 So, now let's take this statement and action it will with a del z bar and a delta v, this I can do. So, del z bar t z z of z and del w bar t z z w is equal to c by 2 del z bar delta w bar 1 over z minus delta w over z. In terms of set this to 0, we must be careful because z equals w is the singularity. So, we must be careful to scale it to 0 with z is not going to be. z equals w is the k. So, let's keep those things around. And we will, we are one of our singularities at coincident points. Okay. Now, the next step we say that from English observation, from different values of the initial theory, this thing is the same as del z of t z z z bar. And this thing is the same as del z z t z z bar delta w. So, what we get now is some, some statement about how the 2 point function, some evidence of the 2 point function of z z z bar and z z bar behave. We integrate it once with respect to z instead of w. That, that integral will reduce this z minus w to the 4 to z minus w to the 2. And you will get lots of factors. There is some statement which dramatically, even try to know factors, is grammatically t z z bar of z, that is t z z bar of w is like del z bar del w bar 1 over z minus w is the whole thing squared. Now, the whole thing squared, and we thought of it as del z del w of log of z minus w. Again, you try to know factors that change twice. So, what this thing is, is del square m of the factors, del square is respect to z and del square is respect to w of log of z minus w. Now, you see why it would have been so dangerous to just say this was here. Because del square of log of z minus w is not 0, it is a delta function. Actually, we are going to put a more, more square here. That is problem, that is a good thing. So, this thing is schematically like del square, del over square of a delta function, which is precisely what we want to show. What I want to learn, what I am going to convince you of, what I would like you to carefully work out for you, I think you will learn factors. It is that t z z bar of z times t z z bar w in flat space cannot be thought of as, you know t z z bar cannot be thought of as precisely the 0. It is something that is 0 expectation value. But it is 2 y function has non-zero contact terms. These non-zero contact terms determine what we know about the theory. And the impact is even like this, which then implies, which then implies that the 1 y function of the trace of the stress tensor in a background that is not flat but a little bit cold, is known as the bottom of this thing now. Go on and think about it at home and we come back. No exercise now, I want you to try, which you get. You go and think about it at home. Yes, it is exactly. So, this is a corresponding thing to how we calculate. But yes, it is it. And if you just proceeded in a standard QFT, very unregulated, unregulated. It is a little subtle though for the following reason, that you must define T mu for corrects. Define T mu as the response of the partition function to a change in consumption. Because if you thought of, if you wanted to ask, what is, you might at first be asking how can I possibly do this calculation in the standard way of doing things. Because I do not have an operator to insert. You know suppose we are working with the free goes on theory for instance, the classical operator corresponding to T mu mu is 0. It is just 0 variations of motion. Since you know since it is not, as we say it is an anomaly and it is all in the measure in some sense. So, if you did it all correctly in some standard quantity thing, we did it, but you have to do it carefully, easier way to do things. Because it is very easy using usual techniques to get control of what T is doing. But then once you know everything about what T is doing, but if you are always an analyst, you know everything about what T Z Z passed. That is the basic point. I know the two point functions of T, T Z Z. Therefore, what is the basic one, the two point function of T Z Z. Exactly. Since we use the energy concentration. Yes. Is this some kind of statement that if it was not so with the energy concentration. Exactly. It is possible to set this T, this two point function to 0 exactly, but then you will violate it. It is a usual statement with an office. You do it. Right. If you are going to preserve some things, you must give up on this. But you often have the choice of what to do. But energy conservation means expectation value of just the divergence of that operator. Right. Energy conservation would not be violated. No. I mean we want, we don't want that. We want the whole board. The operators to do it. Yeah. Right. Yeah. What does it mean to do it? Yeah. If you, I mean basically in order to derive this result, all we have used is the statement that we can replace this expectation value with this. Inside the code. Inside the code. Okay. If that were not true, that statement were modified. You could modify that statement in a way. You set this, this, this is zero. Okay. But this statement follows from some, some requirement. Requirement essentially. That different model is an invariance of the, of the, of the theory to be a good symmetry. And that way we are going to treat as approximate. It's the usual thing. It's like we have a global and gain symmetry. You often run into a situation with anomalies where you are able to keep the global symmetry normal anomalies at the expense of making the gain symmetry anomalies. And you never make that option because the theory of gain symmetry anomalies is inconsistent. Same thing here. I mean the, the t, the 2.1 t trace itself is zero. It's not zero. But okay. So what? Okay. But theory is not. If you walk into an invariant, you're becoming one. Especially in straight theory. Okay. Okay. Fine. So this, this, I, okay. So go through this exercise. Convince yourself of the logic we used here. And what are the words? That's, that's the first exercise. The second exercise, um, maybe I should just talk about the last exercise. Okay. Maybe it's important. It's important now. I'll just talk about it in class. Okay. So we'll do this. And then next time in class, we'll finish up a general discussion of what the normal theory theory is. And then turn into a discussion specific. Very nice. Okay. Next time in class, we will also find that we will, in a very related, but similar, but in a slightly more conceptually interesting way, we will find the same result. We'll also talk about, you know, how, you see this was only step one of the exercises. You know, we found the trace of the stress tensor. We'll also talk about how the whole partition function of the theory is completely determined from the statement. It's done for the same future. Yeah, the Irish program. That's it. Let's put the camera off. Let's get into the discussion we want to have.