 So I want to continue my discussion of the Quantum filters with a global symmetry So what we assume that there is a quantum filter it was a global symmetry conserved global symmetry current j mu which Classically obeys the equation d mu j mu equals zero which in momentum space reads p plus J j minus plus p minus J plus equals zero that's the classical conservation equation which upon quantization should be interpreted as an operator equation to let me write it in words should be interpreted as an operator equation Which means That we're not allowed to impose it at coincident points, but we have to let the theory determine whether it should or should not be imposed at coincident points. So should not should only hold Add separated points separate separated points Now if you have some physical model where it turns out that you can also impose it at coincident points great Then the symmetry would be what we call non anomalous, but in general, it's not possible So this is a well-defined. So the problem of determining the correlation functions Under the assumption that this equation holds at separated points is a completely well-defined mathematical problem Okay, and the general solution to this problem is that J plus J plus I'm going to write it down again because we will now derive some very interesting consequences of These three equations. So the well-defined mathematical problem is that I want to impose the conservation equation at separated points So the solution to this problem is that this is p plus squared over p squared times some general undetermined function plus some constant Here you would get some the same function Up to some constant And here you'll get again the same function up to some yet another constant, okay If you were to impose the conservation equation also at coincident points You would have to conclude that k left equals k equals k right Which would be fine. This would not lead to mathematical inconsistencies But it would not describe some very interesting physical systems. It would just describe a subset of the possible physical systems now Let's see what happens with the conservation equation at Coincident points. Let's try to study the correlation function of p plus J minus plus p minus J plus This is at momentum P And let's say J plus at momentum minus p Okay, so this is just a simple algebra hitting it with this P pluses and P minuses here. I forgot the minus sign And I'll put a minus sign here too So it's a very simple exercise that you're you're encouraged to do by the way. I uploaded some notes Lecture notes which are very very long and they contain much more than I'm saying here But they also contain lots of exercises. So you could try to do some of these exercises. Okay, so this Exercise would lead to the cancellation of this the a would cancel out Because if I did not cancel out that because it's a note It's a non polynomial that would lead to a violation at separate points But k doesn't cancel out and we get k left minus k maybe up to a sign Times p plus Okay That's just a simple exercise Similarly, we can take the same operator and hit it with J minus at minus p And then we would get up to some sign that I can't track now we get k right minus k times by my p minus Okay, so now you see when can we impose the conservation equation at coincident points? This is a polynomial in momentum So that's good that means that we've satisfied the rules of quantum filter Which is that we are supposed to impose this equation only at separate points This is a pure contact term which looks like the plus of a delta function and this is the minus of a delta function So the conservation equation is only violated at coincident points. It's well. It's respected at separated points So you see that are two interesting situations If it so happens that k left is equal to k right If it so happens, it's true in some physical models then We can choose this k To be equal to both of them and then the conservation equation Is satisfied not only at separated points But also at coincident points You should notice first that this coefficient k is just a polynomial So it has no impact on current correlation functions at separate points Well, k left and k right have an impact. I mean they determine the correlation functions at separate points as you'll see in a second but if K left and k right are not the same Then necessarily at least one of those equations would not be possible to put to zero and therefore D mu J mu suffers from So J mu is a good global symmetry, but it has an anomaly and If we tried let's say that if we try to gauge So if we try to gauge This symmetry we would get into a contradiction because this is given by f rho sigma Epsilon rho sigma times k left minus k right So if we were to try to gauge this symmetry meaning to couple it to a background field We would get some non-zero answer and which is proportional to the difference of k left minus k right And it would be impossible to gauge. So instead of writing this equation. You can just keep it for the advanced students A more invariant statement is that if k left is not equal to k right then the current Conservation cannot be upheld at coincident points and that means that it cannot be gauged This symmetry cannot be gauged. It's a good global symmetry. There is nothing wrong with it at separated points It's still conserved, but it cannot be gauged. Okay, so that's how quantum anomalies arise in some models that There there could be this day Obstruction to satisfying the current conservation equation at coincident points. I'm just looking for the eraser Okay, are there any questions about the rules of I mean the rules which I'm using to Constrain this correlation functions Okay, now yes Yes, I'll write a bigger with a bigger font. Okay. Thanks That's trying to see how would you measure k left and k right? I want to make this point that k left k left and k right are completely measurable more clear K is obviously not measurable by separated points correlation functions because it appears only here and It's a pure contact term, but k left and k right are completely physical measurable quantities So you can do some measurements and decide if the theory suffers from a quantum anomaly or it doesn't suffer from a quantum anomaly If k left is equal to right then you can choose in our previous notation a equals b equals c So you can uphold the classical conservation equations even at coincident points Yes Right, so if k left is not equal to k right then you cannot Satisfy the conservation equation at coincident points and once you couple the theory to a gauge field That would lead to an actual violation of the conservation of the current and therefore the theory would not be consistent You would not be able to quantize the theory in a gauge