 Thanks very much for the opportunity to give these lectures. So we have three hours today and one tomorrow, which means you have to bear with me. You might get tired of me already by the end of the day. So I'm going to cover some topics in quantum field theory that are interwoven. There are some intricate relations between these different ideas. So let me quickly go through about the plan for these lectures. So first, I'll start with some basic introduction into how anomalies creep into quantum field theory, mostly in two and four dimensions. I'll discuss three examples. One is for current algebra and other four central charges in two and four dimensions. This results in various applications for a condensed matter physics, statistical physics, particle physics. But in these lectures, I would not have much time to emphasize the applications. I mean, for example, this has played a major role in the quantum hall effect in the past, but I'm not going to emphasize the applications very much because I want to emphasize more the ideas and the formalities of how these things arise. So I'll be mostly talking about the theory, not about the applications. If somebody wants to know about the applications, we can maybe discuss it later in the day or during some break. And so that would be the first part, some very simple anomalies and how they creep into quantum field theory. The second part is going to be more advanced. I'm going to try to define a geometry in the space of theories. And then I'll apply these ideas for cyber-witten theory in which there is some space of models and one can try to compute the geometry in the space of models. And that would be related to some other anomaly, which I will call B type rather than A type. So their first part would be about this A type. The second part would be about B type and there would be an application to cyber-witten theory in which I will show you that you can compute exactly some properties about theory space. So these ideas are a combination of new and old ideas, but this is completely new. And then the last lecture I'll devote to a topic which is completely different about the S matrix of QCD. It has some interesting ideas and it's more like a teaser to I think some progress that can be made in this field rather than a lecture in which I'll present some complete results. Okay, now because we have like three hours today, instead of me speaking, it would be very nice if you ask lots of questions and it would be a more interactive session. Otherwise what I'll end up doing is to basically reciting my notes which I'll put online today. So there are 70 pages of notes. They're somewhat disorganized, but the gist is there. So if you don't ask questions, what I'll end up doing is to recite my notes which would not be very interesting. So, okay, so please ask questions. So let me just start with a little bit of motivations. So obviously the framework of quantum filtering, the framework of quantum filtering has singled itself out as a universal framework, a mathematically consistent framework that describes lots of phenomena in physics, in many, many different branches of physics. It's even hard to enumerate in cosmology, in condensed matter, in statistical physics, and of course in high energy physics, and even in stochastic processes. So even people who study stock market, they need to know a little bit of quantum filtering. And in each of these instances, the reason that quantum filtering emerges is somewhat different. So in condensed matter physics, of course quantum filtering emerges because you can have particles generated from the vacuum and things like that in the same way. And you have sometimes emergent relativistic invariance so you even get to study relativistic quantum filters in statistical physics. It's the old story that statistical systems near a critical point, near a phase transition, they have long range correlations and so you can forget about the lattice structure and you can pass to some continuum description. That's how people understand nowadays critical phenomena, which you must have heard about from Slava. In high energy physics, obviously we have Lorentz invariance and so we can have particles generated from the vacuum so we need to use this framework and so on. So this is a very, very universal framework that can be used to describe many different physical ideas. And so it's obvious that anything that we can say about quantum filtering, any general statement that we can make about quantum filtering is going to be useful for many, many people, many branches of physics and we're still exploring the basic structure of quantum filter even though it's four decades old, the framework we're still trying to explore to what extent we understand the intricacies of the theory, we understand the basic results in the theory and so on and so forth. So the general picture that, the general understanding of quantum filter comes from Wilson's ideas. What Wilson said is that basically you can think about quantum filtering as an evolution process of some sort. So if you look at very, very short distances, so this is going to be let's say short distances. So inconsistent quantum filtering, okay? In consistent quantum filters, the short distance physics is described by some scaling variant model and also the very long distance physics is described by other scaling variant model. So there is some scaling variant model here and in the middle there is some crossover behavior which is characterized by some scale. So there will be some scale which in particle physics conventions you can think about it as an inverse mass scale. So there is some crossover from one scaling variant theory short distances to another scaling variant theory at long distances and the critical exponents here so we characterize scaling variant theories by the correlation functions and the correlation functions in scaling variant theories are some power laws and these power laws have some exponents gamma N in the infrared and here there are some other exponents gamma N in the UV and the crossover interpolates between one set of exponents to another set of exponents. So that's the general framework that Wilson gave us for quantum filter. So what is our job in quantum filter? So this kind of scenario is realized of course in many instances in particle physics, statistical physics, and that's matter and so on. For example, one