There is one more, and here I spent some time explaining what are admissible, but here we should only consider admissible graphs.

However, there is still one detail which I think I didn't explicitly explain, which is technical,

but since people may really want to use this in practice, in some calculations, so it's good to know even technical details.

So actually there is some, I consider oriented graphs, so graphs are oriented, meaning that each link has some orientation.

However, it's a little too much freedom when we define this Hilbert space, because as we know,

our parallel transport has this property that when we change orientation of a link,

then the new parallel transport is a function of the previous parallel transport.

So given an unoriented graph, actually it's sufficient to define the Hilbert space of functions constructed from this graph.

And then regardless of choice of orientation, this is the same space.

So in particular, when I say, for instance, the space, I introduce space of syndrical functions which are constructed from a given graph,

then if I take, so if graph gamma consists of some links, and if I consider another graph which differs only by switching one of the links into opposite link,

then actually this is the same space. So I don't change this space.

And it also applies to this autonormal decomposition and to spin networks.

So we introduced this spin network function which is given by coloring, so it's given by gamma, by this coloring, another coloring,

there was coloring by intertwiners, and there was also coloring by some vectors living on nodes.

And now if I replace this graph by this graph here, then actually there is a one-to-one map.

Then I can find here a spin network function which is exactly the same spin network function.

Namely, what I have to do is I have to change a little one of the labelings.

Namely, what I have to do, I have to, the only change is that if I take this new labeling,

and when this new labeling hits this new link, it should give the dual value to what this labeling was doing at this link.

So it should be this previous labeling, dual in the sense of the dual representation.

So it is equivalent representation, however dual. So dual representation acts in a little different way.

So in this way, change of orientation of a graph amounts to changing, switching corresponding colors into dual representations,

and everything else is unchanged. So here we have, remember that those graphs are unoriented just for this formula.

And there is one more, okay, and now we have a nice, okay, there is one more mathematical, several mathematical comments.

But maybe I will leave those mathematical comments to, I really want to close this subject in a few lines.

So I will first finish with this completely and then at the end give those comments.

So to close this subject, we have some, so we will be using operators in this.

Next subject will be quantizing geometry in this Hilbert space, so we will be using operators.

So it's convenient, so this Hilbert space, even without using any physics or any geometry which will be a new input here,

it comes with some family of operators which are naturally defined here and they are used to express the operators which we will want to construct.

So there are those naturally defined operators which are defined as follows.

So this is our manifold sigma on which everything lives and consider a point in this manifold

and consider something which may be called a germ of links at this point.

I think that I shouldn't overuse this name germ because usually germs, I think, form some vector space and those germs cannot be added to.

But the germ is like an equivalent relation and two links are in this relation if they overlap on some small interval containing this x.

So this is L1, L2 and then L2.

So we will call it germ at x.

So given x, given this germ and also given some element of the Lie algebra,

we can associate an operator which will carry the labels containing the data which we needed to define this operator.