invariant fashion. Yeah, I mean you can think about it in this language What I'm trying to do is to Yeah, this can be viewed as some obstruction But I'm not I mean there are many points of view I'm just trying to do something very simple because there is some physical point that I want to make so I'm going Which I'm going to do now. So let's study Let's try to understand how you would measure k left and k right. Yes Well, this was just a comment for the advanced students what I I can explain what Maybe we can discuss it in the discussions we can discuss it in the discussion sessions The quote how anomalies are related to the obstruction for gauging is something that I can explain In a lot of detail maybe in the discussion section here. I don't want to emphasize it too much I just want to show something Concrete about this k left and k right and how they behave under a normalization group flows But the obstruction to gauging is indeed something we can discuss It's a it's a somewhat Delicate story that we can discuss separately. So let's just make it clear I just want to make the point that k left and k right are measurable completely clear So let's go to very high energies meaning very short distances in the language of Position space. So this is like very very short distances Since this function a goes to zero at short distances or at high energies this can be dropped and The correlation functions that somebody who is doing experiments at very short distances The correlation functions that this person would measure would be j plus j plus is Equal to k left p plus squared over p squared j plus j minus would vanish and J minus j minus would be k right times p minus squared over p squared That's what you would measure if you were to do correlation functions at very very nearby points Now this in position space. Let's if you do the Fourier transform that looks like k left Over x plus squared this looks like k right Over x minus squared and this is basically zero Okay, so you will see that the excitations which have a plus and Excitations, which have a minus r orthogonal at very short distances While those that have a minus minus and plus plus they have this kind of correlation functions So you it's something that you can measure Now Let's discuss what an experimentalist who measures yes K is a constant. So it's a contact term, right? So you could say it's like that It's like k times a delta function But okay experimentally we would just make separated points measurements. So this is like zero Okay So now we have this picture in mind that there is a short distance physics and there is a long distance physics So I try distances. That's what you measure Then there is some complicated Function in that tells you what happens at the crossover scale, which you could also measure. It's interesting But I want now to make measurements at very very long distances where you get a new conformal field theory So we just need to take this function a and Send it and and take the limit where p squared is very small. So let's assume that this function a Goes to some new constant gamma When p squared goes to zero, so what would you measure? What you would measure is Basically, this would be some new constant at very long distances. So it's against some contact term. It's not interesting This however would modify the effective k left and this would modify the effective k right Okay by gamma. So you would say that K right of the infrared theory is given by k right of the UV theory plus gamma K left of the infrared theory is k left of the UV theory plus gamma Right, that's what you would measure if you were to make experiments at very long distances And that's what you would measure if you were to make experiments At a very short distances I don't need the UV subscript because I define them to be such that a vanishes at high energies So there are the ultraviolet k So this is the this is the formula for what you would measure in the infrared conformal field theory Now there is a magic you subtract these two equations and you see the delta k in the infrared is Delta k under this complicated renormalization group flow. We found the conserved quantity The difference between k left and k right remains the same under renormalization group flows So we have a conserved quantity under renormalization groups This is the first consequence that we can immediately draw from this careful analysis of the two-point functions This can actually be measured in experiments that this is true Another very interesting consequence that you can derive which is what I'm going to prove now is that gamma is smaller than zero Always so we have a conserved quantity along the energy flow But we also have an inequality for this parameter that relates The deep UV and the deep infrared So I'm going to give you an argument It's it's a simple consequence of this careful analysis that actually gamma is negative So let's prove that gamma is negative So I I am going to prove that I'm going to prove a certain thing that you could call a sample This I still want to preserve Yeah, this I would erase So I'm going to give a formula for gamma Which is manifestly negative up to some numerical factors that you can fix. It's in the exercise So gamma is minus The integral over all of space Of a positive definite function now this looks like an operate an operator squared So it's the norm of some operator and therefore this is positive This is positive and this is positive So therefore, this is negative Okay, so I'm going to try to explain why this formula is true Are there any questions about why if this formula is true? Then that would follow that gamma is negative So let me try to prove this formula and then I'll try to explain why this formula is true and I mean then I'll try to explain why there is a conserved quantity and why This formula is true to give you some philosophical explanation for where this results about Connecting the ultraviolet and infrared come from So let's first start with the proof Gamma is an integral over all of space Just a number It's a negative. It's a non-positive number So you see immediately that that already answers the question I mean it gives some constraint on which conformal filters could be at the end points Offering normalization group flows because there are some non-trivial constraints They have the same Delta K and it has to be true that each of the case in the infrared is Smaller than the corresponding K in the UV Okay, so how do we prove this this kind of formulas? So it's actually a simple consequence of this of this a business Let's study P plus J minus P plus J minus let's study this correlation function So we just take this guy. Sorry, we take this guy and hit it twice with P plus So we get P squared Times P squared over P squared times