 into this little energy operator. And what I started to do at the very end yesterday was the final derivation of the ANEQ. And I want to sort of gloss over some of the. So the rest, we've put all the pieces into place now. Like all the fundamental things that you need to know are now in place. And I don't want to dwell on the technical details of the last bit of the argument too much. I'll just sketch quickly how you finish it up. The more important part is the building blocks that went into it anyway. And we have those. But let me just tell you how this ends. So this is the correlation function we're going to look at. It's a four point function. But really, oh, it doesn't really have to be a local operator. It could be some superposition or some blob or multiple operator insertions. It's really a general operator. We're going to look at this four point function where we take. We're thinking of these size as some kind of causality probe. And we're asking the physical question, in the presence of these O insertions, can we send a signal faster than light from this psi over to that psi? Notice these are spaced like separated. So if we find that signal, then we've violated causality. So the first thing that I want to do is to evaluate this four point function using this OPE. So that's pretty easy. Psi psi O just gets a disconnected term, which I'll normalize to be 1. And then a contribution from the null energy operator. So that's a v u squared O integral t u u of u from minus infinity to infinity. This expression doesn't quite make sense, as written. Because what we're doing is we're taking these operators, these local operators, and we're replacing them. We're just deleting the local operators, and we're replacing them by a non-local operator that's integrated the null line between them. So we're integrating the stress tensor over that null line. But there's a singularity in that integral. Because when it crosses this light cone, that three point function is singular. And again, when it crosses this light cone, that three point function is singular. So this expression doesn't quite make sense, but you can go through all that i epsilon business that we discussed yesterday. The ordering of these three operators has written this expression tells you how to deal with this. It tells you which i epsilon you're supposed to use when you evaluate the integral. And if you go through that, then what you'll find is there's a plus i epsilon up there and a minus i epsilon down there. So this just ends up being a simple contour integral where you can pick up a residue from circling one of these operators. Oh, yeah, I've dropped some constants here. There's a delta saw over CT, and there's probably some pi's and stuff too that I haven't written. If we had changed the operator order around like this, then we would have gotten a different answer. I've switched two of the operators here. You can still evaluate this using the OPE. And the expression is almost the same, except that the i epsilon prescription is different because we've switched some of the operators around. So in this case, you get a minus infinity minus i epsilon and a plus infinity minus i epsilon. You just deform the contour a little bit. And when you do that, you no longer hit any of the. Now you can close the contour below. And it's just zero. OK, so this is zero. This is just one contribution to the OPE. The correlator is not actually zero, of course. But the point is that this term that we're interested in, this term that's small in the leg cone limit but enhanced by positive powers of u. v is getting small, u is getting big. So this enhanced term is not present in this other correlator. Now if we go back to what I was talking about yesterday, this is the one that's Rindler. This is the one that's positive by Rindler reflections. OK, so let's see why that is. So I've ordered them o psi, o psi. And remember we said that when things are reflection symmetric across the Rindler horizon, there's an ordering that must be positive by interity. And the specific ordering was the one where you just order the things on the right and the things on the left in the same, you just write them in the same orders. OK, so this is the Rindler positive one. And what that means is that this one is bounded by that one. OK, so we're almost there. The last step is to use causality. Just causality, the statement about where this four point function is analytic is a function of complexified spacetime points. And if you have an analytic function, you can integrate it on a closed contour and get 0. I'll actually do something a little bit different, which is to integrate the real part of g minus 1 on a closed contour, which is still 0. We define u to be 1 over sigma. Then the contour that I'm going to pick looks like this. So on the sigma plane, it's not easy to get a picture of this in Minkowski space, but I'll try. So if we go over to our Minkowski space picture, what this integral is doing is integrating the straight part of that integral on the bottom. That's real sigma, so that's in regular Minkowski time. So this is the straight part. It's just integrating these points up to infinity like that. That's this part of the contour. And then it goes out into the complex direction out of the blackboard and connects the two endpoints. So the contour that we're doing, it goes down to infinity, it does this arc out like that, and then back to infinity. It's inside out because u is 1 over sigma compared to that one. This one is sort of inside out. So physically, this is a sum rule. This is a sum rule because you can break up the different pieces of the contour. And they have to cancel with each other. And there's a lot of physics hiding in here because the different parts of this contour are capturing very different physics. This part of the contour here is