 So, topic of this week lecture is causality conformal field theory. So, today I will use the tools that I developed last in the last lecture to impose constraints on conformal field theories, mainly three-point functions of CFTs in D greater than 2. And this is outlined for the lecture. Yeah, and by the way, Sanjipan, the advantage of having few people is that you can be, you don't have to be as pedagogical. Yeah. Because the audience, the average level of the audience is quite advanced. First, I'll review causality, basically the first lecture, but keeping CFTs in mind. Second, now I can be, can give you a lightning review of conformal block expansion because I'll need these notations. Next, I'll introduce shockwave states in CFT and study causality in this kind of states. That will lead to constraints on OP coefficients. And then I'll conclude. I'm vicious. No, the first two shouldn't take that much time, so probably it should be fine. So, let's start with our favorite example. This is some direction y. This is Lorentz and time. There is an operator over here, O1. There is another operator O3 at y equals to 1. And then a fourth operator O4 at y going to infinity. So, this is the light cone. And the second operator O2, it's somewhere here with, so this is just for convenience. Let's introduce complex coordinates. So, to compute something like this, to compute a correlator like this, a Lorentzian correlator like this, again, first we should start with the Euclidean correlator. So, this will be a function of y2 and the Euclidean time tau 2. And then, again, we'll have to perform erratic continuation for tau 2. So, to get to the Lorentzian correlator, we have to analytically continue tau 2 to I t2. So, in the last lecture, I showed that in the complex tau 2 plane, there are singularities and branch nodes. So, we'll have to analytically continue somewhere here. So, we'll start from somewhere here. So, this is the Euclidean time and so this is the Lorentzian time. If I take this Euclidean correlator and perform this simple erratic continuation, let's call this A. So, that will give me the time-ordered Lorentzian correlator. So, this is, in practice, what we are doing is the following. We are starting with the Euclidean correlator writing that in terms of z and z bar. And then, first, we find this function in terms of Euclidean z and z bar. Then, we just replace tau 2 by I t2 and just replace this z and z bar by the Lorentzian value. That will give you the time-ordered 4-point function. So, this is A. I should mention one more thing. So, this singularity corresponds to a singularity at z equals to 0 and this is at z bar equals to 1. So, in terms of z and z bar, that's where the singularities should be. Now, the second thing that we can do is the following erratic continuation. We start from here. We pass this singularity from the right, but pass this singularity from the left. From the last lecture, this is O3, O2, O1, O4. So, O2 is time-ordered with respect to O1 and anti-time-ordered with respect to O3. In practice, what we are doing is the following. We are starting with the Euclidean correlator. So, whenever we cross a singularity from the left, we have to do something with this Euclidean correlator. In this case, what you have to do is the following. Take z bar minus 1 and rotate by 2 pi I. So, take this function, replace z bar minus 1 with this, then plug in the Lorentzian values. So, whenever we cross a singularity from the left, that's what you have to do. Let me give you one more example which will make it more clear. We can also do something like this. We computed this thing last in the last lecture. This gives me something like this. So, if I want to compute this from the Euclidean correlator, we should start with the Euclidean correlator. Now, we are crossing this singularity from the left. So, if you can just check that corresponds to this. So, first, take the Euclidean correlator, rotate z around 0 and then plug in Euclidean values, sorry, plug in Lorentzian values. Yeah, you can just plug this thing in and in Mathematica, you can just check that that's how ZNC bars. No, no, no. What I mean is that it's not obvious to me that this procedure corresponds to what we are supposed to do. Can you convince us a little bit? One easy example one can do to convince himself. It's basically just, you can forget about this prescription. You can just do the I epsilon prescription and find out how ZNC bar behaves when you change the ordering of I epsilon. We'll see that one will rotate z, one will rotate z bar, things like that. So, this is just like this. You can just check. Or I can give you one more, I guess, not example, let me. So, let's say I have a branch cut somewhere here. I can choose either this path or I can choose this path. What I can do instead of doing this, what I can do is the following. I can just go here and do a rotation around here like this and then follow the previous path. So, it's basically, it's, it will give you a rotation of something. It's basically, this is that rotation. Or in I epsilon prescription, it's basically, you're just changing the ordering of epsilon. And because of that, you'll get some kind of, in ZNC bar, if you just check, you'll see that they are not the same. I don't understand is that in general, the correlation function will not have a cut. This line that you're drawing is going to be just the end of the original unitistic. So, what does it mean to take and rotate? Is this a prescription which applies only to CFTs and the some extra assumptions and so on. So, in general, in general, you really have to follow this path