of the first examples was for instance a renormally flow from the tri-critical model to the Ising model and this is something that has been observed experimentally long time ago and there are lots of dozens or hundreds or thousands of other examples where we've seen this picture happening in nature. So what are the interesting questions that we can ask? So the interesting questions could be structural which is what is the connection between these exponents and these exponents? This is one sort of question that you can ask. Can we go from any scaling variant theory to any other scaling variant theory or are there constraints? Are there constraints that if we specify the scaling variant theory here there is no crossover that would lead us to some given scaling variant theory here. You can ask the question of given a scaling variant theory can it be the bottom of this crossover behavior or maybe it doesn't have any way of uplifting it into what we call a flow. So this is called an RG renormalization group flow, a renormalization group flow. So these are sort of structural questions that you can ask of when can a given model be understood as sitting here or here when there is a crossover behavior. You can ask, this is one set of questions. Another set of questions is about what's happening at the crossover scale. So the crossover physics could also be very interesting. For example, deep in elastic scattering which is an interesting experiment in quantum chromodynamics is more about the cross, what happens here, very near the fixed point. This is called the fixed point the scaling variant theory in the UV. This is more about the physics in the crossover regime but not very far from the fixed point. QCD, the hadronic physics is more about the crossover scale. In statistical physics, critical behavior, these critical exponents that you read about in the books in statistical physics are about this regime. So not quite the scaling variant point but very close. That's how you read out critical exponents in statistical physics. So there are different applications of setting the crossover regime, the scaling variant theory, the scaling variant theory. There are also lots of applications in higher physics. Yes, yes, the picture could be more complicated. It could be that for example, you start from some scaling variant theory, there is some complicated crossover regime which actually hovers near some other approximate scaling variant theory and then it continues to some other point. The diagram of possible renormalization group flows could be very intricate and well understanding that global structure is one of the things that you may want to do. Since the framework is so widespread, understanding the global structure of these possible flows is a very important question. So different people have different tastes for what are the interesting questions. In condensed metaphysics nowadays, in fact they've gone to the extreme because what they are studying are this topological insulators. That's one very big theme in condensed metaphysics nowadays. In topological insulators, this scaling variant model in the infrared is empty. So there are no interesting correlation functions. And when this is empty, we call it a gap phase. And what they're studying are some non-local observables in these gap phases. So they're interested in trivial scaling variant theories in which there are some residual non-local observables that are still sort of interesting topological observables. And in statistical physics, of course, people are much more interested in the critical exponents. That's what they measure. They measure local correlation functions. So different communities have different interests and but it all falls under this umbrella which is very nice. There is one picture that encompasses lots of lots of physics. And so different people have different tastes for what is interesting. So let me tell you what I'm going to describe. Of course I made a very personal selection of the things that I'm going to describe. So I'm going to describe a little bit. So the first thing that I would like to describe is the question of when are the scaling variant points actually conformal? So we've observed experimentally, it's an experimental observation or an empirical observation. We've observed experimentally that sometimes these theories are more constrained than just being power laws. They have some additional symmetry and I would like to give you a quick argument that explains at least four two-dimensional models why you have these extra symmetries. That's the first thing that I'm going to discuss and then we'll discuss some very basic properties of the relations between these critical exponents and these critical exponents. We'll discuss some very simple applications of that idea and some simple results. And then I'll try to describe more collectively the geometry of all the possible theories. So there's some geometric manifold of all the possible theories and all the possible renormalization group flows and you can define an interesting geometry and study it and you can even make exact computations in some supersymmetric theories which allow you to learn about the space of theories. And then in the last lecture, I'll be discussing really the crossover shift where I'll be studying the S matrix of a large N QCD. So I'll cover some topics of various sorts. Are there any questions about this introduction and before I start, the first topic which is to explain why the scale invariant models are now to be conformal invariant in two dimensions at least. Okay, so let's start by reminding you about the Poincare group that will be very important to remember. So the Poincare group in D dimensions. I'm going to discuss for now the Poincare group of RAD. So the space is going to be RAD Euclidean. The Poincare group of RAD consists of translations. So we have D translations and then we have some rotations which are in SOD. So this is the Poincare group consisting of translations and rotations. Now what is the scale invariant model? So there is a formal definition. A formal definition is that the symmetry of the scale invariant model consists of the Poincare group plus an additional conserved quantity which we can call delta. That's a conserved charge. So that's a new conserved charge which acts on the coordinates by multiplication. So that allows you to rescale. And once you