 First of all let me thank you for inviting me to give this lecture series. So, this is first of three part lecture series. Today I will mainly talk about causality in quantum field theories and general quantum field theories and this lecture is. So, today's topic is. So, these three lectures are mostly based on three papers. So, these two with Tom Hartman and Sachin Jane and the last paper with Hartman and a PhD student Amir Tajdini. Quantum field theories are integral part of our lives and they have certain properties. For example, if I if somebody gives me a Lorentzian quantum field theory, it should have few properties. First of all, obviously it has to be Lorentzian variant. Secondly, it should be unitary and also it should be causal. Similarly, if somebody gives me a quantum field theory in Euclidean signature, first it has to be Euclidean invariant. It should be reflection positive and also crossing symmetric. So, in the next lecture, I will explain this properties in more detail. So, from the title of the lecture series, you probably can guess that I will mainly focus on causality. So, it is an open question how causality is related to all these properties of an Euclidean quantum field theory. In some sense, this question was addressed in 50s, 60s and 70s by Schwinger, Weidmann and Osterwalder, Schroeder. So, the precise statement is the following. If you have a good Euclidean quantum field theory and if you perform a continuation, that gives you a good Lorentzian quantum field theory which has all these properties. But the point that I want to address is the following. So, what goes wrong if I have a quantum field theory in the Lorentzian signature which violates causality, what actually goes wrong in the Euclidean signature? Before I proceed, let me make couple of comments on causality. Causality is the Lorentzian concept for obvious reason. If you want to talk about causality, you need to have light cones. The main point that I want to focus on these lectures is the following. Causality can impose non-trivial constraints on quantum field theories in Lorentzian signature. There are Lorentzian variant quantum field theories which actually violate causality and I will give you one example. And the next point is in quantum field theories causality is an operator statement. So, a theory may appear to be causal in the vacuum, but it can violate causality in non-trivial states. Because causality is a statement about operators, so a theory should be causal in any state. And requiring a theory to be causal in any state actually impose non-trivial constraints. Here are couple of examples. Please feel free to ask questions if you have any. Yeah, I have a question. Of course the list of the triaxioms there is gauge theories are those triaxioms are incompatible. Lorentzian unitarity and causality cannot hold at the same time engage theories. You are excluding gauge theories for your presentation? Not really. What kind of theories are you talking about? Whatever. Quantum electrodynamics. It doesn't work in quiddiq. It is not quiddiq, so let's start with quiddiq. You see, d is not satisfying the saxons. No, I mean they are incompatible in gauge theories. No, I am surprised to say so. It depends on which observables you look at. No. It's a theorem by Whiteman. No, it's not. Okay. Let me start with a scalar field theory with shift symmetry. So let's start with this action. So in a paper by Adams, Arkane-Hamed, Duoski, Nicholas, and Rathaj in 2006, they argued that this coefficient mu has to be positive, otherwise this theory cannot be UP completed. So obviously this coefficient is non-enormalizable, so this is an effective field theory and their argument is basically if mu is negative, this theory is sick and there is no UP completion. And this constraint actually played an important role in the proof of C theorem in 4D. And let me give you a classical proof of this bound. So classically phi equals to zero is a solution of this theory, but let's consider a somewhat non-trivial solution of this theory. So let's consider this solution where this is a constant vector. Oh, that's a solution of this. I write down the equation of motion. This is a solution. You can just check. So what I want to consider is basically propagation of this perturbation in this background. So again I can write down equations of motion for del phi. So it goes like this. In the plane wave basis, this is simply, okay. Now for simplicity, let's take C to be a time-like vector. So my C is C0 and 0 and K is omega. If I plug that in here, that gives me. So if mu is negative, the speed of propagation is greater than 1. So that's one reason phi mu has to be non-negative. So in this original paper by Adams et al, they had another beautiful argument phi mu has to be positive from by considering 2 to 2 scattering of phi. And they basically showed that unitarity tells you that this has to be positive. An important point about both of these proofs is that they make sense only in Lorentzian signature. So one might ask what goes wrong in the Euclidean theory if mu is negative. So this classical argument is not obviously sufficient for that. Yeah, this is a very rough argument like there is a actual proof of this statement. I know they had the argument which is basically unitarity and I think this was the argument. But this argument is also given in the paper. It leaves some unsatisfactory feeling because you say, okay, this background, I can see this background where phi 0 grows at infinity. Is this actually a very good background? Should I, if something here kept on this background, maybe I should not be very concerned. They tried to answer this