a so we get a P squared times a of P squared over M squared plus P squared times K, right? This is just in momentum space Now, let me take the second derivative. Let me take the second derivative of that guy With respect to momentum squared So let's take D squared over D P So here there's mu mu of this correlation function And let's evaluate the second derivative at zero momentum So therefore it has to act on P squared and P squared it can't act on the function a because if you did Then setting P to zero would annihilate it eventually. So we have to act on these two P squareds So at zero momentum we get gamma plus K, right, which is just the definition which is simply the definition of a K infrared K infrared Are there any questions about this? So this second derivative of the correlation function at zero momentum is K So now the idea is that you just write this in Fourier space How do you do a Fourier transform for something at zero momentum? You just integrate over all the possible positions. The second derivative can be translated to x squared and this is just that So the integral over all of space gives you the difference between the UV and the infrared Now I'm slightly there's a small point that I still need to explain because it would seem that the answer is just K Right of the infrared while here I claim that the answer for this Fourier transform is gamma Well, you have to be careful You have to be careful because You when you do this Fourier transform you also integrate over the point x where this coincides with that and This gives you the K of the UV So when you saw the correct integral is here to actually you're instructed to subtract the point zero where there is a delta function If you were not to subtract this point zero where there was a delta function Then you would just compute K infrared R by definition, but if you subtract this point, then you compute K infrared R minus K of the ultraviolet which is X equals to zero and this is exactly gamma So you have to be careful to do this integral over separated points and then it's true that it's positive Okay, so for this to be positive. It has to be a separate points correlation function So this is the actual correct formula that it's an integral over separated points And it follows from this Fourier transform rather immediately Are there any questions? Was it a little bit too quick? Should I explain more carefully? Gamma is an integral over all of space. Yes The what? Yes, the origin has some delta function one of these contact terms So the origin has a delta function that is proportional to K R So you can write gamma as a difference between K infrared and K UV and The integral computes K infrared when you include this point. So when you subtract this point you get gamma are there any Questions about why this is true. Okay now I want to So there are these two facts about every possible randomization group flow that gamma has to be negative and delta K has to be conserved I want to give you a philosophical Explanation for why these constraints exist and then to give a new derivation that is actually much more general and Can be applied to many other systems this kind of derivations. They are very special to two dimensions where I'm using light one coordinates and I'm using a lot the simplifications of two-dimensional physics, but there is a philosophical reason why this this this kind of constraints exist and This philosophical reasons can be generalized to higher dimensions In applications which are even more interesting than that So that's what I want to do next So we we've observed two facts one is that There is some inequality and one is that there is a conserved quantity I want to give an explanation for what is the physical origin of this of these things So the first is that we have this inequality which has the delta K Inferred equals to delta K In the ultraviolet. Well, the question is where does it come from? Why is it true? one There are several ways to understand where does this come from and one explanation one intuitive explanation is that You see if K left and K right are different then Right-moving modes and left-moving modes are not quite the same Now you may have this intuition that when at the crossover scale You only have massive degrees of freedom and massive degrees of freedom by definition have left-moving modes and right-moving modes So if you have a massive fermion in two dimensions or a massive scalar It has both right-moving and left-moving modes. So what this equation is telling you more or less is that there might be some imbalance in the degrees of freedom that propagated the speed of light because the degrees of freedom that propagates in the speed of light can be either Right-moving or left-moving But at the crossover scale where you encounter massive degrees of freedom They move both left and right in some sense in the same, you know In the to the same extent and therefore the difference between modes that are moving purely to the left or purely to the right Is conserved because at the crossover scale you only only modes that move both left and right This is a very vague physical explanation for what this equation means Now more technically it can be understood according to a tuft So a tuft gave tuft gave a general way To understand where this conserved quantities come from The first observation that hoof makes is that if k left is equal to k right in the ultraviolet Then we can get then the same then the current is conserved also at coincident points and Therefore it has to be conserved at coincident points also in the infrared a More general way to say it is that if the current is conserved at coincident points We can gauge the corresponding symmetry We can gauge that we can couple this current to a gauge field and gauge it and since we've done that in the defining You know since we've done that the short distances nothing can go wrong at long distances because the long distance physics emerges from the short distance physics So that's a general argument that who gave and you can quantify it by saying that even Even if you couldn't you couldn't gauge the symmetry the amount by which you failed should be conserved along During normalization group evolution. So this is an argument due to a tuft That Basically says that if it's conserved at coincident points, it still has to be conserved at coincident points even in the deep infrared so The other the other part is Why do we have this inequality? What is the philosophical origin for this inequality? And I would like to propose an explanation for what's the physical origin of this inequality So I'm going to try to explain that this additional inequality exists because there is an enhanced symmetry There is an enhanced symmetry in the UV And in and in the infrared