 The media would be discussed in the last class. So we studied the theory of the scalar field, given by del by del. So in a moment of space, we studied the part of the takeover of this theory together with a cut-off with a lambda here. And here we put a key in the middle of the function that we chose, such that it was 1 from 0 to 1. So it would be an incidental friendship given by 1 to 0 to the numbers. And we discussed that we looked at the quantity of j and lambda, which was d phi, and we demanded that we get an arthentectual that we generate. So the arthentectual is generally up to overall normalization. It's dependent on j's. The dependence on j will be independent of lambda. We demanded that lambda be by d lambda of the left-hand side to 0. Sir, why do we need that in the final? So you see, what we are trying to do is to make a mathematical equivalence between two theories of different types. So there is a physical question about this theory, and then a purely mathematical question of an invariance of path integral. So what we are observing is the following. That if we have two different path integral that are related by the fact that the cut-off of the first is different from the cut-off of the second. But the action of the first is also different from the action of the second. In a coordinated way, this can generate all the same answers in the space of path integrals. These path integrals are labeled by the action and the cut-off. And what we are observing is that there is one line, one parameter line, one line. Along which we get all the same coordination functions. Notice that we chose our J to have support only in this section. For all the integration elements. So suppose we integrated from lambda 1, some lambda high to lambda low. We choose our J to have support only in this region where T is scaled by lambda low. So what that means is that we are never integrating out momentum modes in which whose correlators we are going to get. So what's the idea? The idea is that we've got this path integral which is some integral. Now this is a picture of momentum space. The ground momentum space, the origin, and so on and so forth. And we are interested in computing correlation functions in this region. For instance, we were doing particle physics. We could, for any exceeding, we could be interested in computing correlation functions from 1 TV meter to 10 TV meter. The actual theory might be defined by a path integral that goes up to 10 to the power 16. G is equal to 10 to the power 13. Suppose. So we've got the actual theory which is defined by some path integral that goes all the way to some ice cream. Now we are noticing just as a mathematically coolness that any correlation function in this region will likely be computed by a path integral that goes only up to this region. In doing that we have to change the action not to claim that the cutoff was actually there. Change the cutoff without changing the action as is happening in this region. But this is just a mathematical observation. Now, the second thing we noted was that if we parametrize this Lagrangian by numbers, then almost all of those numbers had negative dimensions. We also noted the just from dimensional analysis. The just from dimensional analysis, the dependence of physics on these numbers should be in terms of lambda to the appropriate power times an hour. We'll take the philosophy that we're going to integrate down to roughly the moment it's given interest. Whether we non-dimensionalize that number by lambda or by the momentum of the process, order of magnitude did not make a difference. So, this is another way of saying that once we've got the effective action down to the hour scale, then naive dimensional analysis. In dimensional analysis, every number that appears in the actual every interaction and non-dimensionalize it by the momentum scalar process. We will give you a crude estimate how important that process is. That could be a very wrong estimate if we did not bring this cutoff down to our scale because it could instead have been done that number non-dimensionalized by lambda. Now, since lambda is much bigger than the momentum, that could be anomalously higher. An anomalously higher estimate than naive dimensional analysis. You see, one of the very confusing things originally is that, though I don't really know this as a historical factor in a traditional, then people must have tried to do all kinds of naive dimensional analysis. And these estimates actually vary. And that's because there's a hidden scale in the process. It's a linear cutoff. So, the idea of this renormalization down to the scale is that we can get rid of that hidden scale because we make this hidden scale of the same order of magnitude as our momentum. And then there's no additional, additional scales. However, we've done that, we proved it, right? We proved that, well, the correlation function is obtained from this object when differentiating with the strategy. If z of j is the same, all correlators are the same. Because correlators are obtained by differentiating with the strategy. Is this clear? It's actually going to be actually the same. It is that, if you look at Kevler, what we actually proved is that we proved lambda divided by lambda of z of j was proportional to z of j. The z is rescaled, right? But that doesn't have a correlation. It will be the same. So, we compute a high scale in the low scale and the number will also matter. Oh, yeah. That's the point. That's the point that if you use z of j of lambda, then, say, let us take, for now, the attribute that the only thing we're going to use this to compute is correlation functions of phi. Insertions of phi and z of lambda. It's time-ordered, correlators that we discussed. Then because z of j of lambda is the same, all these correlators will be the same. Is this clear? Is this clear? Because of our observation of, you know, in our dimensional considerations, we used this x and d because a and d to the power lambda to the power, n plus d minus 4. And it should become smaller as we go through. Smaller? No, I think it's just this. Ah, sure. That was another question. The way we have to change this is once you go from a higher level to a low level scale, there will be less and less people out there. Oh, no, no, no. Oh, such things. In fact, in the way we're going to set it up, we're going to start with a very simple action that has to be done. Let's say this is just a phi to the power. And then as soon as we start running through the normalization rule, all of the infinite number of possibilities will be generated. There's no claim, there's no claim for simplicity of the effective action in general. In fact, quite the opposite. You could choose to start with a very simple formula. You could choose to start with a very simple formula of the action. And then what would be generated would be obviously complete. Okay, go ahead and tell me. If this is the case, then if you were to do the other way, now if you were to somehow flow through the UV, and you're starting off with the magnetic theory, which had dimension 6, dimension 8, terms, and we had all these relevant operators become more and more important as you go to higher energies. How do you drop them up as you go to higher energies? In general, they will contribute. But, you know, Well, you know, what we said is that we choose to start with the UV theory. And this is true exactly at that one scale. So this is the fine tip. We choose to start with that. If you integrate out just below or just about the scale, even the UV theory will have all the complicated stuff. But let's say, for example, say we have a theory of, say, similar signals, like mesons, which we actually do. And that is, that obviously has 6 dimensions when it comes to operators. And then we, that is obviously embedded in the UV theory, which is starting from the standard model of those operators. So where do those operators go? I understand when you integrate out, that happens, but can you go to the UV theory the other way now? What is the question? The question is, we've got some theory of the scalar field and it's got some effective action at some scale. Yes, and it has all these relevant developments, which could become more and more important as the UV. And in general, it will come. Yes, but that is obviously something, it's like the standard model. Right. But the standard model doesn't have these six dimensions. It doesn't have these six dimensions operators, but let's see. So suppose we took, yeah, okay. So, you're looking at a theory of, we know all the fields of the cycle. So you see what's happening is that some point, your description in terms of these fields are coming back. The correct description has more fields. And flow down from that system to our theory will be, okay, could I suggest that you postpone your questions to the end of this? Does this kind of thing we intend to discuss? Yeah. Yeah. In previous class, we have some expression for the animal dimension. Can it be, I mean, explained in the picture of this, what you're doing is integrating all the elements. Can it be explained in the basis of this? What do you mean? I mean, can I understand what is animal dimension from this? Yeah, exactly. The anomalous dimension itself is a precise concept only near a conformal experiment. But, ignore it that way. Roughly speaking, anomalous dimension is this. Suppose we did this x side, and we differentiate this object with that line. Okay? The derivative there has one term that is obvious. Because we're just differentiating this line. That comes from dimension. Okay? And then there's other term that comes from the beta functions. Okay? The