the top part. This is the v much less than u inverse, much less than 1, regime of the correlator. So there, you can use the light cone OPE. This is probing the infrared of the theory. We said the light cone OPE is given by low twist operators. So if you tell me the spectrum of low twist operators, then I can already calculate for you that light cone OPE. So this is really infrared. I don't need to know the details of the UV. Whereas down here, this is a limit where u and v are both much less than 1, but not in any particular ratio. So they could both be the same size. And down there, the OPE is no good because this part of the contour is really probing the UV. So we can't calculate anything on this part of the contour. And that means that this is relating something we know in the infrared to something we don't know in the UV. So we can't calculate anything, but we can bound it. So the OPE is no good, but on this part of the contour, the reason the integrand is positive is because the real part of g bounded by the absolute value of g. And we said that the absolute value of g in any ordering is bounded by the positive ordering. And we said that the positive ordering was just a wand. The real part of g is less than wand. Technical details, but hopefully the logic is clear. The logic is that in the infrared, we calculate things using the OPE in the UV. We can't calculate anything, but we can bound it by unitarity. So that's the input here. The input is unitarity in the ultraviolet. Yeah, infrared-ish, it's not really the infrared. So if we're doing a, say we have a CFT that's deformed by a relevant operator, then you might think the infrared is some gapped phase or something. It's not that infrared. It's still controlled by the UV fixed point. I really mean infrared in the sense of low-dimension operators of the UV fixed point. Okay, so putting this all together, what that contour integral says, is there a question? So if we order, the two different orderings here can both be calculated using the light cone OPE. So both are the same expression, one plus the integral. But because of the ordering, there are different I epsilon prescriptions. And so there's a minus I epsilon minus and a plus minus. And the I epsilon prescriptions in this one don't, you just don't pick up any poles and you get zero. So putting this all together, we have an expression for the null energy in the presence of the O insertion, which is the integral of positive stuff in the ultraviolet. O is totally arbitrary, but that can be true as if this is a positive operator. So script E being the null energy operator. So that's the end of the argument. We pause there for questions, yeah. Well, they're dense in Hilbert space. Whether the fact that that's not quite the same as being the whole Hilbert space is an issue or not, I don't know, I'm just gonna ignore that. They're dense in Hilbert space. You can make any state by acting with operators and acting with mode. Yeah, but these don't really have to be local operators. I could have inserted five operators here and five operators here, as long as they're reflection symmetric. So that's what allows me to make basically any state this way. Questions about the logic here? So let me recap briefly. So physics in the light cone limit is controlled by the null energy operator. And some rules which come from causality, so this is really, we use causality in an essential way here by using analyticity. Those some rules coming from analyticity relate this thing, you can calculate on the light cone to unitarity in the UV. That's the logic. If you say the ennec was violated, say that you found a state where the null energy was negative. Well, that would mean that somewhere in this domain here, the function is not analytic. So if you violate the sum rule, it means that you must have, well either you have to violate unitarity in the UV, or you have to stick a singularity somewhere in here. If you put a singularity somewhere in here, then what you've done is turned on a commutator where there's not allowed to be a commutator. So somewhere out here at very large U, despite the operators being space-like separated, there would be a singularity and the commutator would turn on. Oh, positive ultraviolet. Is the integral of something positive over the UV piece of the correlator? That's all I meant. Positive UV stuff. Briefly explained the relationship between this and some recent work on quantum chaos, and then we'll move on to talking about holography. So this is a paper from a couple of years ago by Malvisander Schenker in Stanford. And what they did is they studied four point correlation functions in thermal quantum systems. They weren't just doing quantum field theory. They were doing something very general that applies and can it matter and just in quantum mechanics. But I'll tell you how it relates to this picture in quantum field theory. So to connect the two, all we need to do is to reinterpret this picture as a thermal correlator. So the Minkowski vacuum, what we have here is a four point function in the Minkowski vacuum. But the Minkowski vacuum is the Ringler thermal state. So we can reinterpret this four point function in vacuum as a thermal four point function. It's the same calculation. It's just a different interpretation. Ringler coordinates are just u equals r e to the big T. V equals r e to the minus big T where T is ringler time. And now we're interpreting this as a thermal system, inverse temperature beta equal to two pi. This key contribution to the correlator that we've been talking about, which is one plus v u squared times the null energy up to k times the null energy. So this is the Minkowski vacuum. v squared times the null energy up to constants that I won't write plus r cubed. There's a e to the plus two T minus T, so there's a e to the T. So when you reinterpret