so I'll make now a long argument Explaining why these inequalities are deeply connected to some enhance symmetries that appear here and here and If you understand this argument, you will be able to understand why this kind of inequalities exist in many many other situations I believe it's always due to this enhanced hun symmetry So let me try to make the argument but before that are there any questions about the conserved quantities That the arguments for them are due to tuft who said that if it's conserved at coincident points in the defining theory It has to be conserved at coincident points also in the emergent theory at long distances This is the essence of tuft's argument. Are there any questions about it? Okay, so I assume you all know this argument Now I'll try to explain why this inequality is deeply associated to enhance symmetry So for that for that I would like to consider these equations again So these equations you remember where they came from they came from imposing this conservation equation And this was the most general solution to the conservation equations Now what I would like to claim is that if you are an Experimentalist who only observes the deep UV or the deep infrared for example you compute correlation functions at very long distances Are very very short distances. You don't have access to the crossover scale You would actually find that there is another conserved symmetry. Let me write it down at the deep UV or deep infrared There is another conserved symmetry. There is another conservation law What is that? What is the conservation law? It's the statement that not only D mu J mu vanishes, but also D mu epsilon mu mu J mu vanishes So as an operator equation again, this is an operator equation So as always this means up to coincident points So I claim that there's another conservation law at short and long distances and I will try to explain Why this entails an inequality? Why this entails this inequality? So let's see how to prove that this is true So let's go again to very short distances or very long distances It's very short or very long distances. So at short or long distances Recall that the correlation functions simplify dramatically. We have J plus J plus equals P plus squared over P squared times K lift J plus J minus vanishes and J minus J minus is P minus squared over P squared times K right This were the correlation functions that an observer has very long or very short distances would see So if you are at short distances, you use the case of the UV If you are at long distances, you use the case of the infrared Now these equations are consistent with two conservation laws one for the original current and the other for the dual current Because essentially That what this equation means in light-con coordinates So P plus J minus plus P minus J plus Is our original conservation law that we started from and then there is the conservation law that is emergent Only at very short and very long distances. It's an approximate conservation law Which reads P plus J minus minus P minus J plus vanishes. So there is a crucial minus sign here You see that if this is a bed, then this is a bed trivially at very short and very long distances because if you flip the sign of J minus this set of correlation functions Transforms to itself So if you just flip the sign of J minus, which is what you are instructed Sorry, if you just flip the sign of J plus which is what you are instructed to do here This remains invariant But if you are at the crossover scale at the crossover scale this correlation function does not vanish the mixed components And therefore flipping the sign would lead to disastrous consequences. So this symmetry is not conserved at the crossover scale It's an emergent symmetry that appears at very short distances and very long distances Very much like the conformal symmetry itself. Okay, so it's very much like the conformal symmetry. It's an emergent symmetry So it's a symmetry of the short distance physics that is violated explicitly at long distances So and so this is violated by this function a of p squared At the crossover scale, so it's not a good symmetry in general. It's just emergent Okay Now I will try to show that From the fact that this emergent symmetry exists You can derive in a rather direct way this inequality just from this fact and Then you could repeat the same logic in many other examples So this is beyond Hooft. Hooft just defined this conserved quantities But the claim here is that if there are enhanced symmetries at the fixed points Then you could derive inequalities And so you would you could put interesting constraints in the renormalization group flows So let's do that. I would like to use that fact to Prove an inequality, so I would do it in a lot of detail. It's a very interesting exercise, I think Which has a lot of interesting physics to go from yeah So it has a lot of interesting ingredients to go from that observation to the inequality But it's very very general because one can follow the same footsteps in many other examples Okay, so How do we start? We may consider Let's do two in two steps. The first step I use I'll use just letting okay the first step that we will do is to couple this current J plus and J minus to a gauge field So we are going to deform the action by a plus J minus plus a minus and J plus Okay, so this is an external gauge field. It's not a dynamical gauge field. It's the one that Marco Asked about so this is an external gauge field which I'm adding to the theory It's just a source that I can use to study various correlation functions of J Now this this come now comes the part that I promise to explain Now the partition we can define the partition function of the theory Which is a logarithm of so let's define the logarithm of the partition function Now this is going to be some function of the gauge field A plus and a minus and this is defined to be just the logarithm of the past integral of Minus the action over all the quantum fields Plus this the formation term So we have some partition function that we have defined as a function of these sources This is defined throughout the RG flow. It's an object. That's completely well defined throughout the RG flow I'm not specifying that it's defined only in the UV or only in the infrared. It's everywhere defined Now this partition function has some gauge symmetries So it doesn't depend on a plus and a minus in a completely general way If you make a gauge transformation, then it might go to itself So for the sake of simplicity, I'm now going to make a small assumption You can generalize this assumption, it's not going to be crucial I'm going to make an assumption that K left and K right and K are all the same So I'm going to discuss only a subclass of models But this is enough because