 Okay, so today, so today we're gonna do one last, we're gonna do an example of this connection between holography and conform field theory and the light cone limit. And then I'm gonna do some quantum information, tying this to some of the things that Marina has been talking about, and then we'll wrap up. So the first thing I wanna do is it not on? Maybe the volume? Can you get it? Testing, testing. I think I turned it on, but. Is it, oh, now it's working. Okay, how's that? So I wanna re-derive the light cone OPE directly from holography. So we've gotten the ANEC two different ways, one from the light cone OPE and one by studying the propagation of signals through anti-decider. And the idea now is to explain why those two things had to agree and exactly what the relationship is. So consider an operator psi with a large dimension. This is just to make things a little simpler and isn't really necessary. This corresponds in ADS-CFT to a particle, so this heavy operator is dual to a particle with a large mass. I should say that by large, I don't mean very large. There's a hierarchy of scales here when we're doing ADS-CFT and it's still much less than N squared, which is related to Newton's constant in the bulk. So the idea is to look at a particle that propagates through ADS in such a way that it's heavy enough to act like a particle, not a wave, so it's heavy enough to travel on geodesics but light enough that it doesn't really back-react on the geometry. So if we wanna calculate in some state of the CFT dual to some geometry, if we wanna calculate the correlation function psi-psi, then in this limit of a heavy probe particle, you can calculate to leading order, you can calculate that two-point function, the WKB approximation, or geometric optics, which is just to say that the two-point function is calculated as the length of some geodesic. Geodesic lengths in the bulk are related to correlators on the boundary. L gamma is the length of the geodesic connecting these two points. So I'm gonna put the two points, just like I always have been, the plane, I'm gonna put one point here and one point here. And then the statement is that L gamma is the length of the geodesic through anti-decider that connects those two points on the boundary. So this is some space-like geodesic length that calculates the correlator. Now this geodesic length is infinite, so I'm gonna normalize, we could calculate renormalized lengths, or I could just normalize this by the vacuum geodesic length and then I'll get something finite. So this is calculated by minus delta psi times L gamma minus L gamma vac. Here I've replaced the dimension, I've replaced the mass of the field by the dimension. So this, so usually there's a slightly more complicated relationship between mass and scaling dimension in ADS-CFT, but at large mass it just reduces to mass equals dimension. I've set the ADS radius to one. If the geodesic stays very close to the boundary, which it will, so if we're in the light cone limit, then this geodesic is always gonna stay very close to the boundary. And you remember from yesterday that when things stay close to the boundary, you can treat the bulk perturbatively because when things stay close to the boundary, the bulk metric perturbation is suppressed by the distance from distance, by the radial direction. So you can treat this perturbatively, it's approximately one minus delta psi times delta L, where delta L is this geodesic length. This you can calculate very easily, perturbatively in the bulk. So you just find the, you find the geodesic in empty ADS, and then you vary it to first order. Since it's a geodesic, you don't have to move the geodesic, you just integrate the metric perturbation over the old geodesic. So that takes a couple of lines, and I'm gonna skip it, and you get u over two integral minus u to u, d u tilde, one minus u tilde squared over u squared, h u u of u tilde at v equals zero, x perp equals zero, and z v of u tilde, v of u tilde is the solution of the vacuum geodesic equation, u squared minus u tilde squared v over u. I'm working here in our usual light cone limit, v to zero. Okay, does that make sense what I did here? I skipped some steps, but it's a straightforward calculation to calculate the first order change in this geodesic length. Yeah, the question is whether this is for any state phi, and the answer is yes. Even in a black hole, which is a big deviation from the ADS vacuum, you don't notice that if you stay near the boundary, because if you stay near the boundary, there's the metric expansion says that the perturbation is suppressed by powers of z, z being the radial direction. So that's sort of what makes the light cone OPE so powerful is that even completely non-perturbative things, you can treat perturbatively. Actually, this is a general sort of fact about the light cone OPE, is that it gives you a perturbative parameter where you didn't really have one. The approach to the light, like in CFT language, it's the approach to the light cone that gives you a small parameter to expand in. In bulk language, it's the closeness to the boundary that gives you a small parameter to expand in. So even when things are totally strongly coupled and non-perturbative, you still have control. Now, remember the ADS CFT dictionary in four dimensions, I'll write the constants in four dimensions, but this is general, tells you that near the boundary, the metric perturbation is related to the stress tensor in the dual CFT. So we can just plug that relationship in here, which says that HUU near the boundary is four pi g Newton z squared times the expectation value of TUU. So if we plug that back into our formula here for the WKB approximation, it becomes equals two pi g Newton v u squared that v is coming from these factors of, there's just some, a factor of this thing squared changing that formula. So v u squared integral minus u to u d u tilde, one minus u tilde squared over u squared times phi TUU. That's exactly the formula for the light cone OPE that we derived in the first lecture. Familiar? So if you, I calculated this as an expectation value, but we could have been in any state. If something is true in any state, then we can just strip off the, we can just strip off the external operators, the phi's here and just call this an operator relation. So we get something like psi