question in the paper by like taking a bubble inside. So this solution is valid in some bubble and then it goes to 0 outside that bubble. They tried to make that argument in the paper that like, yeah, you can probably make this precise but because they had another argument, I guess they didn't actually. It makes you move space like that from. Yeah, you can do that. Yeah, then it will not change sometimes. You'll get the same result, yes. So another example comes from gravity. Consider gravity in let's say 5D where I have the Einstein-Hilbert term. I have some negative cosmological constant, though I don't need this term. Let me just skip it here and I also have some higher derivative correction terms. So this is the Goswami term in 5D. So this is a ghost recombination of all possible R squared terms and M2 is some mass scale which is far below the Planck scale and C2 is some order 1. So just as a Lagrangian, you can just check that this is partuitably consistent up to some scale M which is parametrically higher than M2. In a paper in 2014 by Edelstein, Kamanhoe Edelstein, Maldesena, and Zubayadab. Can you explain why it is not, why it is a top form of M2? Yes, so it's just, you just take, because there is an overlap. One, two. Yeah, so it's basically something like square root of M2 times M Planck, something like that. Okay, so in the paper by Kamanhoe Edelstein, Maldesena, and Zubayadab, which I'll refer to as KEMS, come to 2014, they studied shock wave solutions in this background, in ADS. So you have something like this, where u and v, they are null coordinates and z is the bulk coordinate. They basically studied shock wave solutions in this background. So you have some function of z and y, a delta function. So this is a shock wave. So let me draw a diagram. So this is v, this is u. So this shock wave is propagating this way. Now let me take a graviton, probe graviton, and send it in this background. So when that graviton hits the shock, there is a jump, and let me call that jump. So bulk causality requires that this delta v has to be positive. If delta v is negative, that means there is some time advance. So in this particular paper, they send gravitons with various polarizations, and they showed that delta v is always positive if and only if C2 is zero. So that means pure Gauss Bonnet theory is a causal. So now from the point of view of effective field theory, this has the following meaning. So as I said earlier, so I have scale m2, m-flank. As I said earlier, this is perturbatively consistent up to some scale m, which is parametrically higher. So nively I would expect that new physics should appear here to make everything well behaved. But this argument tells you that actually new physics should appear here and fix this causality violation. So it's basically if you just take this theory, if I just give you this theory and if you just study perturbation theory, then you'll see that actually it breaks down at this scale. That would be called the strong copy scale, which is not necessarily capable of the field. Yes, so basically m2 is basically like if I have a theory and I integrate out everything above m2, you should get something like this. But just as a Lagrangian, it doesn't require that. But at the classical level, this theory with the square and the curvature, it is not maybe it's the case, but there's no prior reason that the characteristic cone is still the null cone. For Einstein theory, we know that the characteristic cone for the propagation is the null cone. But for such a theory with the r squared, there's no reason not a prior way because if something in front of could have g mu u plus extra term in front of the main part of the field equation. Yeah. So if this classically is not causal? That's right. You can make that argument classically, but again there is another argument which shows the same thing. So this is a very classical rough picture, but there are other arguments which shows that. But what I say is that there is no reason to require the usual causality for such a theory. Yeah. But like if you study, let's say, Graviton scattering in ADS for this theory, that should have unitarity, things like that. Yeah. Classical level, yeah. So this is classical argument, but there are analogous quantum argument which gives you the same result. Yes. This classical argument can have problem, but that quantum argument is there actually. So the goal of today's or these three lectures are the following. So understand causality quantum field theory. This is what I covered today. Then as I said earlier, causality violation in Lorentzian signature tells you that there has to be something wrong in the Euclidean signature. Then I'll consider CFTs in the greater than two dimensions to show that reflection positivity plus crossing guarantee that the Lorentzian theory is causal. Then I'll use that to derive constraints on interactions of low dimensional operators regardless of what happens in the UV. And fourth, I'll show that these constraints are intimately related to the averaged null energy condition. So I'll provide up to of the averaged null energy condition from causality. So this is today. So this is lecture two and this is lecture three. So in quantum field theory causality is a statement about commutators. So this commutator should be zero if they are space like separated. As I mentioned earlier, this is an operator statement. So if I plug this commutator inside any correlator, that correlator should also be zero. So because of that, a theory may appear to be causal in the vacuum but in some non-trivial state, this condition can actually be non-trivial. So in a quantum field theory, whenever