I'm interested in this inequality and the inequality will still say something about the infrared Case compared to the ultraviolet case So I'm going to make this assumption and then prove that the infrared case are smaller than the ultraviolet case But I'm making this assumption just for simplicity So now Inspired by this fact that there is that there are two conserved currents in the deep UV and the deep infrared We're going to define two possible gauge transformations a plus going to a plus plus d plus omega Plus d plus nu omega nu are some functions and a minus Going to a minus plus d minus omega minus d minus nu So I define two gauge parameters One is the standard one that you know from the books and the other one has a funny sign Compared to the other one So this has the funny sign one of them Corresponds to this conserved symmetry to the original conserved symmetry and the other one corresponds to the enhanced symmetry that only exists at the fixed points So we're going to analyze. How does the partition function behave under these two gauge transformations? So the first claim is that under the omega gauge transformation the partition function is perfectly invariant This is the exact claim Why is that true? First because I impose this equation that k left is equal to k right and therefore there are no gauge anomalies So the current the original current can be conserved at coincident and at separated points So it's perfectly fine. These gauge transformations are perfectly fine. They act like you would expect There are no quantum anomalies Under the new gauge under the new gauge transformations the partition function would not be in general invariant There are two reasons why it's not in general invariant One is that this symmetry is just an accidental symmetry or an enhanced symmetry at the UV and infrared fixed points So it's not an actual symmetry of the partition function. So there is no reason why that should vanish and the other reason That it doesn't vanish is that in fact even At the original fix even at the deep UV and in the deep infrared this emergent symmetry is Upheld at separate points, but that coincident points. There is a contact term I would I'm not I'm going to explain this contact her now But let me just write the equation and then we'll explain the contact term so the equation that you get for this gauge transformation is going to look like D to X nu Epsilon mu nu F mu nu Where F mu nu Or we can just write F plus minus for the sake of concreteness and the coefficient is going to be K Which is the same K Plus many many other terms That come from the fact that the crossover scale The physics is not invariant under the symmetry. So it's just an enhanced symmetry. This is an object that would exist even at the fixed points and These terms come from the fact that this is not a good symmetry along during normalization group flow Now F plus minus is defined to be just D plus a minus minus D minus a plus It's the usual field strength that you know So what I would like to do is to explain this term Which is there even at the UV and infrared fixed points where the symmetry is indeed an enhanced symmetry But this is the usual toothed like anomaly Are there any questions about the logic? Okay, so this is the usual toothed like anomaly and I'm going I'm going to explain where this term comes from now in detail So we're going back to our favorite correlation functions and I'm going to write them just at the fixed point it could be either It could be either the ultra valid or infrared fixed point But let's for concreteness right this correlation functions again at the UV fixed point So J plus with J plus is a P plus squared Over P squared times K There's only one K now. So I'll reserve the symbol K for all the three J plus P J minus P is just K. So it's a pure contactor and J minus P J minus P is P minus squared over P squared K So this is the calculation that we already did many times. These are our correlation functions And now you observe Now you make an observation the observation is that P plus J minus plus P minus J plus With I was J plus or minus doesn't matter. This is a P. This is at minus P This vanishes This is the statement that we have chosen K left equals K right equals K And therefore we can respect the conservation equation even at coincident points This is the first observation the second observation is that for the enhanced symmetry this is not true For the enhanced symmetry we get P plus J minus minus P minus J plus with J Plus minus minus P Is this visible to everybody or it's too low Can you see it from upstairs? Okay, good So this is actually not vanishing. You just look at this equation to do some algebra and You will find that this is equal to P Plus minus times K up to some coefficients that I'm not careful about Good, so what does it what does it mean? It means that even though it is physically true that the Troy distances and at long distances There is a new enhanced symmetry and you conserve approximate conserve charge This charge has some contact terms which don't vanish necessarily. They're proportional to K This means this means this means in a You know in the technical term in technical terms that this Approximate enhanced and symmetry has a tuft anomaly as a tuft anomaly So it's a good conserved symmetry But it has an anomaly It's not an anomaly that renders the current ill defined or not conserved it's just And it's just something that means that if we were to couple the currents J plus and J minus to gauge fields Then the constant then this current the enhanced current would no longer be conserved So this implies this equation that I can explain in the discussion that D Mu J enhanced, so this is the enhanced symmetry. It's not the original current This is equal to f rho sigma times epsilon rho sigma Where f is a background gauge field So this is a background gauge field that we couple to the system and there is a coefficient K however D mu J mu For the original current is identically zero because there are no contact terms. It's not anomalous So the enhanced symmetry is there. It's nice and good There is a new conserved current and short and long distances, but it's actually anomalous necessarily anomalous It has a tuft anomaly with this coefficient K What it means is that when we couple it to background gauge fields, then there is this constant equation that violates it So this is why when we perform a gauge transformation of the partition function and they're new And there is gauge transformations that are parameterized by the function new Then we get this term even at the fixed points where this is a good enhanced symmetry and then there are lots of other terms that come from the fact that We also violate the symmetry explicitly by this function A that I've