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So let me first briefly recall what a summarize what I accomplished yesterday.

So we were considering, so we are, the big topic is quantization of a connection of the theomorphism invariant theories of connections.

And we considered on some manifold sigma, we considered space of connections.

So connections were denoted by A.

We also have conjugate momentum to the connection, so we are quantizing this Poisson relation.

And we want to do it in the theomorphism invariant manner.

So we, in order to accomplish this, we have introduced cylindrical functions.

So function, I'm not going to repeat all the lecture, but this is important definition.

I will refer to it often, so I will need it on the blackboard.

So a cylindrical function has this form, where in this way we denoted the parallel transport.

So it is sometimes people write this past ordered exponent of integral of A.

And then we called this cylindrical functions, so we called the space of functions Cil is cylindrical functions.

And in this space of the cylindrical functions we were able to define an integral.

So here we defined, or I told you, actually first I told you that there exists integral, but then I gave you the properties,

and then I explained that in the cases which we consider actually this integral is already defined by those properties.

And I also mentioned that this integral is the theomorphism invariant.

And finally we introduced, so we took for the Hilbert space, we took this space of the cylindrical functions,

is usually completed in the Hilbert space norm, so there are different types of different way to complete this space,

depending on whether we use the norm or whether the sub norm or whether we use the Hilbert space norm.

These are different ways of doing this, but these are subtleties which are obvious mathematically and technically they are not necessary to be realized all the time.

And here we have the scalar product defined as just integral in this space of the cylindrical functions.

And later we also, having those Hilbert space, we also introduced operators.

So we introduced operator for, so I can use sort of abbreviation that we introduce operator for the parallel transport.

In fact we need some function of the parallel transport, so because it's a group element,

so we need some representation and then put this representation, put this parallel transport under the action of the representation.

But we can use abbreviation and say that we have this operator and we have operator corresponding to, for, we have introduced,

so connections were replaced by parallel transports by this, whereas those conjugate momenta got replaced by fluxes,

so we introduced this integral and we call this flux.

So these are our achievements and now let us discuss, so actually you could go on with this framework and start defining other operators,

but at some point it's convenient to have orthogonal decomposition of this Hilbert space into some simpler spaces,

so it pays to introduce this, to go with the momentum and introduce this, some more framework which is defined for this Hilbert space

because then we will not have to go back and then start once again from scratch to introduce.

So let us introduce some more framework, but before this people were asking me at the end about the action of the gauge transformation,

so it's true that one should not forget about mentioning this.

So let us now discuss what are symmetries, which means unitary transformation, groups acting in unitary way in our Hilbert space.

So we have, I already mentioned the theomorphisms and as I also mentioned we introduced some,

we could consider all the theomorphisms, but then we would encounter some technical difficulties,

so it seems more reasonable to, sufficient to restrict ourselves to a class of the theomorphisms which are, briefly speaking piecewise analytic,

which means that they are still can be performed only locally, however, so they are not like analytic the theomorphisms,

which analytic the theomorphism is defined actually globally, once you know what it does here, you know what it does everywhere,

and so we have piecewise the theomorphisms, which means the theomorphisms which are analytic on some patch,

but then they are only small, only differentiable on boundary and then they are analytic in some other patch,

but the advantage is that still they can act locally, they don't have to be extended globally.

So we have the action of the theomorphisms, so consider the theomorphism,

and so to the theomorphism, they correspond some unitary operator in our Hilbert space,

and this operator is defined actually in most natural way, because we have fields on our manifold,

so all the fields are subject to pullback or by the theomorphisms, so you act on our functions by taking pullback of the connection,

but then when we go to those specific cylindrical functions, which are defined by some objects, by some lines,

then actually this action becomes passes to transforming those the theomorphic those lines,