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Let me start with some infinitesimal part of the last segment of the previous lecture

so that we really continue in a continuous way.

So now we are at the Theomorphism Variant Quantization of Geometry.

And I would like to add the same warning which I gave at the beginning of my first talk

that if I was teaching quantum electrodynamics then my task would be to give you in the simplest way

the formula which physicists use to calculate some processes and you would have, even didn't have to think too much.

It would be just enough to learn those final formulae.

Here we are in a different position.

Here our goal is to come up with some examples of how procedure called quantization can give us some quantum theories.

However, there is no uniqueness. Everything depends on many, many choices.

But we don't worry so much about those choices.

The goal is to somehow go to the end.

And if we have many examples then the next step is analyzing how choices which we made,

if they have some physical meaning or how they affect the properties of our examples.

So we are now at this first stage where we just try to construct some examples.

That's why I don't ask why I do this because I just want to find some examples.

Once I have many examples then the next task is to compare the properties

and to see which choices are more physical, which choices are less physical.

So let me remind you also that what we did, we now consider the Diffie-Morphism variant quantization

of theory of connections for gauge group SU2 and for three-dimensional manifold.

And also I wrote how we define, in what way we endow this manifold with Riemannian geometry.

So we still have those canonically conjugate fields.

And we define from one of them, we define orthonormal frame in such a way

that the determinant Q times, so this is the tangent frame, equals this.

So here we can explain the meaning of this index.

So the index is in the tangent bundle.

So in this way, from this we have this orthonormal tangent frame,

and from this we have the metric tensor.

And now using our quantum framework and using our,

for the time being we just used, using those operators, we construct operators

responding to some geometric functionals of Q.

So I will not repeat it in detail, but I just recall that in this way we did already quantum area.

So we considered area of some surface and we found this operator.

And actually there's only one thing I would like to remind.

So this operator has the following form.

It was integral along this surface of certain operator density.

And this operator density had this structure that it was hopelessly looking sum of operators like this.

However, whenever we applied this sum to some specific cylindrical function,

then actually this sum would become finite,

and it would be sum with respect to isolated intersection points between this surface S and this given function.

So whenever we act with this on some cylindrical function,

then this sum would become finite sum.

So then it would become sum with respect to yi, and now these are isolated.

And those operators with this index here, they are not any longer distributions.

They are well-defined operators.

So it maps cylindrical functions into cylindrical functions.

So in some suitable domain, this is a self-adjoint operator.

So the joint operator in our Hilbert space.

So this was this first example.