add this conserved charge to the theory then everything has to be a power law because when you measure, when you study let's say now correlation functions of two local operators phi one and phi two, this would be at x and this would be at zero without loss of generality. If this has scaling dimension delta one and this has scaling dimension delta two which I'm sure it's that makes talk to you about then we have to write the following formula. Delta one plus delta two. Okay, so everything has to be a power law because this is a symmetry. So now what for example, Lando did when Lando sort of described for the first time the theory of phase transitions. Lando realized that the fact that there is scaling variance allows you to derive many interesting relations between critical exponents. And everything that essentially was done back then follows from this idea that two point functions are given by power laws. This is the so called hyperscaling relations that you might have heard about but even if not it's not important. Hyperscaling relations. These are some relations in the statistical physics that just follow from the fact that there is scaling variance at the critical point. Okay, so this is the naive description of the scaling variance theory but experimentally, so an experimental fact, an experimental fact is that what we end up seeing in many of these examples is a bigger symmetry. So what we end up seeing is the full conformal group. So we have Poincare plus this new conserved charge which corresponds to rescaling of space but we also end up seeing the special transformations. Special conformal transformations and all of this together forms the group SOB plus one comma one. Okay, so this, the generators combined into a big group which is bigger than Poincare and bigger than this group the scale group, it's SOB plus one comma one and this group has lots of interesting consequences. So there are interesting consequences. One of the measurable consequences of having this extended symmetry is that two point functions are actually diagonal in dimensions. So a scaling variance allows you to have a nonzero two point function between any two insertions who is well defined eigenvalues under scaling but one of the consequences of the bigger symmetry is that for what's called primary operators you get a diagonal two point function for primary operators and then you have one over X to the power two delta one. Okay, and this has measurable consequences and in many cases we know that this is true. This is a consequence of this bigger symmetry and there are also lots of consequences for three point functions but I'm pretty sure you've seen some of that too. Okay, so the first thing that I would like to do is to explain why there is this symmetry enhancement. So there is the question of, it's a general question in quantum filtering which is why a Poincare plus delta often gets enhanced into a SOD plus one comma one. This is Poincare D. So it's a very general question that pertains to many applications in quantum filtering and we don't know. The short answer is that we don't know why this is true but in some cases we can say a lot. We can more or less give a proof in D equals two. This is what I'm going to do now. I'll give you a proof. In D equals four there's almost a proof. Almost proof meaning that it's not mathematically rigorous. It's a physical argument. So it's almost a proof and all the other cases, for example, three dimensions which is most relevant to statistical physics, there has been no progress essentially. And of course, you know that there are also interesting quantum filters in five and six dimensions and also in those cases there has been not much progress. Okay, so the question is why does this happen? Why does this symmetry enhancement happen? You might think, you might actually recall when you see this formula, that in the hydrogen atom there is a symmetry enhancement from rotations to the Runge-Lenz Laplace group which includes an additional generator. So remember that in the hydrogen atom you have an enhancement from SO3 to SO4. You might think that understanding the symmetry enhancement is very important because for example, the symmetry enhancement plays a huge role in the solution of the hydrogen atom. So understanding the symmetry enhancements is a kind of a basic question. So I would like to give an explanation. So our journey into all of these topics will begin from starting the two point function of the energy momentum tensor. Of the energy momentum tensor. I'll start by doing it in any number of dimensions and then we'll specialize to two dimensions and we'll see something special happening. So that's the object that I would like to study. I'm going to do it in momentum space because it's a slightly more convenient and one doesn't have to worry about various issues that arise in position space. So I'm going to study this fundamental object which is the correlation function of two energy momentum tensors with momenta q. Now if you are a statistical physicist you don't call it the energy momentum tensor, you call it the stress tensor because energy momentum is carried by something else. So in statistical physics this object you would call the stress tensor in higher energy physics it generates energy momentum so we call it the energy momentum tensor but in both cases it's a very fundamental operator that creates some interesting local excitations. So we're going to try to fix this two point function but what are we going to assume? So what I would like to assume is just Poincare. Okay and we'll do it in steps. We'll first assume Poincare, then we'll assume scale invariance, then we'll see that you can prove that conformal invariance emerges. So this bigger group emerges in two dimensions at least and then we will go further to prove various interesting structural theorems about these two point function. So we'll just use Poincare. So if you use Poincare you just have to satisfy the various symmetry conditions. So it has to be symmetric in mu nu, it has to be symmetric in rho sigma and it has to be conserved. The conservation is the big deal that we'll worry about a lot. So remember that there is this classical equation whereby this is conserved and we'll worry a lot about this conservation equations. You'll see that sometimes we cannot satisfy it, sometimes we can't satisfy it. We'll see various interesting