and that's it. And there is no other trick. Okay. So, probably in some periods you can use this trick. No, actually this, so this is valid for any KFTs. Probably the branch cuts are confusing. So, let's, let me just ignore the branch cuts. So, I just have some singularity over here. Then the, it's basically, the analytic continuation is whether you are going, whether you are landing on the imaginary line from the left of the, of the singularity or from the right of the singularity. It doesn't matter there is a branch cut or not. So, these two are not the same thing. So, that's, the branch cut basically is telling you that these two functions, they are not the same functions. So, now the difference between these two analytic continuation is basically, I can just replace this by something like this. It's basically, you can, or these two analytic continuation, they are basically related to each other. So, this, this procedure over here, it's, it's a cartoon for doing something like this. But you don't, you don't have to do that. You, you can just forget about this one. You can just check this one and this one, they are, they are related by something like this. In the shock waves, I'll have one example. Okay. Let me just finish this analytic continuation. The fourth one will be this one. Now, here there is a subtlety. So, first, so this is O1, O3, O2, O4. First, because we are crossing this singularity from the left, you take the Euclidean correlator. So, this is something you can just check with mathematics. That corresponds to this. But then you have to, you are also crossing this singularity from the left. Oh, no. This is not it. Like, I'll, there'll be more things. So, this is the first thing you have to do. Okay. There is a comma. So, now next you guess that you should also rotate z bar minus 1 in this fashion. But that's not exactly true. Because after you rotate z by 2 pi i, this is a different function. So, this might have a singularity at a different place. So, if I take this and then I have to check where this singularity of z bar equals to 1 is for this function. So, let me just define that this is singular at some point which is z bar equals to z not bar. Just, it can be 1 or it might not be 1. So, just, I just use this z not for that. So, in the next case, you have to rotate this. So, that's the fourth one. So, to perform this second step, you first of all, you have to find out where this z bar, z not bar is. Now, the statement of causality is the following. Why can't you just start at, instead of politically continue from negative tau 2, then you just start from the negative, continue from, instead of positive tau 2, you choose negative tau 2, you rotate and then this second plus step is completely equivalent to the first step. It's as trivial as for the first step. You just politically continue and you don't need to rotate anything. Okay. So, if you want to come from- You land directly where you need to land. So, okay. So, that, if you, the whole thing is- This can be, it seems to me that this last situation can be reached in a completely straightforward fashion from a Euclidean correlator without having to rotate anything, just like the first situation. You just have to start from the negative tau 2. That might be the case, but I'm just worried about one thing. I'm not sure in some step, if we assume positivity of tau 2. So, if we start from the assumption that tau 2 is positive or not. So, probably that might, I'm not 100% sure right now. Probably like then something else can happen. So, then probably the, there might be subtleties. I'm not sure of that right now. Okay. So, then the statement of causality is this z0 bar. This cannot be below z bar equals to 1. So, the statement of causality is z bar should be greater, z0 bar should be greater than or equals to 1. So, this singularity can move in the upward direction, but not in the downward direction. Now, if somebody can show that this function is analytic in this region, let's say, then that should be equivalent of showing that these correlators are causal. So, the reason I made the statement is the following. So, in practice, you can translate the statement of causality into a statement about analyticity of the correlator. Let me now move to the second review, conformal block expansion. In conformal field theories, we have operator product expansion. So, what we can do, we can use OP's inside a four-point function, and that leads to conformal block expansion. Just to give you one, just to set all the notations, if I have four scalar operators, conformal invariance tells us that this is, this can be written as in the following way. Okay. So, let me explain all the notations. So, in this case, x12 is x1 minus x2, del 12 is or del ij is del i minus del j. And here this, so basically what we are doing is the following. These are all the primaries of the theory. We're writing down this four-point function as a sum over all primaries. And these are OP coefficients of the scalars with this primary operators, which has dimension delta p and spin lp. And this function is known as, this is a known function in principle. These are conformal blocks. z and z bar are given in the following way. Let me define u and v. Then z, z bar equals to u. So, this conform block is the function of these z and z bar variables. p is basically all the primaries of the CFT. Okay. Now, let me take a special case of the above four-point function. So, now if I have something like this, let me use the notation g of z and z bar. So, what I can do, I can expand in different channels. So, far everything is Euclidean. And for an