what's going on here, so what's going on is we take a null limit and then as we, so let's think of this now as a thermal system. Then these are just, this operator is just the same as that operator, but shifted by time to time plus i pi. But now we're, now this limit where we go to large u is a late time limit in the sense of the thermal system. Because u, so u is getting big, so it's a late time limit from the point of view of thermal system. And what's happening is that, I guess I should write this with, if you do this carefully with the constants, this ends up being a minus. So what we're seeing is that the correlators is one, but then when you go to late times, it starts to deviate, there's an exponential, r is tiny. So it's a small correction, but there's a tiny but exponentially growing contribution to the correlation function. And this is the kind of thing that happens in chaotic systems. So in a chaotic system, if you have two very nearby, in a classically chaotic system, you have two nearby trajectories in phase space, then if you wait a little bit, then they'll start to move apart from each other exponentially. And the exponential deviation between two trajectories in phase space is called the Lyapunov growth. In the context of quantum chaos, this exponential growth, which is e to some coefficient times t, in this case the coefficient is one, that's interpreted as a Lyapunov exponent equal to one. Now the result, okay so there's a similar result that applies not just in Rindler space, but in general thermal quantum systems, and that's the bound that was proved by Matta-Sanichankar Stanford. So what they proved was, you can say it was a constraint on this term as well as constraints on how strong this growth can be. Now I didn't talk about those, but some similar arguments to what I talked about also constrains how strong this growth can be. In fact it can't be any bigger than one here. Into, no, no because it's not literally the same. So the chaos bound that they derive is a bound on this number, and the anach is a bound on, is a sign constraint on this number. If you go back to this contour that we were talking about a minute ago, with a little more work, you can get both of those bounds from properties of that contour integral. Yeah that's right, the Rindler situation is a very special case of the, yeah that's right, yeah. Oh just because there are various constants here, and when we calculate the null energy, we have to evaluate it in the state O and everything. And when you do everything correctly with all the constants, it ends up being a minus sign here. Yeah, no, okay it depends which way we do the contour and everything. So, but it has a particular sign and it's the one that works out to give you the correct sign and the null energy. It's the same epsilon. It's hard to match the signs because they're eyes. So like this, let me not try. They're eyes in these expressions, which are, yeah, but these were just up to constants. Why the, I don't know, I'm not sure what that means. Maybe it's related to what I'm gonna talk about next. I don't know, let's postpone this to the discussion. Maybe I can think about it before then. I don't think it's, the interpretation of this as scrambling and quantum chaos is sort of trivial. Like, it's really just the light con limit. So I know that there's much to learn from thinking of this in a thermal way. It's just another way of looking at it. Things are very different in two dimensions, that's right. Yeah, everything changes in two dimensions. But what, the chaos bound? I don't know, I'm not sure. So I wanna move on to the next point of view on this whole business, which is the holographic point of view. Roughly following Kelly and Wall from a paper in 2014. Before we get into it, let me sort of tell you what the conclusion is gonna be. It's gonna be that in the light cone limit, holography and the OPE are kind of the same thing. And every theory is sort of holographic. Every conformal field theory is sort of holographic in the light cone limit. Not every field theory is truly holographic, but things work exceptionally well in the light cone limit. So first, this is again gonna be a causality-based argument, but now it's not gonna be a causality in quantum field theory argument. It's gonna be a causality in quantum gravity argument. So we're gonna use AESCFT to understand the anac. The first question is, what does causality mean in quantum gravity? That is not clear at all. Okay, so locally, what do we mean? You're not supposed to be able to get past, you're not supposed to go faster than the light cone, but what you mean by light cone depends on the geometry. What you mean by space like sep, whether two points are spaced like separated depends on the geometry. What you even mean by a point at all depends on the geometry. This is not really clear what the rules are. At least locally, I can't give you some simple criterion that says that you're not allowed to travel from here to here in the theory of quantum gravity. There's nothing like that as far as we know. We don't really know what causality means. There are other reasons to think we really don't know what causality means in quantum gravity. One is the black hole information paradox. So when a black hole evaporates, if the information is to get out, that is a causal. I mean, if you draw the Penrose diagram, you have to have some kind of causality violation, at least non-perturbatively in quantum gravity. So whatever the answer is, it must be something very subtle. We can say one thing, which is that in ADS, at the very least, it must respect the boundary causal structure. This is easiest to think about on a cylinder. They'll draw a nice big one. Okay, so that's ADS. And suppose that you send a signal from the boundary, ADS. It goes through the middle, through the