applications but I want to make a small digression to explain what does this equation mean. You've probably seen this equation many, many times but what does it actually mean? So when we write a general equation that some operator, some quantum operator is equal to zero, it has a precise mathematical meaning. The mathematical meaning is that at separated points when you insert this operator you get zero but at coincident points you might need to be careful. So this is an equation that always should be interpreted at separated points. So what it means is that if we insert this operator at some point x1 and then there are lots of other operators and x1 is different from all the other xi which sit here then this vanishes. This is the mathematical meaning of a vanishing operator that at separated points it gives rise to vanishing correlation functions but sometimes you cannot uphold these equations at coincident points. This is where anomalies arise and various other issues creep in. Now in momentum space, remember that when you do a Fourier transform you integrate over all the possible positions with some weight and therefore imposing conservation equations in momentum space is always subtle and this will be a major theme that we'll discuss. You cannot quite impose these conservation equations right away in momentum space, you have to think what corresponds to separated points and what corresponds to coincident points. If you try to impose conservation equations you get wrong inconsistent results as you will see. Okay, so let's start by just writing all the possible point-curring variant pieces and then we'll discuss the conservation equation. So there are two possible tensors you can write. One in which there are constructions across the two energy momentum tensors and the other is when there are only constructions in pairs as you'll see. I wanted just to remind those so that the energy momentum tensor is symmetric in its indices so this is how these structures are constrained. So one possible contraction is q mu q rho minus eta mu rho q squared times q mu q sigma minus eta mu sigma q squared a plus rho interchanged with sigma and this would multiply any function of q squared. So let's call this function f and then there is another possible contraction which looks like q mu q nu minus eta mu nu q squared times q rho q sigma minus eta rho sigma q squared and there is g of q squared. So these are two tensors that you can write down so there are two functions f and g and I've chosen the tensors in such a way that the conservation equation is obeyed but you'll see that it's a little bit subtle. We'll discuss more why did I choose this particular tensors? One is really obeyed, one is not really obeyed and so on. This is just a naive first attempt that implements this equation. So in momentum space this equation can be written as q mu nu q mu nu equals zero. So in general there are two functions that appear here f and g. The first claim that I want to make, these functions would in general be functions of the crossover scale of the momentum over the crossover scale. So in scaling variant models where there is no external crossover scale, this would be some homogeneous function that we'll try to fix but in general it could be a function of the momentum over the crossover scale. So one interesting claim that you can make, you'll see I'll derive this claim later from another point of view but you can already, you can try to prove it also from this point of view is that if the number of dimensions is two, then f and g are linearly dependent. So they're not different functions. They're not different tensors. So the claim is that this tensor and this tensor in two dimensions are actually the same. Not independent, so they're not independent. This is a small claim that I would make. We'll prove it from another point of view but it's important to appreciate it already at this stage. Okay, so now I'd like to talk about what happens if we impose scaling variants. So we impose that if we impose that there is no external scale in the problem. So everything is just some power law of various momenta. Okay, so let's discuss three different cases. Two dimensions, three dimensions and four dimensions. What happens if we impose scaling variants in these three different cases? Now, you have to appreciate that the energy momentum tensor in position space has dimension D and therefore it's Fourier transform has dimension zero and because there is always a delta function for momentum conservation which I've omitted, the right hand side has to be of dimension D where D is the number of space time dimensions. Now we already have four powers of momenta here and here and so in two dimensions it has to be one over Q squared. So F and G, the F to B, some coefficient over Q squared. It won't be important to fix these coefficients. It just follows from dimensional analysis. Now you could ask what about logarithms? So we can discuss logarithms a little bit later. You could add various logarithms here and would still be naively consistent with scaling variants but we'll discuss that later. This is the most straightforward solution. This is the most straightforward choice. Now in three dimensions F and G are going to scale like one over the square root of Q squared and in four dimensions they have to be logarithms of Q squared. These are the three cases, two, three and four dimensions. Are there any questions about why this is true? You could choose either a logarithm or a constant function. It would still be okay but you'll see later that the only consistent choice is a logarithm, not a constant function. Okay, so these are three cases that you could discuss but then there is an interesting computation that you can do. You can try to take the trace over the indices mu nu and rho sigma, both of them at the same time. So let's trace over mu nu and rho sigma. So we're going to trace over mu nu and rho sigma. Let's write down the answer. We get to study T mu mu of Q and then T rho of minus Q and we'll do it again in two, three and four dimensions up to this coefficients that I haven't fixed. So let's start from two dimensions. Two dimensions F and G are one over Q squared. So if we trace over mu nu and rho sigma this would be Q to the four. This would be Q to the four. They would cancel two powers from the denominator and you will get Q squared. This is in two dimensions. Three