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Good morning. Thank you for coming.

I will... maybe I should first tell you what is the goal of what we are doing and in particular what is my goal, what I want to present to you.

So, quantum gravity at this point is neither a mathematical theory nor physics.

It's a little of this and a little of that.

So, it's good to formulate specifically what we are doing and why we are doing.

So, what is the difference between the usual physics course?

So, when we teach quantum mechanics or quantum electrodynamics, then we just give you broader or more condensed outline of what we do in theory which describes physical phenomena which we observe.

So, when I teach quantum field theory, then it's not really so important to formulate why I do it or what we want to achieve.

We have some physical phenomena, we have their description and now we want to understand how physicists can describe those phenomena.

So, we describe the tools, we show how calculations can be made, etc.

In case of quantum gravity, we actually don't have any physical theory which would be confirmed experimentally.

So, what we do is rather abstract, it's like mathematics.

On the other hand, we are not doing mathematics in a sense that we are looking for a theory which could be model of some physical theory.

So, we don't focus so much on every mathematical detail.

We rather try to think of how the tools which we develop can be applied later to describe, to make some predictions which later could be potentially tested in experiment and then our theory can be rejected and then we can start doing something else.

So, for this reason we have to organize our own research, we have to formulate some working goal and try to accomplish this goal.

So, this goal which I formulate for this lecture is that we are trying to quantize gravity and we want to define some model.

So, we want to define something which technically, mathematically can be a quantum theory of gravity and we would like to perform it to the end.

So, actually we would like to start with, so there are many ways of guessing correct quantum theory.

You may not doing this by quantization, you may just be so brilliant that you will just guess correct theory and then you don't have to just explain how you did it.

However, what we do is we have some specific scheme in mind.

This scheme is not the law of nature, this scheme is also not, should be not considered as some set of axioms because it's not that exact either or that complete.

But we just have some scheme which is called quantization.

Specifically, this is canonical quantization and we would like to see how much we can, how far we can proceed to quantize gravity by using this scheme.

The feature of this scheme is that it's full of gaps, so that's why it's not really complete mathematically theory that all you have to do is just have to find examples

because they are just tips for us how we should go on and now we try to fill those gaps and find, so we will be satisfied if we find examples, sufficient set of examples to have something to work on and to confirm with experiments.

So, what I will do today and this week, I will show you how we can accomplish this program and I will show you how we can derive such models.

A couple of models of canonical loop quantum gravity for which one can later raise the question of whether, what is, either what is their mathematics, if somebody is mathematician or what is their physics, if somebody would like to apply it to physics.

So, today I will start with quantization of connection variables.

So, my main, big subject which I want to cover is

in diffeomorphism invariant way.

So, what are the connection variables which we will be quantizing?

We have some manifold sigma, so sigma is n-dimensional manifold.

It will be three-dimensional in GR, in loop quantum gravity, so n equals three in loop quantum gravity.

We have a Lie group, which is a Lie group, compact, and in loop quantum gravity, G will be SU2.

Always with this group we have its Lie algebra, which will be also used, so it's clear what's going to be in the case of loop quantum gravity.

Now, somehow in loop quantum gravity, the aspect, topological properties of principle fiber bundles are not thus far so relevant.

So, we will just, instead of using principle fiber bundles, we will just fix some trivial bundle with some sections, so our connections will be just one form.

There's absolutely no problem in generalizing everything to the language of principle fiber bundles, but then it becomes less explicit.

Each time we have to take some section, pull back forms, and now we will be working just on sigma.

So, A, which is our space of connections, should be identified with the space of one form on sigma, which take values in this Lie algebra.

And we also will be using, typical for physicists' notations, so we'll introduce coordinates, basis.

So, in daily algebra we will use some basis, tau 1, tau k, this is a basis.

In this manifold we will use some local coordinates, XA, and element of this space of connections will be often denoted in terms of its components.

So, we will write it like this.

So, I could actually go on and could be talking only about the connections for a long while,

but to stick to the main subject, which is quantization of connection variables, I will also define another variable which will be used sometime later.

So, the other variable is momentum conjugate to the connection variable.

So, this momentum is not so easy as connection to define what it is, one has to speak a little longer.