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That actually from a logical point of view or from some obvious point of view,

a matter-free case should be the simplest and should be the most primitive one we consider first.

On the other hand, there are some indications that actually matter helps and that with matter it is easier.

So there is this famous... there are those famous papers by Giesel and Tiemann in which they did the quantization with the brown Kuhar dust.

And they showed that when gravity is coupled to matter, which... and if there are sufficiently many degrees of matter

and if matter behaves in a nice way, then this framework may be yet more precise

and you can really derive the theory to the end and calculate... practically everything is calculable in principle.

Modulo is just technical details and some amount of work.

However, I somehow... I have chosen this path, this natural path that we should first start with the case without matter.

And so probably the price I will pay will be that only at the end I will have maybe 10 minutes to outline the results of this model.

So here I should say that shows that the vacuum case is not necessarily the simplest one.

The simplest one. Okay.

However, now we will start... I will be consequent. I will start with this matter-free case.

So we have this Hilbert space built from the cylindrical functions and here this is the Hilbert space in which we defined those operators

which were fluxes and parallel transport. And now there was... I started listing the constraints.

So here we are taking this... so if you have theory with constraints, you can either solve the constraints classically

and then work on the... with the resulting theory. On the other hand, you can try to turn those constraints into operators

and solve them in the quantum theory. So I will be here taking this second point of view.

So we have the Gauss constraint and also here our role often is not only to solve the constraint,

not even only to define the operator, the constraint operator, but even define what a solution to this operator is.

So we have to... because very often it's not really precise without obvious from the beginning.

So the first constraint, the first constraint which we... quantum constraint was this.

And this, as I said, was very easy. We just have square integrable...

we just have functions which are integrable but just gauging variant. So size such that...

and among the syndrical functions there are such functions and we denote them by... so set of solutions is...

I put here capital G for this group with respect to which we averaged.

Okay, and now there was this other constraint which I started to discuss. And this constraint, this operator comes from the quantization

of the classical generator of the theomorphisms. So if we... if one quantizes this operator without anomalies,

then it should again generate a group which is isomorphic with the theomorphisms.

In our case, the theomorphisms act naturally in this space because we use still the structure of manifold.

So it's natural to assume that... but it is assumption. It's natural assumption, there is assumption that in the quantum theory,

the theomorphisms also act just as the theomorphisms. So if this is the case, this equation is... means the theomorphism invariant states.

And I also explained at the last lecture that in the case of... now in the case of a compact group,

solution to the assumption that we are looking for this group invariant states just amount... always give... will give us normalizable states.

So if group is compact, then it's represented by unitary... compact group of unitary transformations

and we can easily find... find states which are normalizable and are invariant with respect to those transformations.

You may be surprised because group of gauge transformations for Young-Mills theory seems not to be compact.

However, this group of... in our case, this group of gauge transformations, this group is actually...

this group can be embedded in a compact group in SU2 to power sigma, whereby this I mean all the maps from sigma to SU2.

And this is a compact group, it has a compact measure. So... and this... and all this group actually... our Hilbert space admits the action of this group.

So... so... and H admits the action. And that's why... that's why even though gauge transformations usually are sort of non-compact group,

in this case, they actually act as a subgroup of a compact group and we have normalizable states which are preserved by this.

Sigma is the manifold... is the underlying manifold on which everything is defined. So this is...

In this sense that this is infinite dimension... this is infinite cartesian product of compact groups. So it has some Tichon of topology and...

So you're just taking it as some kind of discrete of a function or...?

Yes, yes, yes. So for a compact group, there is every cartesian product of compact groups even if it's... if it's infinite, it's still compact.

So if you don't make a huge limit of gauge group, do you think of how to...

Right. That's why I'm not saying that this is the gauge group, I'm saying that the gauge group is embedded in this group.

And this group still... and our Hilbert space admits the action of this bigger group.