dimensions, we will get something like Q to the four over the square root of Q squared. So that's going to be Q squared to the power three halves. Yes, so if you take TQ, P, that's what you're asking, right? Then in general, this would be a delta function in the dimensions of Q plus P. Why? Because if you create an excitation of momentum P it has to be absorbed in the vacuum with momentum P. So this creates an excitation from the vacuum with momentum P and then it gets read. What this describes in the language of statistical physics is that you create an excitation of some momentum P and then it is absorbed sometime later in time. And so it has to be absorbed with the same momentum because we're doing it in the vacuum. Of course, if you do this kind of, if you do this correlation functions in the presence of some non-trivial medium then some momentum can dissipate into the medium or some momentum can be extracted from the medium and then you don't have a delta function. But I'm doing these computations now in the vacuum where no momentum can be absorbed or extracted from the vacuum. Okay, so I get Q to the four. In four dimensions we get Q to the four times logarithm of Q squared. Okay, these are the three cases that we've got. Now, the most striking fact about this result is that this is a pure, this is the only case in which you get a pure polynomial. So this is a pure polynomial. This is not a polynomial. This is a fractional power of Q squared. And this is also not a polynomial because there is a branch cut due to the logarithm. There is only one case in which you get a pure polynomial. What does it mean in position space? If you have a polynomial in Fourier space in position space it's a delta function. The Fourier transform of a polynomial is always a delta function. That's very important to remember. So what it means is that in two dimensions the correlation function of t mu mu at x and t mu mu at y, where y is different from x is vanishing or more precisely the Fourier transform of this polynomial is a laplacian of a delta function of the two-dimensional delta function. So the two-point function of t mu mu in two dimensions vanishes at separated points. In three dimensions it's not a polynomial. It has a non-trivial inverse Fourier transform and you get some interesting correlation function. In four dimensions you have a logarithm. So again you get an interesting correlation function at separated points. It's not just a pure delta function. Now you could have asked, it's important to think, why did I choose a logarithm? If I had chosen a constant which is also consistent with the naive scaling then this would be also a polynomial and then I would also get something like that in four dimensions. But let me explain why this is not the right choice. If I had chosen here, if I had chosen here a constant then all the components of the energy momentum tensor with all the other components of the energy momentum tensor would be pure polynomials because this is a polynomial and this is a polynomial so this would be a constant, you would get pure polynomials and therefore all the correlation functions of the energy momentum tensor would not have support at separated points and that would contradict unitarity and various other things. It would mean that the energy momentum tensor is a trivial operator which is not good. In two dimensions this choice does not trivialize the two point functions. It only trivializes one component which is the trace of the energy momentum tensor and so it's okay. But in four dimensions this is the right choice and therefore it's not a polynomial. Now if you have, yes, which more general situations? Oh okay so there is a great question of why didn't I add a log of q squared? That's the question. Okay let's discuss that. If I had chosen to multiply it with a log of q squared then indeed this would not be true and the conclusion would not follow. But let me explain physically why this is not allowed. So you see what I'm imposing here is scale invariance. Like any other symmetry or operator equation this should be upheld at separated points. So when we do the Fourier transforms back to position space the correlation functions have to be pure power loss even at separate points, sorry, at separate points. They have to be pure power loss. Now if you had the log q squared you would need to divide by some scale, mu squared. Okay just for dimensional analysis. And if you change the scale from mu to mu prime the difference is going to be a over q squared and when you plug it here you get something that is not a polynomial. So you violate scale invariance even at separated points. So if I had put a log here that would mean that there is no scale invariance even at separated points in this model. Okay so there are no critical exponents. This log is different. Now you can start seeing when can we add a log and when we cannot add a log. In this log if we change the scale from mu to mu prime the difference between the two logs is constant. And therefore it corresponds to a pure polynomial in momentum which only has supported coincident points in position space. And therefore it's okay. So this log does not violate scale invariance which is what we're assuming but this log does violate scale invariance. So we are not allowed to put it here. And therefore this result is inevitable. Okay now one final general comment that I wanted to make before I proceed is that in quantum field theory if we have an operator and it's here mission conjugate at let's say x and zero. If this vanishes for all x, sorry. Yeah if this vanishes then for x not equal to zero then the operator is trivial. This is in good unitary theories where there are no imaginary parts in the action. This is like a norm of a state. You can think about it as a norm and therefore if this vanishes then the operator itself vanishes. When it acts on the vacuum it doesn't create anything. For if it created something you would have a nonzero result. You just imagine inserting a resolution of the unity in the middle and you would get that this is true. So if the two point function vanishes the operator vanishes at separate points. And so we conclude that in two dimensions T mu mu has to vanish. We just proved that. It follows from scale invariance. I've not assumed that there is conformal invariance. It follows from scale invariance that T mu mu vanishes. Now when if