we compute some correlation function, generally what we do is we start with the Euclidean theory, we compute the correlators, and then we perform a decontinuation to get Lorentzian correlators. So Euclidean correlators has certain properties. First of all, they are single valued. So let's say if I take a endpoint function. So this is a single valued function. Second is its permutation invariant. That means if I change orders of the operators in the Euclidean correlator, that doesn't change the function. Sorry. You exclude fermions for simplicity. Yes, that's right. And third is this function is analytic. And it's analytic and it doesn't have branch cuts as long as all the points remain Euclidean. The fact that its permutation invariant, that means basically an Euclidean signature commutators do vanish. Then I perform the standard analytic continuation to get Lorentzian correlators. So for that, I take one Euclidean direction arbitrarily and I'll call that tau and I'll analytically continue that to it, where now t is real. So that leads to one question. When do a set of Euclidean correlators define causal theory? So this question was answered by Osterweiler and Schroeder and which goes by the name Reconstruction Theorem. Reconstruction Theorem tells us that well-behaved Euclidean correlators after analytic continuation leads to well-behaved Lorentzian correlators. So let me write down EC for Euclidean correlators. So by well-behaved Euclidean correlator, I mean they are analytic from coincident points. They are SOD invariant, crossing symmetric, reflection positive and obey certain growth condition, which is not relevant for our discussion. And by well-behaved Lorentzian correlator, I mean they are invariant, causal and unitary. So the point of view that I want to take is somewhat different. I'll only assume some limited information about the quantum field theory. So I'll assume something about the, I'll assume something about few low dimensional operators and I want to know if that set of information is compatible with causality or not. So in Reconstruction Theorem doesn't actually give an answer to that question. It just, it tells you that well-behaved Euclidean correlator will give you causality violation, but it doesn't tell you exactly what, like if, if I have a causality violating theory, it doesn't tell you what goes wrong exactly. So that leads to analytic continuation. So this Euclidean correlator G, I said earlier that it's an analytic function of positions. So if my tau is real, then this is an analytic function. So, but as a function of complex tau, so this function G can have intricate structure of singularities and branch cuts. Do you think, do you are talking about only one time or all the times? Are you ordering the times in any way? So in the, in the, in Lorentzian signature, yes, that's, that's what my goal is. In Euclidean. So in Euclidean signature, so this is, I'm just starting with Euclidean, Euclidean correlator which doesn't care about the ordering. So all the points are Euclidean. But when you go to continue? Yeah, so then in the Lorentzian correlator, I'll, I'll care about the ordering. And yeah, I'm going to talk about that, yes. M tau, you have tau 1. Yes, yes, yeah. Yes, and okay, so for simplicity, I'll just, I'll just anecdotally continue one of them, not, not all. So because of this, because this function has branch cuts, so this analytic continuation is actually ambiguous. That sounds problematic, but actually that's not a bad thing. You can actually show that each choice that you make in, in this analytic continuation translates into a particular ordering in the resulting Lorentzian correlator. So basically, that means because of this branch cuts, operator ordering does matter in the Euclidean signature. So these branch cuts are actually responsible for non-vanishing commutators in the Lorentzian signature. Let me give you one simple example. Let's consider this two-point function. So in the Euclidean signature, I can just compute this two-point function. For simplicity, let me call them O1 and O2. So first, I compute the Euclidean correlator. Let me call this G. So complex tau. In the complex tau plane, there is a branch cut at ix and minus ix. So if I want to compute the Lorentzian correlator, I'll start from the Euclidean one and I'll analytically continue. If in the Lorentzian signature, O1 and O2, they are space-like separated, then basically I have to analytically continue from here to let's say somewhere here and there is no ambiguity. However, if in the Lorentzian signature, O1 and O2, they are time-like separated, then I have to start from here and analytically continue somewhere here. And because of the branch cut, so because of the branch cut, you can do it in two different ways. So one will give you the time-ordered one and the other one will give you the anti-time-ordered two-point function. So what are the rules? So first of all, the one important thing is whenever you hit this singularity, only after that you have these two choices. So that means this singularity is responsible for non-finishing commutators and not that is responsible for non-finishing commutator between O1 and O2. So what are the rules for this analytic continuation? So if I have a singularity, I can either pass this singularity from the left or from the right. If I pass this singularity from the right, that gives me time-ordered commutator. If I cross this singularity from the left, that gives me anti-time-ordered. So this one gives me O1 Tx and this one gives me, so this discontinuity across the branch cut, that basically gives you the commutator of O1 and O2. Now let me give you