integrated contribution of that is what we call in the anomalous dimensions. Okay? So, it's the derivative of this term with respect to lambda, roughly speaking, is the anomalous dimension. It's called an anomalous dimension. It's called an anomalous dimension because it's the following. See, suppose we, as we've said, the lambda depends. Once you go all the way down to the scale of interest, it's like the model. Suppose we use an effective action at scale lambda to estimate the contribution of a term to process it at momentum scale. Okay? Then, if you had not realized that there was a cutoff in the problem, your estimate for how important this A and D would be for physics at scale B would be A, N, D, P to the power N plus V at the top. This is not necessarily the case. Because the correct way to estimate it is to look to see how A and D has functioned as lambda scales. Okay? Do you get this now? From here? Do we also get additional contributions? Because of the non-trivial part contributions come from the below equations. So if you've forgotten that there was another scale of the problem, that you actually have to do the integrating of this matter. You'd ignore the other part. Sir, if you've not, why is it all these things? You would just estimate this as a knife contribution. That's why it's called an anomalous. Yeah? No, ask as many as you want. I may ask you to postpone because some of the questions are... Yeah. If you look at a term which is like x squared by x squared, it's just never get rid of it. Yes. The coefficient is just k minus 4 over here. It seems like just because you're renormalizing a curse behind a match, there's already a scheme in the problem because an anomalous definition is like x. If you did not have the master, the master obviously introduces a scheme in the problem. Yeah, that's a really master scheme. Of course, that wouldn't be consistent around the renormalization proof law. But anyway, please go. That's the case when you already have a scheme in the problem is like lambda to the power of n minus 4. So that it seems like you're introducing the master problem just because you're renormalizing a curse behind a match. Because of an anomalous definition. Exactly. It's true that if you were to be naive and cascaded and you would imagine that if you started with a master scheme, it would be a master scheme. So in particular, if you predict the power of n, a function must follow the power of n. And the argument might be, what else could you do because there's no other scheme in the problem. But there is another scheme in the problem. Lambda. You don't see it very clearly, but it's that. Now, if you take the theory which had no mass at some high value to come, flow it down, you would find that there's no mass. And therefore, correlation. It's right. Yeah, right. When you're thinking of a certain where exactly are we doing the where exactly are we doing the You see, we're not really doing the what we're doing is changing the lambda. The problem with this lambda is adjusted so that the propagator of the scalar field is 0 above 2 lambda. So though we're doing the path integral over all fields, the path integral over fields at momentum larger than 2 lambda is 0. That's what we're calling integrating out. It's like we're working with this kind of up to the fact that it runs smoothly. It's the same thing as saying if we, you know, really sophisticated we take a step back and ignore such sophistications. What we're saying is in the path integral at lambda, we're doing the integral only up to lambda. Actually, if we're doing the integral completely up to lambda then partially up to 2 lambda and then stopping. That's what this case is. You understand? Because any diagonal has propagators. The propagator is 0 at the hydrangea. So the contribution of any field any integer field with momentum larger than 2 lambda is 0. So it's like we're integrated out. In fact, the way Wilson presented this was literally by integrating using the hard cut off literally integrating out. But this is a better way to do it. Just because it's easier to do many things. Yeah, it's very important. It's the same idea. Ask as many questions as you want. This is actually a very confusing thing. I could take a step back. I asked this path that I do because actually I haven't even gone through yet. But we're going to discuss that in this class. But I could do that. I could do that. Points of today's class. Hang on, hang on, hang on. Wait, you see, what we talked about last time was the renormalization of no flow. Then we talked about the toy model. I haven't yet told you how it applies to business. Many of your questions are about that. They're great questions, but that's the point of today's class. So hang on to that and then you can ask questions. We're going to be doing the toy model thing. So what we initially did was we took the zero value of x6 and put it in and sometimes it converges to a certain value of x4. So is it necessary for it to converge to x4? It could have converged to any point. When we're doing the general theory, does all x6 need to converge to a certain x3? There's some flow. There's some other. Along the flow, you go to whatever value you want. We want to test whether the flows are coming to be in the company. You see, there is an invariant geometrical idea of it. But the flows are converging. You have an intuitive idea of what that means. This is what you want to test. How do you test that? There are many ways to start up a test. Actually, the one in the push-ins he uses in his toy model is not the one I used. I used it because it actually fits better We could do the one in the push-ins. The many ways to test it. The one way to test it is you see, what you don't want to do is to do the one. One nice thing to do is that suppose I start here and here goes here. And here goes here. And then I'd ask, after I evolve for some time, do I reach nearly the same point? That's a question you could ask, but it doesn't address the question whether the flow is converging. Suppose the flow was converging as in my time. And it's my residue. Still, it could be that you float to here, the first curve, and to there on the second curve. So that these two points of power are far away. That's when you start with it. But the flows are converging to each other. How do you check whether the flows are converging? What you have to do is not flow for the same time, but tune the time of flow of this curve with the time of flow of this curve. You want to have so that now I chose the x-pores for the same time. You don't choose any line. And then many things you couldn't have. Pulsinski chooses something different. I chose this because I said, as you'll see in a minute, better have nothing to do with this. So I demanded that this flow up to the point of the x-pores for the same time. This I can do independent of whether the flow is converging or not. Suppose the flow is not converging. There's a factor. I could still flow till the point that the x-pores would sink. This is independent of question of converging. Just choosing where to compare two points. Then I asked the question, once I've chosen these two times, is the x-x difference between the two points going to zero and large difference? Can't it just happen if the flows are converging? Okay, if it's not happening, the flows are converging. So for more than two variables, we have two more than two variables. Yeah. Now for more than two variables, we will have to know what we expect. So, we have one more variable. Now there are two possibilities. You see, suppose there's one more variable, it's like an x-8. Then what you would expect and what is true is that x-6 and x-8 both converge. So then if you choose x-4, the difference in x-6 and x-6 are both converged. But there's another possibility. There will be some theories in which there's more than one variable. In those theories, you would have to adjust. In those theories, it would not be the claim that flow lines are converged. What would be the claim is that all flow lines converge to, suppose it were two variables, to a surface. Now we would have to know what's in this to get the test. That is, if you, the test would be roughly that if you tune, okay, then you would have to look at your flow as a function of two initial conditions. x-0 and the other guy. And you tune two of these so that again this is going to be what we're going to do. Yes, go on. It's not that the x-6 contribution is small. It's not that the x-6 is small. This curve, a problem like this, it could be going up like that. We're only testing convergence. Do you see? Do you see that? There is no claim that x-6 is small on this curve. That is not even yet natural. Because that's very important. We're not saying that in the Wilsonian effect of action, in the Lorentz effect of action, x-6 is small because it doesn't have that effect. That is not what we're saying. And this is a misconception maybe you have and it's just wrong. That's very important. Okay. The flow line comes to the droids like this. It's going larger and larger. We're not testing that. Our algebra only tested that the two flow lines converge. Whether they converge to large or small values of x-6, no statement. Is this clear? How can we physically interpret flow lines in words? Hang on. Sir, I want to say Yes. Sir, we see that in the Lorentz theory, we have a wide range of factors. But everything is dependent on the scale lambda. But lambda is also important Okay. All these questions, could I request monotony of my questions for half an hour? Now I have questions about what I'm just saying. But these high level questions because I think you will have