bulk. This is not empty ADS. There's stuff in here. There's maybe some black holes, maybe some stars, whatever, in this ADS. It's asymptotically ADS. So it goes through. It eventually comes back and hits the boundary up here. Then if ADS-CFT makes any sense, that signal better not violate causality of the boundary CFT. In other words, if we stick to the boundary of spacetime and draw a null curve along the boundary, a null curve coming around the back of this cylinder, we draw the boundary causal structure, the boundary light cones, then you better not be able to take a shortcut by going through the middle. This is the boundary light cone. And if the boundary light cone makes it to here and the path through the bulk makes it to here, the statement is then delta T should be greater than or equal to zero. Note that this only applies, at least in some obvious way, to signals that start and end at the boundary of ADS. If I send a signal from here to here, I can't say anything, just two points in the bulk. So what we're gonna do is look at the propagation of signals very close to the boundary. I'm gonna do this in the Poincaré patch because it gets a bit hard to work with the cylinder. In the Poincaré patch, the setup is the following. So here's the boundary. Gonna have our usual U and V directions, light cone directions on the boundary. But now we also have a direction coming out to the right, which is the bulk, the metric. So this is ADS D plus one for a D dimensional CFT. And the metric of the Poincaré patch is dS squared is one over Z squared minus dU dV plus dZ squared plus that would be empty ADS. And then we're gonna add to that some matter. So plus some stuff, H mu nu, dX mu, dX nu. The statement that this spacetime has the correct asymptotics, the statement that this is really ADS and not something else, is the statement that H mu nu starts at order Z at the D minus two. So I should have put on here also that we're not just doing three dimensions, there's, we can do higher dimensions, so I'll put an X perp. Okay, so this is our metric. It's not a small, we're not working perturbatively. This is arbitrarily large deviation from ADS, as long as it's asymptotically ADS. But notice that if we stay near the boundary where Z is small, then it's always a small deviation. Okay, so even though we're considering arbitrary excitations, we can work perturbatively in the metric if we stick to near the boundary of ADS. So what we're gonna do is we're gonna try to send a signal along this line here in the U direction. Sorry, my picture is not so great. Does it make sense what that line is? So there's a plane on the boundary. We just went into the bulk a little bit and we're sending something parallel to the boundary in the null direction. So this is a particle at fixed Z naught and X perp fixed Z equals Z naught. X perp equals zero. It's not a G-desic. You can work with G-desics, but we don't need to. It's not only particles on G-desics that have to obey causality. It's any particle that is on a, any particle has to obey causality, even if it carries a rocket along with it, still has to obey causality. So we don't have to talk about G-desics. We could just talk about a causal curve which I'll parameterize as V of U. So if we were an empty ADS, and this was just a null G-desic pointing in this direction, then it would just sit at V equals zero. But in the perturbed metric, the light cones are bent a little bit and we have to change the, we have to change that a little bit. Now causality is gonna require the total delta V, which is like, this is the time delay. It's the deflection as it is going this way as some deflection in the V direction. The total deflection in the V direction has to be positive. If the total, so it's, if you project this onto the boundary, it's sort of headed along this way and it's getting, it can get delayed in the V direction, but it can't get pushed forward. Now I said a few minutes ago that we only have causality constraints if things hit the boundary. So you might be suspicious about this claim because I never, I didn't send a signal from the boundary here, but actually you can't tell from this coordinate system, but actually this does hit the boundary because if you sit at fixed Z in the Poincare coordinates and then you go all the way to infinity in one of the space, in one of the boundary directions, that does hit the boundary. Okay, so it's hard to tell. It's easier to see on the cylinder. So this is the path of that particle and when you go all the way to U equals infinity, it's hitting the boundary up here. So this bound, this causality criterion does apply to this curve because particle hits the boundary, equals plus minus infinity. Questions on the setup? Not sure I understand the question. I mean, what is because of that? I mean, not all particles hit the boundary. So you can have a particle that just sort of bounces around in the middle. Not all particles hit the boundary. So it's important that we chose a path that does hit the boundary. So along this path, which we'll pick to be just along the light cone. So ds squared is zero. And along this path, that's just one over z naught squared minus du dv plus h u u du squared plus higher order terms. These are the only terms that matter along the path. And so we can easily solve for the light cone here. We just want this to go on the light cone. Then we should just pick the path where v prime of u is z naught squared h u u. So this is the path of our particle which is going as fast as possible which is going this way and it's deflecting in the v direction as much as possible according to the light cone in the bulk. So now we can integrate to get delta v the total time delay along the trajectory z naught squared integral minus infinity to infinity du h u u of u v equals zero x perp. Worked perturbatively here