already erased So it's an enhanced symmetry that appears at short and long distances, but across the cross in the crossover scale It's completely violated. So there are lots of other terms Okay, how much more time do I have? Okay, so let's discuss this equation in the discussion section In the meantime, I was planning to Give it as is and proceed to explain how we go from this idea to this monotonicity equations to this monotonicity relations So now we draw this picture again that we have some K uv and some K infrared and our partition function Has lots of terms In divide if we do a new gauge transformation. We get lots of terms. It's very complicated But there is a huge simplification due to the fact that we know that these terms must go away at short distances Okay, this is an input that is very very crucial to bear in mind The partition function is well defined everywhere along the RG flow It can be defined at short distances and then the rest follows because long distances emerge from short distances So at very short distances, we know that the partition function satisfies this equation Plus minus d2x with the coefficient being K This is at short distances and now comes a very non-trivial idea Which is Known under the name of anomaly matching So we know that the partition function in general will have lots of other terms But when we go to short distances, they disappear because the short distances. It's a good conserved symmetry This is true everywhere Not just the short distances and this is true only at short distances But now comes the idea that you know once you define the physics at short distances It should somehow define everything else as well It should define the whole the whole It should define the whole physics at all the lens scales And so all these other terms that you have to add should emerge from renormalization group transformations And they go to zero when you go to short distances or long distances in some sense So the idea is that the partition function can also be studied at long distances and it should satisfy the same rule because this rule is sort of God given at short distances and then at every single at every energy scale This should be reproduced in some way By the dynamics of the theory because once you specify some property of the partition function at short distances The rest should organize itself. You can write this equation schematically that there are some m squared over p squared So it's very very short distances when p goes to infinity these terms die away And so the dynamics would somehow organize itself that this transformation rule would be true at every energy scale How do we make it precise? How do we make this idea that somehow The transformation rules for the partition function should be the same at every energy scale because by definition block spin transformations Or renormalization group transformations. They should not change the partition function They should reproduce the same equations So we need to make this idea Mathematically precise and that's the next challenge. So the idea is that Somehow This equation The d new log z is equal to k and new f plus minus d2x somehow this equation should be Last terms that are of order one over p squared where p is the momentum Would be respected everywhere So the point is that these terms are Very small at short distances by definition because we have a good in hun symmetry But we when we go to long distances when we want to understand the physics of this energy scales These terms become pretty large because they're of order one over p squared And you will have to resum them and it would seem to be a very very complicated process But we know what the answer should be In the deep infrared This symmetry is again conserved, right? So d new log z Should be k infrared times new f plus minus d2x But now there are terms of order p squared So these two equations are mathematically correct We know that the partition function has some complicated transformation rules and it's the same partition function Here we can approximate it by this term plus terms that go away at short distances And here we can understand we can approximate it by this term plus term that go to zero at long distances And this case are not the same This case are not the same and what we're interested in is exactly the difference between this and this We claim that this difference is positive and it has to do with the physics of this term that we've thrown away And we want to understand how they interpolate between this and this So that's the challenge. So the idea is that we need somehow to find the mechanism to control these corrections If we can find the mechanism to control these corrections, we might be able to understand Why the difference has decreased? Of course, I've already given you a proof that this gamma is negative But a proof from this point of view is much more useful because it's easily generalizable to many other situations While the previous proof is very unique. It cannot be generalized to other situations So we need to find them away to control this this Order p squared or order one over p squared terms and this and now comes the main idea That we're going to add another another background field, which I'm going to call the pile For historical reasons. I'm going to add another background field That is called the pile. So now the partition function is going to be a function of a plus a minus but also another field pi of x and I have to tell you how am I adding pi of x to the system. How am I throwing pi x into the mix? So a plus and a minus couple in the way that we have explained But what do I do with pi of x? So first of all, I'm going to take pi to be invariant under under omega gate transformations And under new gauge transformations pi is going to transform linearly So like maybe you've seen in QCD the pions transform under axial gauge transformations in some inhomogeneous fashion So I'm going to add this background field. It's not a propagating degree of freedom It's just another function of on which the partition function depends I'm going to add it to the action with some I'm going to add it into the mix with this specific transformation rule and The best way to explain how this is added to the action is by an example. So I've been doing a lot of I've been doing a lot of formal stuff, but now let's do an example, which is a free fermion I want to do the example of a free fermion in in two dimensions and Show you how this you want symmetry emerges how there is an enhanced symmetry and how do I define pi and From this example, you could understand how it's defined more generally and then the discussion that will follow will be more clear So the example of a free fermion The Lagrangian Free fermion with a mass the Lagrangian is