better understanding of all of your questions once we are in the discussion of the questions. The issue is that you guys are not knowing something. Pretend you know nothing about quality. Only know what I told you. Ask those questions. And then at the end of this class I will say ask any questions. Because you are preempting what I want to say in a more organized way. Okay. Okay, so correct. The question that we wanted to ask Wait, so where are we now? In our review we said that we started with these guys. This guy is a measure of how important physics and scale lambda. And then we discussed how these guys moved as a function of lambda. These guys moved as a function of lambda. We wrote down flow equations for that. That was the exact renormalization group equation that we wrote down. That could be regarded as equations for this argument. And very importantly they took the form of beta and d, of functions of x and d. There is no expression of lambda. That's very good. It's like an autonomous set of equations. You know what x is. Then you know what they will be the next time. In respect of where you came from there is no memory left. But where you came from. Then we looked at the toy model. Then we looked at the toy model in which we had one irresistible operator of the same things. And we proved this convergence that we have discussed. And we proved it as we said by the process that we have discussed. We started with two points at some initial time. Which were having x equals 0 and no x4. And now we will be moving on. Now suppose the many things that I wanted to say. Suppose instead of setting x equals 0 at some x4. We could instead have had any one pattern intersect of initial conditions at t equals 0. We could have had that at t equals 0 we have x6 0 put some function at x4 0. Which is to say, instead of starting here we could have started some surface and this and that. So, we would have started this flow. This flow instead of starting here we would have started this one here and this one here. We would of course improve the same thing. See, where we started what initial conditions we started from being no difference. Because what we will always do is adjust the time of flow of the second curve to be the same point. What we are trying to check is a geometrical property of flows which don't need the x6 equal 0. This is a project. Exactly. x6 equal 0 was just an example. Choose any slice that we wanted and we would clearly prove the same property. So, that's one thing to say. Although I chose x6 equal 0 that was utterly in this engine for what we are actually doing. Is this clear? The other thing to say about this is that about the joint model, that's all I want to say. We come to the question of how all of these analyses apply how all of these analyses apply to each other. So, let us first go to the result that we actually proved in this paper. Use an analysis that's reminiscent of what we did. But we actually let me try to prove this. So, what are the questions we actually prove the following? You see, even though this renormalization proved to be similar to what we started with the Lagrangian that is, he starts at t equals 0 with the Lagrangian L of y put a mass here put an additional mass in here and we could have a fight. Let's look. Suppose we got some flow. This is now some flow in some infinite dimensions. And the claim is that suppose now you choose so now your flows are labeled by what? Your flows are labeled let's call this flow f is equal to flow labeled by m naught by lambda and by lambda. So, we start at let's say that we start at t equals 0 with m naught and lambda then the flow lines are labeled by m naught and lambda. This is the analog of the two flow lines the flow lines in our drawing which are labeled by one packet then the lambda form these are flow lines labeled by two packets it shows as the formula. Suppose we do the kind of variation that we do that is we suppose we do the formula suppose we take this flow line and we adjust m naught and lambda naught lambda naught with let's say we have this flow line and sorry not now some flow form let me put this one and we got some flow from lambda naught down to lambda and now what we are going to do is a little different from what we did now what we are going to do is the work over here and we take another flow that starts at lambda naught plus delta then also choose to start with m naught plus delta and little lambda naught plus delta naught so we got two different flows one starting at lambda naught with m naught and let me call this G soil the first flow starts the second flow starts with lambda naught plus delta lambda naught G naught plus delta G naught and delta naught plus delta delta G naught and delta N naught in the following we choose the delta G naught and delta delta N naught in the following way such that X2 and X4 the part of the quadratic effective action for which you can set p equal to 0 sorry for the part of the quadratic effective action for which you can set p equal to 0 and the p squared and sorry and the part of the p the part that is p squared of the quadratic effective action that is X2 and X4 these coefficients okay such that these do not change as you change okay so we demand that B lambda naught B by B lambda naught of X2 of lambda so X2 is now a function of lambda naught M naught G naught and of lambda there is some fixed scale lambda some given scale not every lambda but at some fixed lambda two different flows first one labeled by lambda naught G naught and second one labeled by lambda naught plus delta lambda naught G naught plus delta G naught N naught plus delta in general these two different flows in general will give you at scale lambda at scale lambda I will give you different values of of all parameters but we chosen to change not by a certain amount and then we given ourselves a real the real rule is that we can also change G naught and N naught in a manner that is convenient for us degrees of freedom which we can use to ensure two conditions so these two conditions are that I will choose the starting G naught the variation in the starting G naught and the variation in the starting N naught so that two and X4 at scale lambda do not change because I made this change in that it depends you know you started at some scale lambda you are going to some scale lambda I have but if I want to come to this it is like G naught and lambda and you are choosing two values of this is like this point right no it is not it is a flow yeah sir can you please what is X2 and X4 X2 is what okay you remember we have this parameterization of the Lagrangian X2 simply X2 0 and X4 is equal to X4 in N delta G naught so let's say this way let me say it faster G naught is functional so that this is the case lambda naught of X2 at lambda naught equal to lambda here I am getting 0 yes exactly at lambda equal to lambda not lambda naught it is not a flow it sparks at lambda naught and goes to lambda now we are changing the variation I am changing the variation condition and demanding that these two coefficients out of the infinite number of coefficients in the Wilsonian detective action do not change I always have the freedom to do this because these are two conditions two objects and have two functions to adjust so as to make them so I choose this to be the case and let me call these fixed values of since I demand this okay I am in particular demanding that my Lagrangian at scale lambda has fixed values of X2 and X4 let's call them X2 IR and X4 IR all the flows that are at scale lambda these two numbers X2 IR and X4 IR the same okay my flows are just these three parameters and flows labeled by lambda naught, m naught and g naught I have got some fixed scale lambda okay and I put the conditions that all my flows are have this property now because I put this condition I have only got one parameter set of flows labeled by lambda r because m naught is function of lambda r has fixed and g naught is function of lambda r has fixed so I have got one parameter set of flows and the first thing that Polchinsky proves in perturbation theory in this you are fixing the value of X4 at scale lambda to something order by order in that quantity okay in perturbation theory in that quantity he proves that once you do this lambda naught d by d lambda naught in fact there is to zero like 1 over lambda naught square times log r at any given order in perturbation theory in this X have you watched this script 20 seconds okay now what is the geometrical significance of this theory in the space of flows I am going to make a thing and you see if you can see it what this is saying is that in perturbation theory in this parameter what this is saying is that in perturbation theory we have the set of flows tends to a two-parameter why is it saying that it is saying that because if it is true that a set of flows tends to a two-parameter surface if that is the case then if you specify then knowing nothing else you don't know where you are on the set of flows you don't know which flow line you are on where you are suppose that tends to two-parameters problem is it is hard to draw to nothing but imagine I am going to ask you a question because this is what are we saying what we are saying is that I know where what X2 and X4 is at point level then I know what every other value every one of these other infinite infinite entries are at point level in the analog in the toy example was that if I knew where what X4 was at a particular point then I knew what X6 was so that in the toy example we had flow to a unique flow line or to a one-parameter set of points one-parameter set of flow lines or a two-parameter set of points is this