 Look at the gamma ray bursts of the center of the galaxy. And the gamma ray bursts, there are sources in the center of the galaxy which give you bursts of gamma rays and you can measure them in observations. And some, it's apparently a random process. But you calculate the Earth's exponent, it's self-similar in the sense that if you take a small section of it, you cannot tell that it has been, it's a small section of the whole. But the Earth's exponent is 0.75. It's not, so it's not one half, it's super diffusive. So this is what I'm saying, what I meant to say, that there are processes in nature which have a fractal structure. And the fractal structure is such that you just have to measure them to see what they are. And it probably has something to do with the way they are produced, but since I don't know how they are produced, I don't want to go into them. Sorry, lots of, lots of something I cannot show you. I cannot show you. The other, yeah? You can calculate the exponent two ways. One is by just sticking balls on this in the same way that I gave you the definition of. The other one is by calculating the variance. I'll come to that in a second to how the exponent can be calculated from the variance as well. So here is a random fractal Brownian motion which as I said has exponent of one half. And it's a good place here to point out that this is a random fractal. It is not exactly like itself when you scale it. It is almost like itself. So this graph is not like the whole. But if I show you a graph like this, you cannot tell if I have taken a section of another graph beforehand or it is the whole. Now here is the basic mathematical problem. If I have a graph Y of X and I scale it with A and then I suggest that it is self-affined or not self-affined, I face a dilemma. So for example, let's say Y A of X is A Y of X. Clearly, this has one solution only and that is Y of X equals X. Or if you try and put in a B here, then Y of X equals to X to the power N such that B to the power N equals A. There is no other solution. So what am I talking about? What is a fractal? How can I draw the graph of a fractal? These are giving me very simple functions. So there must be a mathematical way out of this dilemma. And that is by looking at the graph wire stress graph. Wire stress function is a function which is continuous everywhere, differentiable nowhere. So it is a function which, how I've given its formula, no. This, a virus has completely destroyed my slides. Let me see, sorry. I didn't check this far in my graphs. Completely mixed up. Okay. So I try and remember what a wire stress function is. There's. A to the power minus N by two or something. A to the power minus N. I think that might work. I wanted to put a B here. B to the power. Okay. Now you see that the wire stress function is under a scaling G of A of X will go into cosine of pi A N plus one X sum over B to the minus N. But I can re-escalate it why because this sum is over infinity. I can re-escalate N by one. So what happens is that this will go into minus B minus one. This goes out and then a B comes out. So it is equal to B minus one, G of X. In other words, B, G, A of X is G of X. Yeah. The problem is that not having your slides is going to be difficult for me because you then have a fractal dimension in terms of B and A. Let me quickly search the, all the slides and see. No chance. Completely, completely destroyed. Okay, I have to keep this qualitative. So what happens is that the wire stress function is continuous, but if you take a derivative of it then a factor A comes out and then it's no longer, this series is no longer convergent. So DG DX will be something like A over B to the power minus N and whereas this was convergent, this may no longer be. And hence it's nowhere differentiable because this series is always divergent. It's everywhere continuous because this series is everywhere convergent. And you can get a condition on A and B as to what they should be to make this divergent and to make that convergent. That is what I was looking for and couldn't find it. E bigger than B, both of them bigger than one. Then doing a little calculation, you can work out the fractal dimension of this graph. You don't happen to remember that as well. I think it's log of A divided by log of B, maybe. I'll put a question mark here. I have to correct it by the next lecture. But the fractal dimension of this shape can be calculated using the methods I told you because it is self-similar. You look at one part of it and the part looks like the whole. So the solution of this dilemma which I posed is this. This scaling property of functions cannot really hold for a smooth functions. There has to be something wrong with the differentiability of it or not wrong something unusual in that, for example, something like the virus stress function, which is continuous everywhere but differentiable lower. That's the only way you can get away with this business of scaling and it gives you a graph. Okay, so this is all I'm going to say about fractals. We will see fractals as we go through the course and we come to various ideas on fractals and where they appear. And when we come to them, I will deal with them as we come to them. Okay, now if I have conformal invariance, let's go back to the conformal invariance idea. I can now invent a conformal field theory, which would be a field theory with fields which have, which are representations of the conformally invariant algebra. So let's take a field which would be defined on the complex plane. So it's a function of z and z bar, both of them. And then you transform z to w of z. So in this we'll transform to a new field, which is phi of w, w bar now. Question is, how does this field look under the new look? And this is the relationship it's going to have. Let's see what this transformation implies. It looks a little strange to begin with, but let us say, let's see what it implies. Suppose w is just the scaling of z. Then the w dz is lambda. Hence this is lambda to the power minus h, lambda to the power minus h bar. So I'm saying that scaling the complex variables gives you just the scaling of the field. Now I can divide h into two components, and h bar as delta minus s divided. This shows you that I'm not really thinking of h bar as complex conjugate of h, but just a different number. The number by which z bar does its scaling. So this now looks like lambda to the power minus h plus h bar. D, sorry, minus delta because I get one delta from here and one delta from there, and I have a lambda squared, and the s's cancel each other. How would I get a different s? How would I get s coming to this? If I also say that h bar scales with its own scaling function, which would be lambda bar, and then this would be lambda lambda bar to the power minus delta over