psi plus d minus minus psi plus plus psi minus d plus plus psi minus Plus a master. This is just a free massive fermion complex. Let's take it to be complex Plus a complex conjugate Okay, so this would be a single complex fermion with right-moving degrees of freedom and left-moving degrees of freedom And there is a master. You see that the master couples right-moving modes to left-moving modes if you don't have mass then right-moving modes and left-moving modes are independent and that's What I said before that you would have in balance of massless degrees of freedom Now my u1 symmetry is going to act like psi plus and psi minus Going to e to the i alpha psi plus and psi minus This is a good u1 symmetry That is conserved by all the terms in the Lagrangian because this is and this is also invariant This symmetry is the one that has this kappa left kappa right and in some conventions in this example There will be kappa left equals kappa right equals one So that in this third it's parity even so there is a One left-moving mode one right-moving mode. So kappa left k left and k right are the same and then we have this u1 Axial symmetry which is not exact But it becomes conserved the Troy distances. So how does it act? It acts by taking psi plus and psi minus 2 e to the i alpha for psi plus and e to the minus i alpha for psi minus Okay, so you see that if you throw away the mass term it's a good symmetry So Troy distances it becomes a good symmetry But once you include the mass term The mass is not invariant under this symmetry. It has charge minus it has charge 2 Under this new symmetry. Let's call this parameter beta not to confuse with alpha Okay, so you see that once we add the mass this symmetry is broken But if the theory is massless then this theory is preserved These two Symmetries are related exactly like what I explained that this is generated by some current j plus j mu And this is generated by the current epsilon mu nu j nu So this is the approximate symmetry that I was speaking about Now the current of this axial symmetry Is not conserved if you gauge this u1 and you are starting qed Then this current would not be conserved. It would be given by k by f By star f where f is the gauge field. So this is the anomaly equation that I wrote This is in the massless case if there is a mass then this current is not conserved even before you gauge the symmetry good, so What I wanted to say now is that there is the following trick So we want to understand this enhanced symmetry a little bit better So we see that it's conserved the short distances and it's violated by the mass term So when you go to the crossover scale, it's completely violated But the trick is that Let's couple This Lagrangian to some To some background field. So I'm going to put here e to the two I pi Okay, so I'm going to add in front of the mass some function This is like making the mass into a space time dependent mass If you don't like to add this additional parameter, you just imagine that you instead of starting a fermion with constant mass You study a fermion with a space time dependent mass And then formally If we make this transformation And we accompany this transformation By a shift of the pylon then we get back the same Lagrangian Of course, it doesn't mean that we have we have the symmetry Because i'm changing some coupling constants, right? So there are some coupling constants which break the symmetry But what i'm doing is to assign to this coupling constants some transformation rules that would As if restore the symmetry This is what sometimes is called sporean analysis that you think about coupling constants that break the symmetries as functions And then you assign these functions some transformation rules and then you can restore the symmetry The simplest example of that is if you do quantum mechanics of a particle moving in an electric field So let's just think about quantum mechanics where the Hamiltonian is like something of that sort some particle moving in some electric field Then the electric field breaks rotational invariance But it's still useful to think about rotational invariance that acts on x and on the coupling constant e Because then the energy levels of the systems of the system are just function of e dot e right So the energy levels are still just functions of the scalar product of e with itself So there are s o 3 or s o d invariant even though this coupling constant breaks s o d So assigning coupling constants some transformation rules in this way is a very useful idea to keep track Of the symmetry of how the symmetry is violated So that's are there any questions? I I I didn't hear the question but what I'm trying to do Is to make all the math so the disenhance symmetry Is violated by two sources One of them is just this anomaly that I explained there So one of this one of the sources is just this anomaly Which exists at the fixed point and you can't do anything about it The other source which violates disenhance symmetry is the crossover scale physics like these mass terms And those you can get rid of or a book keep off By adding this additional background field that transforms under axial symmetry And it helps it it helps to just keep track of the symmetry violation So from this comes the main claim, which we will analyze in the next lecture So what's the main claim? The main claim is that if you do that Then the transformation property of the partition function, which is now some function of a plus a minus n pi This is now given by k F plus minus D to x and this is now true exactly Where by definition delta nu of pi is nu So this is the main claim that by doing this trick The partition function, which now depends on the background gauge fields and this pylon background pylon field Now just says the anomaly all this explicit breaking term due to the crossover physics Have been booked by this additional field pi that transforms Is that Right, so this is an excellent comment. So how do you couple pi to the Lagrangian so that this would be true You take your Lagrangian There's like a bunch of mass terms and each mass term you just multiply with some power of e to the i pi So that when you do a transformation axial transformation, it's soaked away It's soaked up by this pi Now you could say that this is model dependent, but you can actually give a model independent description You can You can say that pi couples to the divergence of the axial currents The axial current has an anomaly and it has some violation due to these mass terms. So you just couple pi to the Divergence of the axial current in flat space So this is the general description But this trick allows you to get rid of all these terms that violate this in hun symmetry And now you have a good now it is as if you have a good symmetry along the flow