Euclidean correlator, you're allowed to change orders. So, in other words, you can expand in any OP channel that you want. I can expand in s channel, which is basically the channel where I'm expanding in this OP. Then, g and g bar is given by, or this correlator is written in the following way. No. As long as there are different operators, that should be fine. In picture, you can draw a diagram just to use as a cartoon for this full expression. So, this is s channel expansion, where we are summing over all primaries, primary exchanges. Similarly, you can expand in this channel, which we'll call t channel, which will be almost identical to this. So, I'm not going to write that down. I'll just write down. I'll just draw the corresponding cartoon. So, this one is basically this. And similarly, there is u channel, where you expand in this OP, o and psi. Let me not write that down. It's almost similar to s channel, but it's radius of convergence is different. C alpha, what does it mean? C alpha. Is it alpha? No, on the top. This one? No, no, no. Oh, sorry. This is infinity. Sorry. Crossing symmetry tells us that these different ways of expanding the collator, they should be the same. It's basically permutation invariance. So, the statement of crossing symmetry is s channel plus t channel. This looks simple, but as Slava and his collaborators showed us 10 years back, that this crossing symmetric can actually be a very powerful tool to constraint safeties. In recent years, people use crossing symmetry both analytically and numerically to derive various constraints. But what we have to do is a little different. We have to use crossing when some of the operators are time-lapse operated. And that's that can be subtle. Now, let me move on to deriving actual constraints. For that, I need to define something like a shockwave state. This is the motivation comes from gravity, mainly from the paper by Kamann Ho, Adelstein and Malasina, Xavier Dove. But it doesn't have anything to do with holographic safeties. This is true for any safeties. So, what we want is something in safety that looks like a shockwave. So, we define this state. So, we take some scalar operator, insert that operator in Euclidean time delta, but at the center of the spatial directions, connect that on the vacuum. So, then I'll take the limit delta goes to 0. The reason for this offset in the Euclidean direction is because the norm is now finite. And the reason this is called a shockwave state is the following. What I can do, I can take the stress sensor operator and compute its expectation value in this particular state. So, what you can show is the following. So, this expression value in the limit delta goes to 0. It has support over an expanding null shell. So, it's if this is origin, this is my time, this is any of the spatial direction. It has support only over a null shell. So, basically it's a delta function in the limit delta goes to 0. So, that's why it looks like a shockwave. That's why it's called a shockwave state. And what we want to do is basically study causality in this particular state. And as I'll show that, this actually gives you non-trivial constraints. Excuse me? What's the coordinates for the stress tensor? Oh, I mean, is it Lorentzian or Euclidean? Yeah, it's Lorentzian. Yes. So, but in your first papers, I think you did not use this shockwave state. You just used this kind of correlation function. No, we used the shockwave state. Yes. The reason is basically you need to probe z and z bar. Even in the Lorentzian case, you need z and z bar to be complex, not real. But there, you insert the papers at 0 and infinity. So, it's a very different situation. And still, you said that there should be a constraint that the singularity occurs at z bar less than 1. So, that constraint should be ready also for this correlator. Yeah. So, there is a constraint for that correlator, but that doesn't give you anything non-trivial. This one gives you some, like this one actually gives you some condition on the OP coefficients. So, this shockwave state. So, that condition, you cannot translate that, like that's, the reason is basically you can actually check that that condition is, it satisfies, it's very trivial actually. So, for shockwave state, that condition is non-trivial. Okay. Now, what we want is the following. We want to, we, so this is psi insertion at the origin. Then I have, then I insert another operator O over here. Let me use the coordinate x2 for this one. And this is the light cone for that operator. And there is another O, which is somewhere here, let's say, at x3. So, this O is time lag separated with respect to size, and, but space lag with respect to this O. So, if I compute this correlator, that should vanish because x2 and x3, they are space lag separated. So, that's the statement of causality in this particular state. Now, one might ask, since we are in a CFT, the precise value of delta doesn't matter. Since we are in a CFT, the precise value of delta does not, you are not, you don't have to take a limit, delta goes to zero, you can keep delta on zero. Yeah, you can keep delta non-zero, but all the other distances should be large compared to the delta, yes. Yeah, this should be true in any distances, but because we want something like a shock wave, otherwise, this condition is true, but for shock wave background, it gives you something non-trivial. So, delta goes to zero is kind of like a technical thing that we need. In a CFT, you can map this four-point function for some regime of z and z bar. Let me again use the symbol g for this. So, what does this causality constrain means for this