clear that is one way of parametrizing there is an infinite dimensional space of the universe and we are flowing to a two-dimensional surface somewhere in this infinite dimension space so that if I tell you what X0 X2 and X4 are on the surface because we are on the two-parameter surface the equation of the surface then tells you where everything is is this clear okay let me try to say it again what should you imagine you should imagine this flow in this infinite dimension space infinite is at level some high level what have we what is the claim the claim is that we proved the formula okay we proved that infinite dimension space of parametrizing of parametrizing of the X6 so you are saying that all these terms present all these terms present okay but I am saying the formula I am saying that if we do this variation we do one particular kind of variation but we do not prove it for any other kind so the statement is geometrical state then if on the space of flows we change initial conditions somehow but we changed it in a way that made sure that we did not change X2 and X4 then once we flow for a considerable amount of time okay all other parameters are also unchanged another way to say it is that all other parameters let's call all other X's other than these two Y's all other parameters X2 and Y3 are determined by these two so Y is all of the Y's infinite number of them our Y is of X2 not and Y sir so if we say infinite dimensional surface so we need NX yes exactly exactly exactly is this clear sir but what are you starting with that is stroke the width is as in essential as it was in the drawing model in the drawing model again it starts with the 41 parameter but you remember once we come to the result we can start with anything else it is a geometrical statement here values for these two little parameters yes which we are proving to just one parameter you can say it this way sir in the drawing model how would you say it you will say the what suppose I have X4 at some scale which I bold fixed I have this curve now I change X6 but I also change the lambda not I just said different over here here I said I change lambda not but I also change X6 that's not right X1 I change X4 not but I also change lambda not so that X4 and scale lambda it gets fixed X6 and scale lambda tends to do you see it is exactly the analog of what you were doing in the drawing actually that's why I said I am the analysis of the drawing model in this way because it is the exact analog of what we are doing is this clear completely clear so I answer X not I changed the amount of flow and then I change X not tuned to the amount actually the way we analyze the drawing model we change X not X1 not and then we change the amount of flow tuned to the amount doesn't matter when we have many variables it is more convenient to say it there is one distinguished thing mainly the amount of flow and many different parameters it is more convenient to say it in the other way doesn't matter so let's say here this way I change the amount of flow and I change X not in a way that insures the X4 at some scale that's a fixed scale that's not changed then as I take the flow time to infinity now what we are taking the flow time to infinity is taking lambda not because the flow time is between lambda not that's I think flow time to infinity the difference between X6 on the two flows goes to Z now you should try to imagine this somehow try to imagine this in three dimensions try to imagine that in three dimensions we have got these flows and it is going to have two dimensional surface try to imagine that what we are changing lambda not but changing X4 not and X2 not in a way that insures that X4 and X2 at scale lambda do not change then in this three dimensions a proof is that X6 at scale lambda also this is the statement of paternal to renormalize it let us say what we have proved okay what we have proved is the following what we have proved is that the effective action at scale lambda the full effective action at scale lambda goes to something that is lambda not independent if you choose a Lagrangian you take a Lagrangian at scale lambda choose X4 and X2 as functions of lambda in an appropriate way then there exists an appropriate way of choosing X4 and X2 as functions of lambda so that the effective action at scale lambda is completely insensitive to which value of lambda not you started with in the limit lambda not your statement notice that it is a very non-trivial statement because you had only two parameters to play around with that is X4 at scale lambda and X2 at scale lambda the full effective action at scale lambda as an infinite number it is not data-filming counting would tell you that it was possible it is this flow convergence that tells you now notice that once we know the effective action at scale lambda that is sufficient to compute all correlation functions therefore all scattering quantities we have most of what we want in old physics below scale lambda so what we have in India proved is this that there exists at least in perturbation there exists there exists a way of choosing so you start with a positive integral and scale with a cutoff scale lambda there exists a way of choosing X2 at lambda the mass as function of lambda and the final four terms are function of lambda so that in the lambda not goes to infinity limit every correlation function at some fixed scale at middle has a good lambda not goes to infinity limit with the discussion of infinity limit you see you remember that we found that in the fight of the four in the effective fight of the four coupling we compute a correlation function of four fives so that there is some effective fight of the four coupling it blows up because of the one with the graph because of the contribution of the correlation function now of course you can cancel that flow by changing your bear action the action you started with by putting a term in the fight of the four coupling that was itself divergent that was itself a function of lambda not tuned so as to cancel this divergent function you could do that but now you might get very worried you might get very worried because you might think well now if I use this crazy action which has an infinity lambda not depends in it maybe I maybe I can adjust because I maybe I can adjust g not as a function of lambda not that's a function and n not as a function of lambda not because I've got two functions in my hand I could adjust these two functions to make sure that the physical pass and the physical fight of the four coupling is fine but then what tells me is that six fights that the coordinate of six fights also has an lambda not tuned expected divergent what about 8 fights what about 27 fights what about 87 derivatives of 92 fights you know it's about 26 because this is an infinite number of couplings in the historical ineffective action corresponding to the fact that there are infinite number of observables important okay whether we approve or but I have not proved the class just because it's very painful can you prove because you have to be qualitative to estimate that nothing goes all the main ideas are and it's very clear once you think about it then you can prove this the action you're doing is a pain in the neck okay I'm sorry you're not saying to remember what 8 short and then three are converges and more remember in the toy model in the toy model what we had was that the right hand side had minus 2 and then you got contributions from the original data now so all you have to do is to show that the contribution of the original data are not large enough to overwhelm the minus 2 why do you say that's the point of doing this calculation because suppose you such that G0 at scale lambda was 0 now that is a completely quadratic effect of action now there is one way to ensure that that is to start with G at scale lambda not is equal to 0 because in our in our renormalization proof there were no interactions there will never be generated what was the renormalization proof there was del L by del phi times del L by del phi so suppose you start with something quadratic del L by del phi is given del L by del phi is given so that generates 5 to the 6 times then we also had a term which was del 2 L by del phi squared to 5 to the 6 can then back react to the 5 to the 4 then suppose you start with the only quadratic things del L by del phi times del L by del phi is always quadratic del 2 L by del phi squared is 5 to the 3 so you start with the quadratic action and the renormalization proof flow will always keep you within quadratic actions because renormalization proof flow for quadratic actions is trivial ok so suppose we demand suppose we demand suppose we demand that G G0 at scale lambda G at scale lambda this is x0 x0 the final order at scale lambda was 0 there is an obvious way to satisfy it that is to start with x0 0 and then adjust the masses ok it's the nature of the flow it's the fact that x4 0 is fixed that if x4 is 0 it will stay that's what we just showed that adds evidently to the equations from the equations, from the nature of the equations x4 equals 0 we don't need to solve some questions complicated renormalization proof flow equations we know the answers very trivial then perturbation theory is perturbation about that but since we are going to perturbation theory so in the trivial flow all quantities, all the other higher axes if they were generated at all actually they weren't there they would be 0 suppose we perturb them a little bit they would adapt according to their classical dimensions just by dimensional axes so if you look perturbations around this trivial flow the anomalous dimensions will clearly be