two, lambda over lambda bar to power s over two. Lambda over lambda bar clearly is just an angle. So this is just e to the power i theta s. Because of this, in literature, this is called a spin because it shows you by what factor of theta the field changes under this transformation. Now, if I define such fields, if I have such fields, then phi z z bar is a field over the complex plane. And I can hope that to make a quantum field theory with these operators, then this quantum field theory which is constructed with these field operators will have conformal invariance, so I will have a conformal field theory. This is what I'm essentially going to say about conformal field theory. So I have constructed things which have both a quantum nature and conformal invariance. Are there these exponents at minus h and minus h bar? At this level, why they exist is because this is really a density, not just a function. So they would be more complex if the w is more complex, but I chose this simple example to see that they are not all that unusual. You can see that this is showing the scaling. But I will use this property soon and you'll see that this is necessary to have. Okay, now what are the generators of the conformal symmetry? I mentioned this guy, the energy momentum tensor. Let's go back to the energy momentum tensor, T of z. Where does this come from? We have in our any two-dimensional theory what is called an energy momentum tensor. It is mu and nu range from zero to one, zero for time, one for x. So this is essentially a matrix. And it is symmetric. Therefore, T01 is equal to T10. And it also satisfies a conservation law which is the energy momentum conservation law. If I write it down, it's something like that. T0, zero plus T0, T1, T10. So if I interpret T00 as the energy, then T01 has to be the momentum. So that now I have a conservation law like that. The second line of this conservation law will be the conservation of momentum which is related to the stress or the flow of momentum. So these are the two conservation laws which I'm getting out of my energy momentum tensor. Now what I will do now next is to go to the complex plane. So complex if I. That means you set z equal to T plus Ix. And you also construct a T which I have here and I construct this out of T00 plus I T01. Now let's see what does D bar T give us. That is D by DT plus I D by Dx. This guy there. Why I do that is because I have the experience that in the complexification procedure, I can get analytic functions by looking at the D bar derivative on them. So what I am hoping for is if this T will satisfy the Cauchy-Riemann conditions. So this comes out to be DT T00 plus. D by DT T01 plus I Dx T00 minus D by Dx of T01. So you see that this things cancel each other because of that. The only thing which is left is here. And this does not cancel as far as I have in my equations. Nothing will help me here. Unless if I postulate something and say, suppose T00 is equal to minus T11, then I have this equation and they will also cancel. Yes, T00 T01, if T00 is T11, then I have this second conservation. So this second conservation law can be used to cancel these extra terms. If T00 is set to T11, minus T11, they cancel immediately because of that. This is T00, this is T01, but it has to be minus to cancel, okay, minus. So this is a very interesting result that D bar of T then is equal to zero. This means that automatically I have a function which is holomorphic. So T is only a function of Z now. I can alternatively invent a T bar which will be only function of Z bar by taking a minus here. But for this to be true, I have had to impose this condition. This condition which reads T00 plus T11 equals to zero means that the energy momentum tensor doesn't have a trace. So suppose I have a theory with trace less energy momentum tensor, then that theory will enjoy this property that's energy momentum tensor over the complexified plane is holomorphic. Holomorphic meaning that's only function of Z, not function of Z. Now for all analytic functions over the Z or holomorphic functions, I can make it Lorentz expansion. Lorentz expansion is a version of the Fourier expansion except that it is over the complex plane and it covers all powers whether positive or negative. This extra minus two which I have here is for convenience reasons and that is because I already know how T will behave under a scaling of Z so I've stuck in a two to be ready for it. Thank you. Where ln exactly by Lorentz expansion are defined by contour integrals of T around the origin. Now what I then can do is then I note that these are operators. So these will be also operators in a field theory. This is an operator. These will be operators and I can ask what is the algebra of ln and ln. And something very exciting happens. I note that the algebra of ln and ln are exactly the same as algebra of the Witt operators which we introduced sooner. With the difference that I have this extra term here, note that this extra term disappears. It's zero for m equal to zero plus minus one. So it doesn't change the SL to C nature of it but it does change the infinite version properties. How this comes about is in fact a complex calculation and I don't really want you to be able to derive this. It's just for you to see that the Witt algebra is actually extended to an algebra which has a term which is called the central term because it's a constant and it commutes with all the generators of the algebra. Okay now, therefore I have this algebra, the Virasura algebra. I have those fields which I explained and what I want is that I want a theory, the quantum field theory over the complex plane to have this as its symmetry which means that I have to somehow construct things here which are representations of this algebra. I will do that in the next lecture. Thank you very much. Questions? Probably the second equation there is the minus, let me first. The question of the system. This one, yes. So, I am considering p into the max. I am considering p into the max. That's what I want to do. I want to do one more equation in the same direction and I will give you an example of this. And very quickly I'm going to call it a vector of x square, y square has two снова naught. And I'm going to call it the square phi. I'm going to take you to the center, this is your second minus W1, you have the second minus W1, and you give a one, and this is the second minus W1, and this is the second minus W1. And this is the second minus W1, and this is the second minus W1. There is a confusion about how you define the derivatives. If you take derivatives like that, then these are g mu nu of d nu. The question is which of these derivatives you are taking. And because g mu nu is one minus one, so there is a minus coming from, I have to be careful to get the minus one right.