function g? So, that's a precise question that you can ask, and I'll try to address that question. First of all, let me use a different symbol probably. Okay, no, let me use g. First of all, let me compute or let me tell you what this four-point function means. So, we can just show that again, what you should do, you should start from the Euclidean four-point function. Okay, let me not use g here. Let me start with the Euclidean correlator, and as a function of complex tau 2 pi tau 2 is the time coordinate for the second operator, you can show that there are singularities here. So, this is for psi i delta, this is for psi minus i delta, and you want to adequately continue to some point over here. Now, see, they are already shifted because you insert these operators in the Euclidean time. Now, what you can do, you can start from the Euclidean time, you can perform this iatic continuation. And in terms of z and z bar, if you just plug this thing in Mathematica, you can show that this is basically taking the Euclidean correlator, so let me. So, okay, z and z, yeah, z and z bar, they are known functions. They are functions of tau 2, and so, you know these functions already. So, z, you have z and z bar. No, you don't need that. So, just take z and z bar, plug this iatic continuation in Mathematica, you'll see that that corresponds to basically or yeah, it's just, you can just like, if you want a picture probably, you need Mathematica. So, you can just show that this is basically z, that gives you this particular Lorentzian correlator. Let me now use z hat for that. Yes, yeah, yeah. Now, let me introduce some new symbols. Let me now write down z as 1 plus sigma and z bar as 1 plus eta sigma, where eta is real, and it's much, much smaller than 1, and sigma is complex. So, this is useful particularly for the case we are dealing with. Then, the condition that this commutator is 0, you can just translate that into a statement about this function, g hat. Sorry, sorry. This is to make a Lorentzian configuration about z bar, this one as before. No, so basically, you first adequately continue, then just plug in this. This is just a redefinition. So, then you can just show that this statement is equivalent to saying this function in the complex sigma plane. So, this function g hat is analytic in the upper half plane close to sigma equals to 0, so somewhere here. Because if this function is analytic in this particular region, then while you do analytic continuation, you will not hit any singularity or branch cut, and that means all the commutators should vanish if it's analytic. Let me also make a couple of comments. So, the limit eta goes to 0 with sigma fixed. It's a well-known light-cold limit, and there is one more interesting limit where you keep sigma or eta fixed and take sigma to 0. That goes by the name gradient limit in CFD. So, I'm a bit lost. Can I be expected a bit? Because at first you said, you told us that 4-point function is useless because it does not give you anything interesting. It gives you some causality constraints, which are obvious. Then you said we should study the shock wave, but then you said that the shock wave again reduces to the same 4-point function which you, two minutes before told us was useless, and then you discussed analytic continuation, which are very similar to the ones we should discuss before. So, what happened? So, what is the new agreement here? So, CFD 4-point functions, they are not useless. So, the one we studied first, where like all the points were on the real axis and z and z bar, they are real. So, you can map anything to do this 4-point function, but the values, the regimes of z and z bar, they depends on what you started with. So, in the first example that we studied, you can show that the z and z and z bar, they are always real. And in that particular case, when z and z bar are always real, you don't get anything out of causality. So, what shock wave does is basically, even after analytic continuation, they keep z and z bar complex. So, you are basically exploring different regimes. And in that particular regime, you get constraints. All right. So, now I am going to use the fact that it is, okay. First, I will argue that actually reflection positivity and crossing symmetry guarantees that this function is analytic in this regime. To do that, let me briefly say what reflection positivity says. It says the following thing. In Euclidean plane, if this is my Euclidean time, this is another Euclidean coordinate. If I have some operator phi 1 here, phi 2, phi 3, I can have any number of operators I want. And I have another operator which is a reflection of this along this axis. So, I have phi 1 star, but it's a star, phi 2 star, this star. And again, I can have any number of operators I want. Yes, basically, yes. So, this correlator, the statement of reflection positivity is, this correlator is always positive. You can think of this correlator as a norm of some state in Lorentzian theory. So, this is a positive correlator. So, this is the statement of reflection positivity. This is a powerful statement because you can actually smear all these operators with any function you want as long as you do the same thing on the lower half plane. And even then, this should be true. Sorry. Even the smearing doesn't have to be positive because the same function will appear here. So, it's true for any function. So, this is a very powerful statement. So, using that, what you can show is the following. You can use the S channel expansion for this function G, for the Euclidean function G. So, that will go something like this. There will be