small they will be small at order by order in this perturbation so that's the point perturbation theory anomalous dimensions are order g or g2 or g3 cannot compare with 2 or 4 or 6 so it's obvious if you think about it it's obvious if you think about it it's got this structure in perturbation theory now you know that in infinite number of couplings there are lots of things that you have to think of think through and you know people wonder that there's distance between taking what I just said and making it a proof in the Pulchins case down there and I invite you to read the paper it's very beautifully written but I don't think you will feel after reading that that you understood much more physics than you would have you will feel sure that there was no gap in our history okay that's the case but I'm not going to take 2 lectures I'm going to take a class time to go through that this is not good there's no idea beyond that it's just that if you start with the quadratic action you will always remain quadratic let's remember what were the two terms of the right answer one was del f and del phi phi of phi of phi that was del l the other was 2l and del phi of phi of phi del phi of x let us just check okay let us just check whether you can ever generate a non coordinate this is linear this is linear so the product is linear this guy is in direction zero so you can never generate anything non coordinate any other questions or comments sir this idea of a calculator we normalize it firstly there are two steps that's proving what Pulchins case proved do you understand that if we have proved what Pulchins case proved then we are happy then every correlation function that we compute at scale lambda i r r will have a lambda not close to infinity where? a non divergent a non divergent lambda not close to infinity that's what I mean I mean something that is why we found this is this clear because what have we proved that the effective action at scale lambda is fixed you know has doesn't change just we change lambda and therefore clearly there is no divergent we take lambda not this we take some fixed lambda not and then we take lambda not from there push doesn't change then if there was a lambda not to somehow then we will be true there is no sensitive dependence on the scale if we have proved this then we are true ok now there is the issue of how to prove that that's the thing I am inviting you to bring to Pulchins case the only thing that we have is basically a drawing model except we have to prove that in all these dimensions do not overwhelm the cassette and that I am saying is obvious because all the contributions from that are suppressed in powers of the paternal environment whereas class you are comparing with some classes which is 2 or 4 for 6 so what do I want to prove I have questions about this in this case I will be saying that if we start with x40 equals to 0 then it will remain 0 class of 0 in the drawing model in the drawing model of course depends on the equations of the term I am not going to want the equations but I am saying that the analog of real quantities is going to start with equations such that they have the property that if x40 equals to 0 then we are just going to say so that beta 400 and beta 64 has a power of 0 if you start with some equations x45 y45 then at the limit of the there will remain x2 that is the line implying from the fact that using this equation we could prove the statement yes other questions about this it is left a distance to go this is just the statement of some perturbation but number lies it is a great statement it is a great statement and gives you a sense for how quantum field theory a continuum quantum field theory is defined it is defined as a limit a limit we are cut off as the cutoff is taken in infinity with some parameters chosen as functions of it and the great thing about that is that it is always possible to choose the equations for how those parameters vary with your lambda naught scale so that all correlation functions are lambda naught independent no sense independent this is the result of course the physics behind it was convergence of these renormalization established this result only by the way Poichard's case paper was written in 1984 because renormalization was understood and there were many proofs of renormalizability of particular theories well-informed we look at it in Zickson and Zuber it will give you a proof involving trees studying the detail structure of perturbation theory skeleton graphs trees, overlapping divergences many many complicated issues how do we start thinking about the question just at the level of impact if we ensure that there are no divergences in the fight how do we know that there are none in fight 6 many issues have to be addressed those could be addressed directly but you see all of that obscures the nature of the world what is actually going on what is going on is this convergence once you have the convergence renormalizability is automatic now how about this we chose to parameterize our initial grand scheme by x naught sorry x2 and x4 being non-zero and everything else being zero this was not necessary any two parameters set of initial grand schemes would have never been we could have chosen to parameterize our grand scheme by x6 a 5 to the 6th company a 5 to the 8th company a 5 to the 10th company a 5 to the 96th company as some arbitrary function of the mask once you have made that choice you would still find it always possible to choose x2 and x4 is functioned the ground amount and therefore the x6 or the x10 so as to ensure this progress okay the original quote-unquote renormalizable form of the action that we traditionally choose in our study of quantum physics is just a convenient choice it's nothing more than that okay, it's nothing more than that there was nothing sacrosanct about choosing this particularly simple form of the action in the ultraviolet just like in our toy model there was nothing particularly sacrosanct about solving this problem the essential point is that of convergence of the flows because of the convergence of the flows when we change lambda naught choose a two-parameter set because the things converge to a two-parameter two-dimensional surface the details of the two-parameters set up basically in this action that choice means that there is this category which have all those x6, xz is equal to 0 we calculate those two 0s yes is it necessary we set these to 0 in the u on the flow they don't integrate yes is it necessary that this is the case you see that this is possible it is true because in perturbation it's true you see that choice was always possible because we know that at 0 coupling is possible and therefore perturbation yes in perturbation theory it was obvious right because it is an answer as a question to accident and in perturbation theory the answer can only be correct in perturbation theory because that's not just some caveat for mathematicians it's a real caveat we make a wrong answer because we are reading perturbation theory I am very excited but in perturbation theory the number of things you have to choose is clearly just the number of couplings per z is this clear now that we understand this we are going to broaden our minds ok now that we understand this we are going to broaden our minds and study and try to imagine various possibilities from all kinds of actually put the fixed values on their practice surface that's a very important thing but in the IR all those values should be 0 not 0 the value is determined by the practice surface whatever it is is this clear practice it's like in our toy model x6 was not necessarily 0 it's not the claim that x6 and x8 is not low to 0 that's not the claim the claim is that they flow to some value determined by x2 and x4 they may have been generated in the flow they may have been generated in the flow when you see when we consider ordinary line up by flow theory what we do is define the theory in the UV and that's the point you see we need a two parameter set of possibilities in the UV that may take us to the attractor flow and then when we do calculations we will generate the values then we would have got using these attractor flows you see we don't actually in real life we don't actually use the Wilsonian effective action it's not like we actually compute the Wilsonian effective action up to scale lambda that would be very cumbersome what we actually do calculations is just do calculations with the high scale countoff this was useful for us to get a way of understanding that it's a way of choosing x2 and x4 at high scale so that if this goes to infinity in the right way it will not depend on them because of this attractor nature of the fields keep asking till it's clear because you are very confused I know when I was a student I found this very hard time not only because I learned the way to produce these paper tools I got the proof of this you know it's from the PPA it's the proof of the normalization you know I don't want an exercise in muscle building it's great this stuff nobody should or will teach you what to do in a muscle building it's totally irrelevant the way complicated way the way unphysical way what does that mean something simple I can just read from the yes we can in the IR you have the space you find where there's two parameters but we also have all the possible couplings which flow through these in the lambda fiber theory in some U which is at a higher level where we could choose all the other where all these other higher couplings could actually be center zero exactly that's a choice now something I should emphasize is the problem why is this damping down happening not only in