T mu mu vanishes in conjunction with this condition you get far reaching consequences. Namely you get the full conformal symmetry of two dimensional models. Let me review why this is true. So since we're in two dimensions let's choose light-con coordinates. They will be very convenient in what follows. So I'll choose the coordinates x plus and x minus and correspondingly there's p plus and p minus for the two components of the momentum. So we can write this conservation equation in light-con coordinates and p plus minus coordinates. So it reads q plus, q plus t plus minus. q plus t plus plus plus q minus t plus minus equal to zero. And q minus t minus minus plus q plus t plus minus equals zero. These are the two conservation equations in momentum space, in the light-con space. T plus minus is just T mu mu. So if T mu mu vanishes, which is what I've proven, then these two go away and we remain with these conservation equations that q plus times t plus plus and q minus times t minus vanishes. Or in position space, that means that the d minus of t plus plus vanishes and d plus of t minus minus vanishes. This is something that we've proven based on scaling variance. But if this equation is true, I can multiply t plus plus with any function of x plus. So therefore it's also true that d minus of any function of x plus times t plus plus vanishes. And d plus of any function of x minus of t minus minus vanishes. So you get infinitely many conserved charges if T mu mu vanishes because I can multiply t plus plus with any function that I want and that generates the full bero soror algebra. So that's the infinite dimensional conformal algebra that you certainly heard about in two dimensions. So you see that we proved in 15 minutes a very far-reaching result in quantum filtering which is that in two dimensions, if you just assume the scaling variance which is just saying that you have power laws, you get the full bero soror algebra. You get the full bero soror symmetry. And the only ingredients that went into the proof were unitary, meaning that I assumed that the resolution of the identity is present and also that all the states have positive norms and one current variance with scaling variance in the usual sense. So this is a very strong result that we know is true in two dimensions and it explains why in quantum filtering, the scaling variance points that appear at long distances at short distances are actually bero soror invariant theories which are much more constrained and there are many experimental consequences of having this bero soror group. So unfortunately you see that this argument does not carry through to three dimensions or four dimensions because we get two point functions which are non-vanishing at separated points of T mu mu. So there doesn't seem to be an easy way of that sort to prove that scaling variance implies conformal invariance. You can assume that conformal invariance is true and then T mu mu vanishes and then you can proceed but you can't prove it in this way from first principles. You have to do something much more elaborate, presumably. Okay, are there any questions about this part? Yes, right, this is like a local operator. This is just zero as an operator, as a local operator. Typically you mean when there is conformal invariance or what? Right, right, indeed. So yeah, I'll repeat the question. So suppose, so in four dimensions, we can't prove that scaling variance implies conformal invariance. Let me just repeat the question. The question was, what happens if you don't have conformal invariance? That's my interpretation of the question. Then this is not true but maybe some weaker form of this equation is true, okay? So indeed, indeed. Let me quickly tell you what do we know if there is no conformal invariance. So if there is no conformal invariance, then T mu mu does not vanish as an operator. Rather, T mu mu, we know, is equal to the gradient of some other operator. So it's not a completely general operator, it's just the gradient of something. I mean the divergence of something. How do we see that? There is this charge that I defined, which exists in scale invariant models. This is the integral over d, d minus one x over some space like slice of some current, which we can call a, let's call, let's denote this current by b. And this current b is given by x mu T mu mu plus v mu. This is a general form for this current, which is conserved and whose integral gives the scale charge. Now, for this current to be conserved, you need to satisfy this equation. Just hit it with nabla mu. Use the fact that the energy momentum tensor is conserved and you'll get the T is the gradient of v. Now, if T vanishes, then the gradient of v vanishes and then you get many more conserved charges. But if you don't have conformal symmetry, the only thing you know is that the integral of b naught is conserved and that's it. So the integral of b naught is very close to what you said. It's a v naught plus x i, x mu T zero mu. So if there is no conformal invariance, this is all we can say. That there is some operator v whose gradient is, whose divergence is d and that's it. But if you have conformal invariance, you have a much stronger result because you know that this vanishes and this vanishes. You can throw it away and you have lots of new conserved charges. So conformal invariance is indeed much stronger than the scale invariance, but it's not always easy to prove that fixed points have conformal invariance. In fact, Landau did not anticipate it. Landau never used conformal invariance. He just used scale invariance in his theory of second order transitions. But I think in the late 70s or maybe mid 70s, it already became apparent that all the known models do happen to have conformal invariance. And many people wrote early papers on trying to explain that, but still the only case in which it's completely well understood is two dimensions and to some extent four dimensions. Okay, now I'm switching to the new topic. Well, it's not completely different. I'm just going to talk about gravitation anomalies, gauge anomalies and all that and how they evolve on the renormalization group transformations. So let's have this picture in mind. Now we are smarter. We know that this is a conformal filter. This is some other conformal filter. I'm going to do two dimensions. I'm going to discuss two dimensions for now. So we're a little smarter. We know