one more involved example. Let's now consider four-point functions. Probably it does tell you something about the CFT but for us only thing that matters is if it's non-zero or not. Or it can be non-zero or not. That's the only thing that we care about because that's the signature of causality violation. Let's now consider this four-point function. This is T, this is Y. So this T is the Lorentzian time and this is some arbitrary direction. So my O1 is here, my O3 and O4 they're here. Let me draw the light cones and my O2 is let's say somewhere here. So all of the O1, O3 and O4 they are space-like separated and they are at T equals to zero. Only O2 has some non-zero time component. So as now this coordinator in the complex tau-2 plane has this structure of singularities and branch cuts. So first singularity appears when this O2 hits this light cone of O1. So that's I times O2 minus O1 and the next singularity appears when this O2 hits the light cone of O3 branch cuts. And there is one more singularity because of this O4 but let me ignore that. Similarly there should be singularities on the lower half plane. So now consider this case when O2 is time-like separated with respect to O1 and O3. So it's somewhere here. So because this is inside the light cone of O1 and O3. So in this language so I should start from somewhere here and I have to analytically continue to this point here. So when it's basically above both of these singularities. Let me just somewhere here. So there are obviously you can immediately check that there are four different analytic continuation that you can perform. First one is this one. Let me call that A. So this goes this passes this singularity from the right also this singularity from the right. So this is when this operator O2 is time-ordered with respect to both O1 and O3. So that's O2, O1, O3, O4. The second thing that you can do is this. So you can analytically continue this way. Right of this then left of the other singularity. So as I said according to our rules O2 should be time-ordered with respect to the operator responsible for the singularity that's O1. But it should be time anti-time-ordered with respect to O3. So this is O3, O2, O1, O4. The next thing you can do is this one. Left of this singularity then right of this. So this is B, this is C. So you are crossing this singularity from the left. That means O1 and O2 they are anti-time-ordered. So O1, O2, O3, O4. So O2 is anti-time-ordered with respect to O1 but time-ordered with respect to O3. And the fourth one is this one where the collator is completely anti-time-ordered. So there are four choices for analytic continuations. And in the Lorentzian signature you can write down these four different Lorentzian collators. Now let me make the statement about causality. Need this board. So let's now consider a special case of the last example. Okay, so again I have four operators. So this is time, this is y. So I have an operator psi and operator O, another psi. So this is the light cone. And I have another operator O which is time-line separated with respect to psi. So it's somewhere here and it's approaching the light cone of other O. So I have another operator O. So in the language of analytic continuation, so this is what I'm doing. So again I have two branch cuts. So basically I am approaching this singularity. So I'm approaching this singularity. And again I can do it in two different ways. I can choose to go from left or from right. Now first consider the simpler one. So this is the coordinator O, O, psi, psi. So when I hit this singularity along this path. Yes. That actually doesn't matter. Yeah, I can do that. I care about the commuter. So it will give you a different sign. That's it. It's approaching it from below. Yeah, from below, yes. So it's basically like, I basically want to study this singularity by approaching it from the below. So when you hit this singularity, at that point the O, O commutator becomes non-zero. So this thing becomes non-zero when I hit this singularity. Now if I follow this path, as I said earlier, this singularity is fixed by light cone. So as long as I'm below this singularity, this is zero. And that means this is, this coordinator is causal. And that's not very surprising. This is, so this coordinator is kind of fixed by the light cone structure. Now let's look at this coordinator. This is psi, O, O, psi. So this quantity psi commutator of O and O with psi, this becomes non-zero when I hit this singularity, but along this path. But because I'm crossing this branch cut, the position of the singularity is no longer fixed. So basically I'm going to the second sheet of this Euclidean correlator. And on the second sheet, this singularity can actually move around. And this commutator becomes, or this coordinator becomes non-zero when I hit this singularity, but on the second sheet. So now the statement of causality. Now let me erase the trivial one. Let me now look at this analytic continuation. So two things, when I do this analytic continuation, two things can happen. So on the second sheet, this singularity can, let's say, move in the upward direction somewhere here. So if that happens, then this coordinator becomes non-zero when I hit this point. Now in this diagram, that point is somewhere here. So that means this coordinator becomes non-zero at a later time than you expect from the light cone. That means there is some kind of time delay. And that's all right. There is nothing wrong about having a time delay. However, if this singularity, so this gives you, however, if this singularity moves in the downward direction, that means this correlator with the commutators becomes non-zero. So