the toy model it's because of x6 had the minus 2 there are some things that you cannot if you chose your initial conditions such that x6 as a function of suppose we chose x6 and x4 such that x6 you know we chose x6 not to be any function of x4, lambda, lambda whatever you want suppose I chose x6 to be equal to lambda not squared then it won't work it won't work because now you're choosing your initial conditions to grow as fast as you expected to death is this care the changes in initial conditions were of order lambda not squared higher then we have no reason to expect that that will also tend inflow to some fixed scale lambda that will also tend to say attract the surface to say it again attract the equations need time if that's why I'm repeating lambda not infinity always close to fixed surface because you can't infinite amount of time but if you also change the initial conditions by infinite even stretching as you take lambda not infinity then you don't give enough time to initial things okay now say that this way it sounds totally crazy why would anyone choose the initial conditions like you can do that on squared sounds like you're beating down a straw man building up some opposition which nobody would ever build up and then shooting is done but but it doesn't surprise because you see we were very careful with the name of this non-dimensional x-axis if you look at what x-axis is getting like lambda not squared means in terms of the original a6 this just means that a6 is equal to constant because a6 had dimension minus 2 a6 had dimension minus 2 a6 would be equal to x sorry x6 is equal to a6 times lambda not squared so if you just hold a6 constant if you just hold a6 constant then x6 will automatically scale like that so you know naturalness is a bit subtly the mathematics tells us what we have proved we have proved that we choose the non-dimensional guys to be any functions that don't involve lambda x0 x4 and x2 then we are fine we choose the dimensional fellows to stay fixed as we take lambda not to infinity I expect this is not such a crazy thing to do it's the bottom what we should imagine let's suppose that we imagine in the bottom way let's suppose when we actually use quantum field theory quantum field theory is used never because we are actually taking something to infinity if for no other reason that there is the Planck scale the quantum gravity department the quantum field theory is highly unlikely to be a correct description of physics after we are on the Planck scale okay what the applicability of quantum field theory both in particle physics and in lens particle physics is always because of the separation of scales between two kinds of between new physics that's something you are interested in so suppose there is some scale in lambda at which you got new physics okay this could be like w goes on for instance in this technique where lambda not going to be effectively like mass this physics generates some effective action in scale below for instance we generate a four-ferme interaction from the w goes on by integrating out the lambda the reason it's happening on some scale some initial scale lambda all the dimension full quantities will be generated with that scale lambda something that's the case if we integrate out the w goes on from the particle formula which tells you that at this scale in lambda the non-dimensional axis will start out at already will start will be of already say you generate a cycle of water which is a dimension six operator in your negative you will generate a healthy generator you got some formula you got some w goes on other formula alright this w goes on probably is complicated but in the level of momentum small compared to the mass it's very simple it's just one over mass okay so schematically you get psi bar, psi bar, psi one over mw is one so you see that this a6 that you generated is automatically of order one over mw so a6 at scale mw is one over mw square which means that x is at scale mw is one over mw this is in many examples you can look at this is how it works and this generically are the exact things that there's some physics at some scale and that generates some new physics like new particle string theory something that generates some that below which above that scale you cannot use the language of your quantum field you cannot use your portfolio below what you get now so you got some reorganization workflow that starts out okay because you integrated the new physics okay this happens at some scale let's say mw square what that does is generate initial conditions for x's that are over over and then the flow operates and that's everything other than what you start and you have to have some information about the new physics very few meaning you want to know you want to test whether it's true that generics or you want to if I tell you I don't know anything about what is going to happen after that scale what new fields do you think are coming then how are you sure what operators will be there the claim is that generically you take all operators over don't know anything about the order numbers but generically they will all be all in it the idea is that the physics at some scale that is operating to generate your individuality and so that that scale in the later the very important point is that if this is the heart of predictability the important thing I said in our earlier lecture suppose it happened in the case that in order to make predictions about I don't know the fine structure lamp shift we need to understand quantum gravity likely human beings will never have this kind of physics it's too much for any one person or one group of people to do it in one shot because does do the predictions of the lamp shift depend on what happens in the class yes because it gives us the starting point of our improvisation but do they depend in a very complicated way you see the point is that if you have determined but if you have determined that if you have determined that you have something happening at this scale and you know only that no matter what it is at lower scale it's different generate their action which is of a single scale and you believe that all the X's are the current scale you generated then you know that you will flow somewhere into this two-parameter set of flows it's actually inflow tube depends on what exactly it takes but there are only two parameters you need you need to function of all the complicated things that could happen at high energies you don't care about the details you only need to know two combinations of those that you cannot know from quantum gravity you don't know quantum gravity but that you can just treat as fit two-parameters the predictability of quantum gravity is that you fit these two-parameters by experiment you cannot predict the experiment because those come from something new but you fit these two-parameters by experiment but in terms of these two-parameters then you predict everything else okay so the renormalizable route renormalization route flow is a blessing and a curse it's a blessing because it allows you to deal with things well you can deal with things one at a time you don't have to know everything about everything in order to make simple predictions but it's a curse because it means that if you want to make detailed measurements at our energies to deduce what the right physics is at high energies it's very difficult because to excellent accuracy what happens at our scale depends only on these two-parameters and not about all let's say that we knew we could figure out what all the X's were at the blind scale that's an infinite ordinary information that would be huge amount of information that would allow us to get a lot of the experiment input on the question of what the theory of quantum gravity is what all the X's were we only know two patterns two functions of those that's some data but it's very little data okay so once again it's a relaxing because it allows you to do things at low energies without putting the details of what happens but it's a curse for the techniques that once you got to the energy theory and get more ambitious you have limited data in your perspective so the high energy is that all these X's will depend on just these two parameters which essentially means that you don't have a unique UP theory no, no, no, X's depend on these two parameters that's one no, no you see the X's could have had any dependence on these two parameters so they could have been anything you wanted and the difference between one way of depending on these two parameters and another is suppressed by lambda I R by lambda it would be 2 sub power 2 so if this suppression was very high in order to detect how the other X's depend on X2 and X1 we need to do measurements to the center just data Y squared in our case it was squared because you see all operators that we ignore what I mentioned 6 or higher 6 minus 4 that's good same that's good another question so we are saying that in a certain scale we can introduce some operators if lambda U V if lambda U V will be in that stream where on the order of lambda I R if lambda U V where on the order of lambda I R then we would not be very misquoted we would not be very misquoted then this observation of the convergence is useless because convergence helps you when you have enough time to do if you plot two scales that are nearly separated then everything I have said in the last lecture this is true but useless okay so and then you would have neither the blessing nor the curse but yeah when you have started with some the arrangement is more serious than 5 in the same order if you will have the same the arrangement consists of the same