that this is the picture and here there is some crossover in the middle between the two theories. So the question that I'm going to ask now is are there interesting quantities that have to be the same here and here? So maybe the critical exponents may change. Many things can change, but some things may be preserved under this evolution. Or maybe there are some inequalities along this evolution. So I'm going to study two examples of things that are preserved or have a controlled evolution. And you'll see what kind of ideas go into that. So I'll start with something simple, which is to suppose that this whole evolution preserves a U1 symmetry. So there is some global symmetry and then we'll generalize it to central charges and so on. So I'm going to assume that there is some global symmetry. So we have some current, which is preserved. Everywhere, it's preserved everywhere, not just at the fixed points. We just have a complete global symmetry, a complete honest to God global symmetry that's preserved along the flow. And there are many examples of that, sorry. For example, take the O2 model, which is an interesting, which has interesting applications in statistical physics. So we have a honest to God conserved current and I'm going to study its correlation function in the same way as we did with the energy momentum tensor and prove some general facts about it. So we will use light one coordinates in which there are three interesting correlation functions to study and I'm going to use momentum space again because in momentum space anomalies in general are much easier to discuss in momentum space. I'm going to write down the most general possible answer that follows from one current variance and then we'll use the conservation equation. We'll discover that sometimes it's anomalous, sometimes it's not anomalous and we'll try to see how does this story go. So just from one current variance, you have to put a P plus squared over P squared times some function, which is a dimensionless function of P squared over the crossover scale. How did I get to these sunsets? So the fact that you have to have two powers of P plus is obvious. That's just boost invariance. Now the fact that A is dimensionless follows from the fact that A current has dimension one into dimensions. So the dimension of the current is one and therefore its dimension in Fourier space is minus one and we have a delta function so this thing has to be dimensionless. This is given by some function B of P squared over M squared which is some dimensional function, a dimensionless function and this is P minus squared over P squared times some function C of P squared over M squared. So far I only used dimensional analysis, I did not assume scaling variance, just dimensional analysis and boost invariance and of course the full point correct group. That's what I used so far. Okay, now, so now there is a very interesting and subtle story about what does the conservation equation imply? Okay, so this is a very interesting and subtle story. Perhaps if you haven't seen these kind of things before that would be extremely confusing and you'll have to think about it but I'll try to explain it to the best of my ability. So suppose you just impose the conservation equation. Suppose you just impose the conservation equation. So you impose that P plus and J plus is equal to minus P minus and J minus. So the conservation equation in the momentum space is P plus P minus J plus plus P plus J minus equals zero. So that's the conservation equation in the momentum space and you can just impose this equation on these three possible functions. So the conclusion, the main conclusion that you find is that these three functions have to be identical. Okay, now if you impose this condition you'll end up getting lots of paradoxes. For example, you won't be able to reproduce these results by Feynman diagrams. There will be some contradictions with unitarity and so on and so forth. So the name application of the conservation equation fails and the reason is that you have to be very slightly more careful. What the conservation equations actually imply is a slightly weaker equation. They imply that P squared of A squared over M squared is equal to P squared of B and equal to P squared of C. So if you allow yourself to divide by P squared then you get this equation. But there may be some subtlety with P squared at the origin, okay? Another way to try to describe where the subtlety is is that when we impose the conservation equation then we basically want to demand that this would be true as an operator equation. Now operator equations cannot be imposed at coincident points. That sometimes may lead to inconsistencies and this is one example where it would lead to inconsistencies. So what you actually have to require is that this equation is true at separate points and you should allow yourself some freedom in what's happening at coincident points. So the fact that there is a P squared here means that you should only satisfy these equations up to things that would correspond to polynomials in momentum space after you hit them with the derivative. So the actual general solution of the word identity is not that A is equal to B equal to C but it's the following. So the plus plus correlation function, I'll just call this plus plus plus minus minus minus to make the notation simpler. So the actual solution is P plus squared over P squared and then there is A, which is some function of P squared over M squared plus some constant which I will call K lift. Why is this constant allowed? Because when we try to study conservation equations because we hit everything with P squared that corresponds to a polynomial in momentum space which is a contact term. So it doesn't actually change the fact that this equation is upheld because it should be only upheld at separated points, not at coincident points. Now plus minus, for plus minus we get B of P squared over M squared plus some other coefficient that we call K. We don't know what K is, we just put it there because we may. And minus minus is given by, sorry, this is the same function A. And minus minus is given by P minus squared over P squared, A P squared over M squared plus K right. So this is the most general solution to the worded entities. I've included three new constants beyond the naive solution. So the classical solution would be A equals B equals C but in fact we