some pre-factor. So, in the S channel expansion of this correlator G, it will go something like this. And this is a complicated expansion because this is actually an infinite series. It doesn't simplify in the light flow limit. But using reflection positivity, you can actually show that it doesn't matter what this expansion is. These coefficients, all of them should be positive. So, the statement of reflection positivity actually tells you something about the S channel expansion of the correlator. From there, you can show that this function G hat, which is the function after attic continuation, that's bounded by its first sheet value or bounded by the Euclidean value of the correlator with some pre-factor, again, which is finite. But we know that this Euclidean correlator, because of the S channel expansion, this thing converges in this particular regime. We know that this is finite. So, I understand this. So, I think I understand this bit. So, I understand this statement in terms of Z, Z bar coordinates. So, if you turn Z, Z bar, like on the left where you wrote G, Z, Z, and continue because G hat. So, then I understand, it will be true with pre-factor one, absolute value of G hat would be less than G hat. But when you go to this sigma A, the coordinates, this is less clear to me. If you want, you can just write Z and Z bar here. It's just re-leveling. It doesn't do anything tricky. It's a, what's relevant? Yeah, but why is there a pre-factor? Why is it not just one? Yeah. So, that's what you, generally, that's what you expect that this pre-factor should be one. But there is a problem with convergence of this expansion. So, we want something close to Z equals to one or Z bar equals to one. So, this expansion is not convergence. So, we have to use the row expansion. So, if you use the row expansion where everything converges and everything looks nice, then there should be a condition that the G or we call it H of row and row bar. This is less than this, something like that. So, then if you just go back from row to Z and that gives you some pre-factor. That doesn't do anything but like the only thing that you have to check if this pre-factor is large or small. It's basically a finite thing. So, that means because the Euclidean creator, that's convergent. So, that means even the G hat function should be analytic in this particular regime. So, that gives us the slogan, Reflection Positivity plus Crossing that gives you causality. Now, I'll show that it's bounded but you said that it was analytic. I'm using this term analytic in a loose way. So, what I mean is the following. So, if this G is finite in some regime, basically we don't expect to have a branch code over there. For a function, you can have that but probably I can say it's more of an assumption for us that all the branch codes, they come with a singularity. Most analytic functions like square roots Z, they are bounded. Yes. Yeah, they have branch codes but generally we expect from an Euclidean creator, we expect to have a branch code whenever we hit a light cone. So, in that sense, basically we don't expect whenever we hit a light cone, the creator should also be singular. You can have a subleading singularity which is like a square root like singularity in the light cone. I think such examples cannot be cooked up. Yeah, I'm sure, yes. So, that's why it's an assumption. So, we are just talking about singularities which we are talking about branch codes which comes with singularity. So, that's something we have to assume. But can you explain why do you take this assumption? This assumption seems to be completely ad hoc. Why you said that somehow CFT makes you suggest that you should take it but can you say a few words? I don't see any connection whatsoever. Why because we have a CFT, this assumption is some kind of reason? It's basically like we are only talking about singularities which come from we are basically talking about singularities that come from light cone and branch codes which also come from light cone. So, that's why for us it's sufficient to discuss this. It might not be most general thing, but what we are basically... We are trying to make a very general statement which is going to apply to any CFT. So, like in the end we are going to try to argue that you prove this collider bounds. In the end you prove collider bounds not for all CFTs but for just some special class of CFTs which satisfies some special ad hoc assumption then why is this interesting? I mean you should try to justify this. Why is it physically reason? I don't see any physical indication. Okay, so that assumption in this picture is basically this one. So, for the Euclidean Coilator it will see a singularity in a branch cut when it hits this light cone. All I'm assuming is the following. So, this singularity over here the after I perform this Z rotation only thing I'm saying that this singularity doesn't move on the left of it. So, this light cone singularity over here it doesn't move here because if it does then what will happen when I approach the other operator or after approaching this point I have to perform two different analytic continuations. So, that... What have you proven this? I don't see how you prove it. So, prove that this function is bounded in that region. So, I prove that there is no singularity here. Yes. It can have non-anteriorities but only thing I'm saying so this singularity this one didn't move here and in principle there can be other non-anteriorities we are ignoring them. So, only thing we care about is this light cone singularity. Why do you say singularity? We are not interested in singularity. We