and higher potential operators that's the assumption but in this case in the W case we see that really W because one has disappeared in the long range it is not W here as you said the number of years are the same this is the heart of the reason why in general continued renormalization group flow up to me you see as some of you know you should do 5 to the 4th theory and you continued it to beyond the W Poisson scale at the level of formal renormalization group flow equations maybe you could but what you would find is for instance what you would have to do 5 to the 4th theory above a certain scale gives you nonsense ok in some other examples if you try to calculate the uv you would get some infinity in sum hitting some singularity or some inconsistency or something like that is a sign that what's actually happening is new physics that your equations are not equal to 2 so there are 2 kinds of things that can happen when you float let's take the paradigm paradigm is like you've got new kind of feeling with a particular concept when you integrate out when you go down the scale is below it you completely integrate out that new feeling then you all learn to compute any correlation function in working the new feeling completely integrate it out and then after that you can deal with you know you can deal with just the old just the like possible but integrating up you would have never seen this because you could have it has not been your format so what you would do is to keep building a theory of the low mass field and typically I don't know if there's a theorem here but often you run into problems because consistency is required having that high mass field in the case of 4th one degree theory the problem is well known if you learn how to write the theory yes yes you cannot you cannot that's why renormalization is the one way renormalization root flow is the idea that you know physics and all scales you can in particular use it in the more convenient way of law you cannot you cannot do the other way around there's one practical thing the practical thing is this that even if the violence stuff that we are talking about is happening small differences you would have to know things very precisely at low energies to know them with any precision at all with higher energies because why almost all operators damped up it flows towards the higher almost all operators blow up it flows towards it that difference is that we have this convergence to this flow surface it's the converse that we have a divergence so this is the first practical aspect then this is like see suppose as we were saying that let's suppose okay quantum gravity is too abstract let's suppose that there was new physics let's suppose 10 degrees to 100 degrees okay suppose okay I don't know how light you have that is but suppose say that super symmetry at 100 degrees and you want to know what we use what we know to figure out the details of the super symmetry we can't build a 100 degree it's going to be like 10 degrees we want to use the details of what we want to figure out that's still tough it's still tough because you see let's say that until this new particle appears we can use our old framework with all the feelings that we have so we try to run the RG flow onwards a lot of information to know what the Lagrangian was just below the scale at which we integrated out the mass function we have a lot of information now it's equivalent to knowing how X6 was X6 not was a function of X4 X8 not was a function of X4 but in order to know that you have to measure things very precisely at the energies at which you have to measure so that's one element and then the second element is that if you go beyond that scale you're like in terms of yeah this is people that you have some budget part of the loop which you haven't discovered and they show up because if you can't figure it out then you get something which is not a great experience but this is the kind of thing exactly now this by the way this singularity in RG flow you go up it was an argument for things that's like 440 if you really hand the renormalization to higher and higher energies without any exposure this kind of model without any exposure some point you need unitary information in W-scan ok so you see this is the kind of this is the kind of way in which one way of using renormalization will leave pretend you can do it and see if you run into trouble if you run into trouble lightly it's very different logical status from using it to the higher because if you know the physics in this game of course you can there's no barrier in principle to knowing the physics but you know the physics in this game there's a barrier in principle to knowing the physics because we have new stuff happening how could you know ok once you say you run into a negative violation you introduce a new freedom to overcome that violation and can you start with a new RG with 3 parameters and do the same thing you can try it very many such things ok our question about this in our derivation we assume that x4 at the low scale was small so then put it in the user scale so suppose we use some other value the same sense and say then the interaction will be as long as you all that we are using is that the quadratic action now if you choose x2 and x4 and then choose all the others as a function of x2 and x4 but for the convenience of using perturbation theory you will choose them in such a way that if x4 was 0 x6 was 0, x8 was 0 and that would be sufficient all that we require is that if x4 is 0 then we get a quadratic action see it is certainly true that you could do more general things that would make it harder for you to function but again there are two statements one is the abstract statement that you can choose any kind the second statement is that you want to use the facility of perturbation theory so if you want to use the facility of perturbation theory you should use a cut that is well suited to perturbation that cut basically is that there is one line in that flow that is just a free flow how do we see that the predators at the moment are suppressed because x and b are the most important mostly problems with the entity and the level of the 6th predator for some reason lambda square and then what you change make lambda, look smaller x6 and so the intuition is that if lambda becomes smaller this becomes smaller now this intuition would suggest that x6 itself becomes smaller this has been repeatedly emphasized not necessarily true because that is the right hand side see what is the point right hand side can generate running in this x6 which makes x6 not necessarily smaller but what we have shown what we have shown is that this additional power to the fact that it looks like it has been done in the past actually needs to converge that is the only correct way how is the inverse problem this is an inverse problem meaning that is the only correct way how is the inverse problem this is an inverse problem this is an inverse problem meaning as you go to lower lambda it becomes smaller how long we want for this we do not want minus minus would be the other thing become more the inverse problem you ask how we get minus and minus t was equal to minus because our fluid time was flowed to the IR to lower that that was put in by hand that is what we need for this then I will have x2,4 actually which is not now which has a positive and this is the other way wait wait x if we do x A comma t what is the dimension of this in the other d that is n comma p what is the dimension of this in d dimensions it is v minus 2 by 2 like that plus now go let us do t equal to 6 so 6 minus 2 to the 4 by 2 so 2n plus scared dependence that was that matter 6 minus n minus p now what will it be it will be 6 minus 2n d dimensional analysis where will we get that n we got the n from the dimension of 5 goes on otherwise it is a loss we will discuss only half of what at length that is ok otherwise it is a loss ok now all the people who had questions in the earlier part of the class where I pushed it to the end any questions now ok but what do you do ok confused you enough I was saying that in the role everything is dependent on but the whole dependence the whole ignorance of what happens is contained in this case what is x for what is x at low any other question ok now last 5 minutes basically in the last 5 minutes I am just going to introduce these equations these equations x n dot is equal to beta of x the equation what we can sometimes call a dynamical system a dynamical system is just some some in this language is just some a system of equations there are autonomous how something evolves in time does not depend on time explicitly only depends on what it exists and of course in dynamical systems just as mathematical equations are much studied and so let's see what could happen in these dynamical systems one thing that can happen is that you got a fixed point so that you have x dot so I am not writing the n's this is abstract is equal to beta of x and there is some value of x so let's call this beta tilde which includes the classical pieces x dot is equal to beta tilde of x ok such that beta tilde of x vanishes and a particular value of x so suppose you got these these equations and you got some value of some x fp such that beta of x fp for all of the x's beta 1 beta 2 beta beta n comma d of x fp is equal to 0 this is called a fixed point we have such a fix let's look at the flow equations beta tilde the whole thing are of the form that we admit such if that's the case then we can linearize the flow equations so let's say that we say that x is x fp plus y we will get equations of the form y is equal to m that is y where m is a column