let's say somewhere here. But that means, so this operator O and this operator O, they are now space-like separated. But even then, this is non-zero. That means this is a causal. Or you can say that there is some time advance. So the statement of causality is basically this singularity on the second sheet cannot move in the downward direction. So this is not allowed. So let me summarize. The first statement is I don't know of any example where it moves at all. So the state like the causality tells you it cannot move in the downward direction. But in principle, it's allowed to move in the upward direction. But I don't know of any example where it moves. In fact, in CFT2, we can actually show that this singularity doesn't move at all. So starting from the Euclidean correlator, causality on the first sheet is trivial. The non-trivial statement about causality is a constraint. Singularities move around in the complex tau plane when we other light cone branch curves. So in the next lecture, I'll use this setup to derive constraints on interactions of low-dimensional operators in conformal field theories in dimension greater than 2. And next, I'll comment on how these constraints are related to the operational energy condition. So let me stop here. Can you remind us what determined like there was some tacit assumption that the first sheet corresponded to both branch cuts go in like you drew them. Why then? We have to note on the right. Okay, so this is kind of a cartoon. Where did this get broken? So it's not like it's not like you can actually forget about these branch cuts altogether. So what you can do, let me, so let's just consider the two-point function that's easier. So if I have a single, if I have this singularity and if I just do two things, I compute this function and I just compute this function and I can just match these two can check if they are the same or not. So you can just check that they are because of this singularity that they are not the same function. So we draw a branch cut somewhere here. So I'm drawing these branch cuts. You can actually just forget about this branch cut. You can just follow this path and find out that function here. So all I'm doing, I'm just saying I'm approaching this imaginary axis from the left or from the right. So that's the only thing I'm doing. But why, where was the symmetry between left and right broken? So it's not as... You said that when you approach from the right, from the right it's trivial and from the left non-trivial. So why is this symmetry broken? So it's not like, okay, probably there is a symmetry between which I call trivial and which one I call non-trivial. Yeah. So there has to be some formulation of this, not in terms of right and left, but in terms of some other. So, okay. What you can do, you can... Somebody can just choose to draw them on the right side. Then basically you can just do the whole thing. So in my rules of anticontinuation, I can just move the left to right and right to left. It's basically, it's just identical. So the reason I'm drawing in this particular way, so that the trivial anticontinuation, that gives you the time-ordering. By trivial, I mean like just I'll take a point from here and I'll just adequately continue this way. Start from positive tau and then you rotate. Yeah. So that gives me the time-ordered collator. But I don't have like, I can just choose the branch curves in this direction and then I'll just call this one time-ordering. That's it. You made a comment before concerning that you are going beyond the of the van der Schrader continuation. Reconstruction, what is the point here? We don't have to re-exploring the other regions. So for the reconstruction theorem, basically you need to know all the collators and everything. So here I'm just, I have a single collator and I can talk about causality of that. Like if that collator is, will give you a causal Lorentzian collator or not. So it's basically, I only have this, if I start with this limited information, even then I can talk about causality. That's my point. So in other words, basically I don't care if this comes from some quantum field theory, which has all the good properties. Only thing I can, the thing I want to show, if I have some collator and it has this, and it has this attic properties, if it's consistent with causality or not. I don't care if it comes from some. What do you assume on the Euclidean side? So on the Euclidean side basically, okay, so I assume everything on the Euclidean side, but only for this collator. I don't care about all the rest of the collator that you can have on that particular theory. So it's basically, in that sense, it's some limited information. Can't you say that, can't you say that you will, unlike Astrovaldi Schrader, who reached some qualitative conclusions about what is good and what is bad, you will reach some quantitative conclusions, like you will be able to derive some constraints on the data of your theory. Yes. Some numerical answers about what can happen and what cannot happen, and this was not contained in those theorems. Yeah, that's right, yeah. So next week I'll move on to CFT, when I can actually use this setup, where this setup will actually give you non-trivial constraints. So what that tells you, it doesn't matter what kind of theory, what kind of CFTs you are looking at, all of them should obey those constraints. You mentioned that you will use also reflection positive up to this point I haven't seen. Okay, so this is now, so far this is a statement about like what does it mean to be a causal Lorentzian correlator. So next week basically what I'll show that if I start from Euclidean theory