are interested in non-anteriorities. Did you prove that non-anteriority did not move? No, the singularity is the beginning of a branch cut. So, did you prove that the beginning of the branch cut did not move? No, that statement I didn't prove. That statement I didn't prove. I just proved that singularity didn't move that I proved. Perhaps this can be proved. I don't know. It seems perhaps it just follows by a simple extension of your argument because like instead of saying that this function g hat is bounded by something could you just argue that the series which defines this function converges and so the function is analytic? Could you make such an argument? Because this inequality is also true term by term. So, you know that the term which defines the function g converges the fact the series which defines the function g converges and x to g converges and and this function g hat is defined by the same series with some phases and so if that series converges the series also converges and so it defines the analytic functions. So, basically not only do you have this inequality but you could in fact indeed argue by the same argument that the function isn't analytic. So, I retract my objection. There might be subtleties but probably what you are saying is true but there might be subtleties. When shall I stop? How much time do I have? Okay that will do. Okay now let me just use the fact that on the upper half plane this g hat is analytic. So, I will take some semicircle and perform this contour integral and that should be 0. So, this circle is of radius r. I am working in the limit eta much much less than r much much less than 1. So, I take eta to 0 first then I take r to 0. So, now what you can do is the following. So, along this arc we actually know this function g hat. So, this arc is the light cone limit. So, what you can do? You can just start with g as the identity exchange plus let us say just the stress transfer exchange. So, the light cone limit you can so this is the leading non-trivial contribution. So, then you can just compute g hat in the limit eta goes to 0 then sigma goes to 0. This goes some as 1 plus eta over sigma i times some p factor lambda t. Where lambda t is up to some coefficients c psi psi t c o o t this p coefficients. Now because I know that I can just perform the integral over here and what that gives you gives me is the following. So, this is the integral along this line and this is basically dominated by the regi limit. So, you can think of this as some kind of an optical theorem where you relate the light cone limit with the regi limit. But we do not actually know much about this function g hat. Is this pi the first symbol? Before lambda t what is this symbol? What is written there? This one this one. No no below the equation lambda t eta but what multiplies it in front? Pi x pi. So, again just from reflection positivity from this you can show that this g hat. So, whenever we are I am talking about the 4 point function I am actually talking about the normalized 4 point function where I am normalizing with respect to the 2 point functions. So, from reflection positivity you can also show that this g hat eta sigma on the real line is bounded by 1. So, that means this lambda t has to be positive. So, that is the constraint that you get from causality. But if I stop here probably Slava will yell at me because this lambda t is basically the product of this OP coefficients and from what identity we know that we know this OP coefficients and this product is automatically positive. So, by doing all this we basically derive something which is which is which follows from what identity. But we can do more we can make this statement little stronger. What we can show is the following it does not matter if this operators if this operators O they have spin or not as long as this g hat can be written in the following way. So, as long as this second term is growing in the limit sigma goes to 0, but small because of some parameter there is a constraint on the coefficient that appears in front of it. So, that that is the statement of causality. So, whenever you have something like this where you have a term which is small, but growing there should there is a constraint and this there is a there is one more condition that you can derive which is actually very important. You can also also show when this second term is growing it cannot grow faster than 1 over sigma. It can grow only as fast as 1 over sigma that also follows from the fact that this function g is analytic in this region. So, let me now some some let me now summarize all the results. When I have external scalar operators and let me assume that so this O's are scalars and let me assume that there are obviously we have the identity operator. We have stress tensor, but let me also assume that you have many other conserved currents exactly conserved yes because in the light goal limit only operators which are conserved will contribute because everything else will come with a higher power of eta. So, these are the things that did that follows from our causality constraint. First of all if you have a spin 1 conserved current let me call that j there is no constraint. The reason is you can just you can just show that if for a spin 1 exchange g hat goes like this 1 plus i or for any spin spin exchange it goes like this l l minus 1 times 2 is 2 over 2. So, for l equals to 1 there is no sigma term. So, this second term is not growing. So, because of that there is no constraint. For t it is best as I showed earlier it is c for O t c psi psi t it is positive which as I explained before it is trivial and