in matrix rotation space of all the couplings is called y and m is a matrix of the space of all the couplings on that column is this clear? no confusion about my notation we get some equation like this we will be using arbitrary parametrization of these couplings but but if you mean that this m system is nice enough and actually we can prove that there is nice enough the similarity transformation can be used to diagonalize this m this m is not somehow bad this m can be diagonalized by some similarity transformation we will assume that's the case we will come back to that ok let in that case this we have got some y i is equal to lambda i y i y dot is equal to lambda i y i give values or i give values that are ok about such a fixed point relevant operators could be some totally crazy linear complication in terms of in the next few minutes before we finish this class is imagine a renormalization root flow in the neighborhood of such a fixed point ok where these marginal operators are operators can be marginal then there are these things called exactly which really is the question of whether suppose you are a marginal operator there are two possibilities one possibility is that the beta function is not 0 as you leave as you go away from the fixed point that's also not a linear order but quadratic order that would give you a margin another possibility is that actually there is a one parameter set of x's for which the beta function vanishes as an exact thing if it's the case that there is a one parameter higher some parameter set of x's for which the beta function vanishes then the change of these parameters along these directions is exactly marginal and then you know the quantum field theory at a fixed point is called a conformity and if you've got exactly marginal operators then you've got a one parameter or two parameter n parameter set of conformity things this happens now it was thought to be very uncorporate it's now going to be less uncorporate capital for the mass dimensions of the same lambda especially when it's marginal oh yeah what is lambda? this is lambda these dimensions these dimensions even an exact marginal series can become non-marginal if you have the normalization just because of the normalization just as a matter of terminology the statement about whether a theory is exactly or not is a statement about whether the exact beta function is actually that includes all anomalies suppose we've got some renormalization okay imagine that there's a renormalization root flow and one relevant let's say that the relevant directions okay and one irrelevant direction let's suppose there are no ones okay when the flow is in the neighborhood you say this guy will just flow to the fixed direction because if you want an equation we could imagine that the equation is d y by d p is equal to y that's the relevant direction but d y by d suppose this was the start okay the numbers because we normalize now in this x y space this is after line x y is after line now how does this look well if x was zero y just decays to a zero y always decays to a zero because these equations aren't coming at the time that y is taking to a zero x starts flowing so you see that the flows look like this the structure of renormalization group flows in the neighborhood of a point where there is one relevant and one irrelevant let's study these flows because there is no more relevant there is no more if they were both relevant of course we would just take the both of them and produce the same variance and we would just flow along now of course so you might think that these diagrams are very little but the point is that drastically we have seen that the number of relevant operators is not very large some small finite number and all these dimensions can change things a bit but our experience in quantum field theory is that unless you go to some very clear crazy regions it has changed things very much so that this general qualitative factor the number of margin in the dimensions the number of marginal relevant dimensions is a finite number usually a small finite number there is a good theory remains true diagram of that point we have to generalize it in mind but it is paradigmatic for renormalization group flows around fixed points with no marginal directions the point is that there is an irrelevant dance away but there is a relevant direction statement this kind of if a renormalization group flow in this situation would generate another one parameter set of quantum field why? because suppose you were here the statement here is that if you wait long enough every flow boils down to which one excess has damped out and we are just playing on the axis in the context where there is a fixed point without marginalizations there is a very clear characterization of this fixed surface actualization goes as follows this fixed surface here defines one parameter set of quantum field theories in this case one parameter how would you reach this fixed surface what you would have to do is something quite strange if you started with arbitrary conditions of you arbitrary initial conditions where would you end up you would never reach the neighborhood of this fixed point because you would be far away on the x axis if you wanted to reach this neighborhood at all what you would have to do is to find you your UV conditions so that you started out in the neighborhood of the of the y axis if you did that you are very near the fixed point before being sent to you if you did this kind of fan tuning then you lie somewhere on the field theories defined by this one an example of the kind of operator that does this you know this kind of this operator in the mass operator of this kind of thing the mass operator of this kind of thing is a random operator the fact that in order and this the fact that in order to get a theory that you know a small mass is the problem of what is called naturalness defined if you took the kind of point of view that we took through the bulk of this lecture that there is some physics or some ice game and we generate every biometer we generate every biometer of order lambda naught in particular we generate the mass of order if you started at some generic value we generate very large value some very large value of x2 on the renormalization loop flow in the real world or in condensed analysis we are often in situations where this is what you imagine should be happening but it doesn't actually happen okay in the real world this applies to the mass of the Higgs okay if physics is being generated by some some stop at some very high end that's in Planck's behavior you would expect the Higgs mass to be a border the fact that it's not happening the fact that we are actually getting a quantum field theory with Higgs much much more than the Planck's is somehow apparently some things are fine to me either by some physical mechanism or by hand a clear example of this happening in condensed matter physics is in theory of phase transitions which we have studied in detail for instance in the Isaac okay how phase transitions are related to quantum field theory in this study I won't say much but just to say when you have a phase transition the phase transition point is governed by a conformally married field theory and the relevant operator the relevant operator takes you away from the phase transitions like tuning the temperature things so the fact that it gets to the scale in any point you need to do a tuning is simply the statement that the phase transition happens not at every temperature but at one temperature and the physics in the neighborhood of the phase transition is governed by this what they measure okay let us stop now we will have a more systematic discussion of these general issues okay I'm stopping now then for questions I understand that what you said about the relevant operator but I mean I asked the question what the marginal operator is because say the question the question about the marginal operator is that on this I mentioned hence the marginality you have to say about the marginal operator so it seems to me that since an operator dimensions an energy in effect this status then do we have to adjust every order of operation here because we don't need to change this parameter so right so the question of whether an operator is marginal in order to be answering the question suppose you have a fixed point that occurs at some non-zero value of an operator then you cannot address the question of marginality of an operator in an operation you can only address the question of whether it's small or large you see because because as you say marginality of an operator is something being seen something being 0 to the first 20 orders in perturbation theory does it guarantee that it's 0 to the 21st it's important because theories as far as I know the theories can be re-normalizable or non-re-normalizable depending on whether the company constant is either marginal or irrelevant or if you have irrelevant company constant then often those theories are not re-normalizable so for example QB which is the theory we have alpha alpha is non-marginal and so what happens to that what happens to those two okay the kind of stuff I was hoping to talk about in the next class so we could address it better it'll take us to lunch yeah so in the next class what I will try to do is to study various kinds of re-normalization group first and talk about how they try to study the re-normalization group first UCD we'll put the final form do we do a lot of such things okay the quick answer to that question is that basically what we have here as you said next time next class I'll try this one no other day I'll try today yes we should