and let's say if I start from this collator in the Euclidean signature, if it obeys reflection positivity and crossing symmetry, then this is guaranteed, that this will not move in the downward direction, that's guaranteed. I have a question about the rules you assigned, is it just there and a deeper reason for how you read all the rain from the way your analytics continue? So there is basically reconstruction theorem does one more thing. It actually tells you how you can analytically continue one Euclidean collator to get different Lorentzian correlators. So the rules that I described today, they are basically equivalent to those, the one that you get from reconstruction theorem. So in reconstruction theorem what you do, you use some I epsilon prescription, which basically shifts this singularity either on the on this side or this side. So you can do this analytic continuation in two different ways. Either you can choose to move the singularities or you can choose to move the path which you take to do the analytic continuation and both are equivalent. So all I have to check is basically I am not all I have to check is basically this prescription is in agreement with the known I epsilon prescription. And it agrees with the I epsilon prescription. Yes. Since there was a question, can you clarify, which I think is possible to clarify, so to, in which sense, to which correlation functions, to which observables would your discussion apply if we had a gauge here, something like young meals in 40 degrees, some really complicated. Whatever I said should be like, if you managed to give me a four point function in the position space, everything I said that should go through. Well, which of which operators? Any scalar of, any Bosonic operator will do. It shouldn't matter. It will not be reflection positive. It should be gauging their interpreter, right? Yes, yeah, yeah, that's right. It should be gauging their interpreter. Yes, it has to be gauging, that's right, yes. And probably it's, we can still allow for myonic operators, but you have to be more careful. This is for a scalar, so I'm sure there is a easy generalization of this for fermions. But fermionic or otherwise, I think that's easy to fix. Yeah, it's more important to see if it gauging the other problem. Yeah. And I believe that this will apply really engagement otherwise you will never have it. Never have it, I think. Just some question. Yeah, even to those kind of long angles, this discussion of what is the main of analyticity for wind functions here, because we have configured this in the space and it's kind of unknown for many points. And it's about geometry of this, all of these discussions about analyticity and convexity envelope for this domain. It's anything else? So this analysis question is that you get. So because the cases that I consider today, they're kind of simple, because I'm adequately continuing only one point. So but there are known subtleties, if you just try to adequately continue all four points, let's say, there are known subtleties. It's possible that this one-to-one correspondence between number of analytic continuation and number of orderings in the Lorentz signature, that one-to-one correspondence might not be true if I adequately continue all of them. So there are subtleties, yes. No, but mathematical questions, you have some domain in that function, I want to see you as a values of point. Yes, so I'll actually comment on that on the third lecture. That will actually be important for the discussion. Because you are assuming that you have only cats here. This is not known in general. Yeah, that's right. Yeah, but you see, this is a minimal set for flow. You see, it's the other way around, the way I understand it. If you find that this particular commutator is not causal, the theory is not causal. It does not tell you that every other commutator will also be causal. And what you ask is a much more general question, is whether the entire theory, but what he wants to see is, if you have a constraint, this one commutator is not causal or not. If one commutator is not causal, the theory is not causal. But even for this particular thing, it's assuming something concerning the analytic structure of the four-point balance, which is not known. Yeah, but you can do it for the two-point function. Two-point, okay. But there is an old theorem due to a Bartman-Coward-Weidman, which tells you a domain where the Euclidean correlator is a single-valued analytic function. So can you see this domain in your consideration somehow? Not in this. To find the domain of the model, which is the angle of this, the film. So not in this construction, but so when I'll prove average knowledge condition, there it will be important. So I'll discuss that. So this is a minimal setup. So this is, I guess, too restrictive. So we might not be able to see everything from this setup. Yes, yeah. You're talking about local operators. Yeah, yeah, they are local operators. Yes, that's right. Yeah, I should have mentioned that. Yes. So you can see the account to filter. It's okay. For example, define Euclidean where it can define gauge variant operator. So because I'm not studying from a Lagrangian, it's basically all I have is basically some gauge invariant local operators in the theory. That's it. Because I guess there exists quantifier, so we don't have gauge variant local operators. Then you have to worry. I was wondering how much of this would go through. Yeah, I haven't thought about that. So I'll probably, yeah, I have to think about that. I'm not sure. Okay, well, if there are more other questions, then I'll be preparing to clap. So let's thank speaker again.