And the next step is now that we have an exact equation. We can try to compare it with the approximate equations So this equation is true approximately at high energies and this equation is true approximately at low energies So we have to understand how this exact equation Uh is related to this approximate equation and this is how this constraint on gamma will emerge I have five more minutes, right or not Okay, should I take five more minutes? Okay So this is just to bring to finish this point So this is now an exact equation and We have a paradox now So now we have a paradox Because we have an exact equation and it seems to not agree with this approximate equation So our exact equation agrees with this relation because at very high energies This is what we get but at very low energies. We should get k infrared Well, our exact equation has k of the uv So now there is a paradox because the approximate equation Which becomes exact at low energies doesn't agree with this exact equation And the resolution of course is that we added another background field So The fact that we added pi of x Explains y Delta knee of log z Which just depended on a plus and a minus had the k infrared in front f plus minus d2x So this explains why this is true before we added the pi field the point is that the log of z of a plus a minus and pi becomes in the infrared Some In the infrared it can be decomposed into two terms One of which is just the partition function of the infrared CFT This is the partition function of the infrared CFT So this is the partition function of the infrared CFT whose variation under nu gives exactly that But then you need another term because the exact Transformation rule for the partition function. Where was that? Yes, the exact transformation rule for the partition function is that so you have to add another term Which would make up for the difference So it has to be proportional to k minus k And this is exactly what we were looking for right some constraint on this guy And so we have to write some action here we have to add some term Which would soak up the difference correctly So let me just write it abstractly We have to add k u v minus k ir times some action some local action That depends on the pion And a plus and a minus Such that its variation such that delta nu of s local Exactly nu times f plus minus So if we could achieve that then there would be no paradoxes Because when we turn off the pion this will go away This will remain and this would correspond to this approximate transformation rule that we found there But with the pion the full the full transformation rule would be what it should be That this is the exact result So the only remaining part is to determine this action this local action which soaks up the difference You can think about this term as coming from this massive degrees of freedom that violated The axial symmetry along their g flow And so they generated some local effective action for this background field So that they would soak up the difference in the anomaly the way to determine this local action Is uh simply from this transformation rule this transformation rule fixes it uniquely So your first guess would be to write pi Times f plus minus D to x right this is the first guess Why this is a good guess? Because remember that d nu pi is nu So if we do a d nu of this equation we get nu times f plus minus which is more or less what we need But this cannot be the full answer Because uh remember that under gate transformations D nu a nu of a plus is d plus nu and d nu of a minus is minus d minus nu Therefore f plus minus is not invariant that there are new transformations because of this funny minus sign D nu of f plus minus Is just box of nu So this is not a good answer. We have to fix it a little bit We fix it by adding another term pi over x box of pi over x And now you can check that all the mix terms cancel out and this local action exactly Reproduces this transformation So now everything makes sense now we have uh Some theory where there is an approximate symmetry. It's violated But we have managed to understand the extent to which the symmetry is violated by some local effective action That depends on a new background field pi And now it's more or less straightforward to derive that this difference is positive why because Let's now take a plus equals a minus equals zero This is the original flat spacemen coughs can theory that we're interested in after all So then this goes away and we have something that looks like a kinetic term So in the action we have k u v minus k in for red multiplying a kinetic term Something that looks like a kinetic term, but it's for a background field So the point is that the kinetic kinetic terms for background fields And the kinetic terms for dynamical fields obviously they have to be positive definite Because they correspond to some correlation functions at separated points And so this has to be positive. Otherwise you violate your notoriety I can explain more rigorously why you have to Demand that kinetic terms for background fields are positive But they have to be positive because they correspond to some separate point correlation functions But loosely speaking it looks like a kinetic term. So it has to be positive And this is how it follows. So the so I just no more equations Just to recapitulate recapitulate the idea. I think it's a non trivial not not completely trivial story. So The general idea is the following It works in many many other cases, but I've just explained this case here You have some complicated rg flow some quantities are conserved the quantities that are conserved were more or less classified by a thuft They correspond to some Conservation equations that hold at separate points versus coincident points But then some quantities Decrease And the cases in which we can control this decrease Is when there is an extended symmetry When there is an extended symmetry We can try to account for the violation Of this extended symmetry due to the crossover scale by adding new background fields And then finding a local effective action for this new background fields And in many cases it turns out that they have to be with positive coefficients So the general physical picture I think is that If there is some symmetry along the rg flow you have conserved quantities due to a thuft But even if you just have extended enhanced symmetries at the fixed points You can still do something which is very similar to what a thuft did But you get constraints which are inequalities rather than equalities So that's I think the general picture and it works for many examples that I will list I'll make a list in the next session of all the cases where this logic can be repeated and it gives new results And then we'll discuss a new topic Okay, thanks