because we're working near the boundary. It's not the smallness of h that's making this possible. Well, h is small near the boundary. Is it clear what this quantity is calculating? So from a boundary point of view, we've sent a signal from here. This is kind of way down near infinity, but we've sent a signal from here. This is u direction, it's headed in the u direction, but when the signal hits the, when it lands on the boundary again, it needs to land positive. There needs to be a positive deflection in the v direction, so this has to be positive. Okay, so that's our, in gravitational language, that's our answer. This path that we're talking about is causal if and only if this integral is positive. Now I wanna translate this into CFT language, using ADS CFT. So ADS CFT says that the graviton H be new in the bulk, the T be new in the CFT. Each field in the bulk, we have an operator in the CFT, and this is the mapping for the graviton, and this geometry is dual a state phi with some expectation value for the stress tensor. Okay, so if the graviton is dual to T mu new, and we're talking about a geometry where the graviton is turned on, then that means we're talking about a CFT state where the stress tensor is turned on, and there's a well-known formula relating the two, which is just that the T in the CFT is D over 16 pi G Newton, times little T mu new, where that was the beating term in the metric near the boundary. So if we take our expression for the Shapiro delay and translate that into CFT language, integral du, the integral of the metric perturbation, and you can see how it's gonna turn out, that's just gonna be the integral of the stress tensor. So this is the integral du, the expectation value of T uu CFT of u v equals zero, x perp equals zero, and we said that this had to be positive according to causality, this boundary causality criterion for particles going through the bulk, that's exactly the average null energy condition. That's it, that was the whole argument. Okay, so part of the point to take away here is that what took us one to two and a half hours to do from CFT took like 15 minutes to do from ADS. This is a very general argument and it's pretty easy. I mean, once you know what to calculate, you can prove the anac in 15 minutes. Now, holographic derivation, you might wonder whether it's totally general, of course. Okay, so not all CFTs have holographic duels. Certainly not all CFTs have some Einstein gravity living in ADS that describes them, but a rough statement, which I can't really make this statement totally precise, but we'll see some ways in which it's true, is that in the light cone limit, all theories are holographic. In the light cone limit, when things are, another way of saying it is the following, if things are controlled by the OPE, that means they're being controlled by conformal invariance, and if they're controlled by conformal invariance, you might as well calculate them in ADS because anything, if you calculate something in ADS and it's controlled by symmetry, then it's true not just in theories with an ADS dual, but in all theories, as long as it's something controlled by symmetry like these OPEs. I think this is any dimension, although I can't remember if there might be something special that happens in three dimensions. I can't quite remember. I think it might be in any dimensions. Well, if there's non-uniturity CFTs and causalities violated on both, the unitarity is violated on both sides of the duality, and none of this has to apply. Okay, I could start something new, but we have four minutes, so let's do, the question is, does this prove that non-unitary theories can't have a gravity dual? Cannot, right. I think that it can't have a consistent, it's not a consistent quantum theory, and it can't have a consistent gravity dual, whether you consider that a perfect match between inconsistencies on the two sides, maybe there's a way of thinking of it that way. Gravity dual? Well, I said that particles in the bulk were constrained by the Lake Con, so I've assumed some consistency of the theory in the bulk in writing down, in solving this equation, this was a bulk calculation of where the particle is allowed to go based on the bulk line. Based on the bulk light cone. So if you allow the bulk theory to be sick, then I don't know what the rules are here, maybe we could just make V do whatever we please in the bulk, and then we could get things to be violated on the boundary. I think near the boundary, the answer is yes. I think in some sort of a near boundary sense, the answer is yes. I don't have a, maybe it's not yes in a completely precise way, but it's yes in the sense that everything seems to match if you can do the calculations, and it's only when higher operators, so if you think of it as a field theory calculation, it's only when higher dimension operators and double trace operators start to enter your calculations, it's only then that you notice whether your theory is truly holographic or not. So as long as you do calculations like this that are only sensitive, say, to the stress tensor or the stress tensor two point functions, or even the stress tensor three point functions, then you can't really notice that your theory is not holographic, so that the ADS calculations give you all the right answers. Okay, so we'll stop there for, oh, one more question. Yeah, that's right. So the question is why does this work, it seemed to work in ADS three. I think it works in ADS three, I'm not 100% sure of that, I have to think about that, but I would say the answer is that although in two dimensions our CFT, the CFT derivation that I gave does not work in two-dimensional CFT, but the ANAC is still true in two dimensions. So it's okay if we can derive it from holography since it's still true, but the derivations don't map onto each other perfectly.