may not be able to impose this equation at coincident points and therefore I allowed three new constants, K lift, K lift and K right. Okay, and without loss of generality and without loss of generality, we may assume, I'll be done in a second, we may assume that the function A of P squared over M squared goes to zero at very high momentum. Because if it doesn't go to zero then we can absorb the constant in this case. So without loss of generality we can assume that this is true. So what I claim is that these three constants are necessary, they're physical and you can't determine them from the conservation equation. Okay, they are completely free. Now one comment before I'm done. This K here is a pure number. It doesn't multiply any function of momentum. So this is a contact term already before you hit it with a conservation equation. So this is just some delta function, a completely uninteresting delta function. While this guy only becomes a contact term after you hit it with a derivative. But before you hit it by a derivative it's a non-trivial function P plus squared over P squared. P squared is just P plus times P minus in light cone coordinates. So this is basically P plus over P minus. It's a non-trivial function and it's Fourier transform is not a contact term. So this actually does have some implications for separate points physics. So does this. So these two K left and K right are completely physical measurable quantities because they have some effect on current correlation functions at separated points. This is completely ambiguous. This you can play with whatever you can play with to your delight. You can try to set it to be one, zero. Maybe you can try to set it to be equal to this or to this or some geometric or some average of these two quantities. So this is completely ambiguous but these are completely physical. And the fact that they can be different is in violation with this naive classical conservation equation. So if K left and K right are not the same the wider than D, D mu, J mu cannot be satisfied at a coincident points. And that's where an anomaly arises. So what am I going to do next? What I'm going to do next is I will prove that if we have some RG flow or normalization group flow of that sort and we know something about K left and K right here. Then we know something about K left and K right here. So these are completely different physical regimes but you will see that some information about this case is preserved. The difference is preserved and the sum gets decreased. And we can prove it rigorously using some ideas that will take me 15 minutes to develop. But maybe I'll finish by asking if there are any questions about that. This is a very subtle business and you should be confused. So, right, so that's a great question. This is at the core of how an anomaly is creeped into quantum filtering. The conservation equations, these conservation equations classically just mean that like charge is preserved or the amount of charge that enters some region is equal to the amount that came from somewhere else. So basically this just classically means that charge is preserved. But in quantum filtering, this is promoted to a local quantum operator, a Hermitian operator in this case. And so when you write this equation, it's an equation about some operators. Hermitian operator is detecting some Hilbert space. And you have to understand what does it mean mathematically to write this equation. What it means mathematically is that when you study correlation functions of this operator, this quantum operator with other operators, this will indeed be true, but only as long as this operator does not hit an operator that sits at a coincident point. Why is that? Good, so I'll give you, I'll answer the question of why is that, which is a great question by the examples, okay? I'll give you a quick example, which is the Klein Gordon field, a free Klein Gordon field. So the equation of motion of a free Klein Gordon field is that, agreed? Classically, this equation is true. It's a set in stone, it's a true equation. But quantum mechanically, we have to interpret it as an operator equation because there may be some examples where we would not be able to satisfy this equation at coincident points. So let's see why. Remember the propagator. In four dimensions, this propagator is one over X, oh, sorry, I forgot. Is one over X minus Y squared, right? Now let's try to hit it. This are some quantum operators. Let's try to hit this equation with a box. So if this is a true operator equation, then this better vanish when X is not equal to Y. But if you take the depletion of this function, you get the delta function. This is the green function in four dimensions. So what you get is a delta of X minus Y. So you see that quantum operator equations, sometimes you just can't satisfy them at coincident points. You can satisfy them at separate points, but sometimes it's just forced to new that they have to be violated. You see, this is the simplest example one can give. Or it's just impossible to make sure that this equation would be true at coincident points. Now, usually these coincident points are dismissed as unphysical, but you see that in these cases, if I'm really careful about the meaning of this equation of imposing it only at separate points, you get this new coefficients K left and K right, which correspond to separate points physics. Only when you hit it with another derivative, they become coincident point physics and uninteresting. But before you hit it with a derivative, that's a genuine non-zero correlation function at separate points. And it was measured in the quantum Hall effect. Okay, so in the quantum Hall effect, there is such a K left is equal to one, K right is equal to zero in some sense. Okay, so this coincident point physics may some, most of the time, it comes from some other correlation function, which has the support at separate points. Like in this example, the fact that this is non-zero at coincident points can be traced to the fact that there is a propagator, which is physical. So one has to be really careful with this operator equations, because if you were not careful, you would miss the fact that there is K left and K right. And in some models, like a right moving fermion, you would not be able to write consistent correlation functions. Okay, so just to summarize, are there any other questions about that? Okay.