for higher spin exchange any finite number of them ruled out. So, you cannot have finite number of higher spin conserved currents. This is in d equals 3 this was was proved by Maldesena and Zavedov and then it was generalized for higher dimensions, but for us it is basically a simple consequence of causality. Can you explain how it follows from? From here. So, if you have a spin let's say 4. So, this will go as 1 over sigma cube, but it can never grow faster than 1 over sigma. It is basically that you can you can just show from the fact that this is analytic in this regime you can use something like maximum modulus principle to show that only thing that is allowed is basically 1 over sigma if it is analytic in that regime. So, if you have infinite number of them if you like if you what you can do you can basically you can eliminate the one with the highest spin first. So, if there is a high spin you can you can eliminate that, but if you have an infinite number of them so basically like that argument will not work. So, you can just sum them up and after that so individual terms can be actually more singular, but after summing them up they can be they can grow at a slower rate basically and that is allowed. What I am confused about is that how the theory with infinitely many currents is going to satisfy your constraints. Which constraints? So, I ruled out only the. So, I just ruled out there cannot be a high spin. What will you have look like if you have infinitely many currents? That depends. So, then you have to tell me about everything about the theory. So, even if you have an infinite number of concept current and you sum all of them up the entire thing cannot grow faster than 1 over sigma. So, that is the condition. Now, for external spinning operator let me just quote the results. So, if j is a spin 1 concept current and t is the stress tensor just from conformal invariance this 3-point function is fixed up to 2 coefficients. So, these are known tensor structures and they come with some coefficients which are not fixed and apparently they can be anything. Similarly, stress tensor 3-point function they come with 3 coefficients. In d equals to 3 they come with there are only 2 structures but in general there are 3 structures in any other dimension. But in d equals to 3 you can have a parity odd structure. So, even in d equals to 3 you have 3 structures. So, these coefficients are this s. So, when I write down anything with a subscript s here or these things they are basically known tensor structures. I am just not writing them down. So, they are just known structures. So, the point is all these 3-point functions they are fixed only the only thing that you can change is basically this coefficients. This is not hard to explain. So, this s f refers to take a theory with free scale and with free fermion and then this is a particular example and tells you that it can be used as a basis. Okay, then if you just perform the same procedure but with spinning external operators you can show the following. Okay, let me just use n s. So, all of them should be positive. So, conformal inference does not tell you about the sign of this coefficients but causality tells you that they should be positive. Yes, yes. But these constraints they are actually very interesting to lot of people. So, we and another group we derived this from causality in 2016 and in the same year these were derived using unitarity. In fact, these were derived using deep inelastic scattering. So, basically these are unitarity bounds by Kulaksev, Pernachev, Kowat-Gorski and Zaveyadov in 2016. But interestingly these constraints were first derived by Hoffman and Muldesena in 2008 using averaged non-energy condition. So, this is a condition. So, the averaged non-energy condition is the statement that t minus minus square x minus is a null coordinate. This is a positive operator. So, if you ask or expression of this is positive. So, if you start from this condition you can just derive them. So, that made us wonder if we can actually derived this averaged non-energy condition directly from causality and the answer is yes and that will be the topic of the next lecture. So, let me now summarize because we are already out of time. So, we studied shock wave states in safety and from there we showed that causality is equivalent to some statement about analytic structure of the correlator. Then the second statement is if you start with a reflection positive and crossing symmetric Euclidean theory that guarantees that the correlator is analytic in the regime where it should be. So, that gives you causality of the Lorentzian theory. Then using the same property of the correlator we derive constraints on the interactions of low dimensional operator irrespective of what happens in the UV. Then also we ruled out finite number of high spin conserved current and finally we showed that these constraints are equivalent to and this constraints are related to the averaged null energy condition which I will discuss in the next lecture. Thank you. Questions. See, is it important they are exactly conserved those currents or if you have a large n models the currents are just better than not conserved? Sorry, that's it. It was important to rule down to conserved current they are exactly conserved. If you look at theory which are you know like we have 1 over n deviation from an exact conservation. Yeah, for us they have to be exactly conserved. But if there are operators which are nearly conserved probably you can use all these tricks there. We haven't tried that yet but I would guess that you can actually say something non trivial.