psi is one plus some constant v u squared a girl from minus u to u d u tilde, one minus u tilde squared over u squared squared TUU. When we were talking about the, when we did the OPE and we kept all the leading twist operators and we added them up and we calculated the OPE coefficients and turned that into an integral and so on, this is exactly the answer that we got. So this is the sort of thing that I meant when I said that holography is in a way just capturing for you properties of the OPE, this is not true in general. In general, holography is the dynamics of the theory important and I'll come back and say a few words about that later today, but in the light cone limit, it really is just the OPE. Yeah, I can't hear. Sit again. I can repeat the question. So the question is, what about the double limit? So when we talked about CFT, we took a double limit. First we took v to zero and we got exactly this formula and then we took, so that was taking these to be like cone separated and then we took u to infinity and we could do the same thing here and the interpretation is the same. The reason we took u to infinity before was just to turn this into the ANEC operator. But in fact the relationship is more general. That you don't even have to, you don't have to take the second limit in order for this to agree with CFT. It already works at this stage and we could now take the double limit if we wanted to. There are lots of other cases where these light cone expansions and near boundary expansions are very easy to do on the gravity side and pretty hard to do on the CFT side from correlators to entanglement to other things. Okay, so now I'm gonna change gears for the last 45 minutes and talk about this coefficient. This coefficient works out. It comes out exactly right. So there's a, this G Newton is related to one over CT and if I kept all the delta size and CTs, then we would get exactly the same coefficient that we got from the CFT side. What was M? In the small mass limit, good. No, it really is delta psi. So that's what we saw in the CFT that it was really delta psi. If we wanted to do this at this, holographically at small M, we'd have to use Witten diagrams instead of GDZX. And then we would get delta psi again. No, just from it being not a, just from the fact that you can't use GDZX for light fields. So you could do the same thing that we just did. You could do with a Witten diagram instead of a GDZX where you just draw a Witten diagram connecting this two point function between these two points. If you use that Witten diagram, then you would end up with exactly this formula with the delta psi and not an M. Well, the way to think about it is that these, it would be a diagram that looks something like this. That would be the Witten diagram. So there would be a point connecting here to there and here to there, and there would be a free graviton point connecting you to whatever it was creating the state in the bulk. So you have to do sort of this stripped Witten diagram. It's a single graviton because it's still true that whatever Witten diagrams you do are still gonna be dominated in the light cone limit. So it's still gonna be a perturbative Witten diagram calculation. Okay, so I promised three perspectives on these energy conditions. So the last one is quantum information. I think this is really one of the, this is really one of the key reasons for being interested in these energy conditions recently, which is that they connect together the kinds of things that we've been talking about so far with quantum information and the connection between gravity and entanglement. So at this point, I'm gonna describe the result of Faulkner-Lee-Pardkar a couple years ago. The starting point is the relative entropy, which we heard about a little bit from Marina, relative entropy between two density matrices, rho and sigma is the trace of rho log rho minus the trace of rho log sigma. If that expression doesn't give you a whole lot of intuition for anything, then let me write it a little bit differently. This is h sigma in the state rho minus s of rho minus s of sigma. h sigma is defined as minus log sigma minus s of sigma. Okay, so I haven't done anything. I've just added and subtracted the entropy of sigma because this expression here is minus s of rho, the von Neumann entropy, and then the trace rho log sigma is this term and I've just added and subtracted the entanglement entropy of sigma. But notice that the s term here is just a constant. And so h is called the modular Hamiltonian. It's just defined by pretending that the density matrix is a thermal, just pretending that the density matrix takes the thermal form, but in general, doesn't have to have anything to do with the thermal state. Okay, so now that it's written this way, it looks a bit like a free energy, right? This is the energy in the sense of the modular Hamiltonian and this is an entropy term. This minus s of sigma, that's just a constant. So we don't really have to worry about that. We just, there's freedom in how we define the modular Hamiltonian and that's just a constant. Okay, so this is sort of like some kind of free energy and obeys some similar properties. It's a measure in information theory that relative entropy is a measure of distinguishability. There's a precise statement, although I don't quite remember what it is. It says something along the lines of, the relative entropy tells you how many experiments you have to do to figure out whether you're, to tell the difference between sigma and rho. Like if somebody hands you a system and you wanna know if it's in the state sigma or not, then it's how many experiments you have to do to decide. It obeys lots of nice things. One that I'll use is monotonicity, MPT. MPT is for monotonicity under partial trace. So this says that for A prime subsystem of A, S of rho A prime sigma A prime is less than or equal to S of rho A sigma A. This is pretty intuitive. So if you have a measure of distinguishability, it can only get harder to distinguish two states. If I show you less of them. Okay, so the less you can see, the harder it is to distinguish them. But it's also just something you can prove mathematically. So now we're gonna apply this relation, a generalized Rindler space, which I started drawing before the lecture because I can never draw these pictures properly. So this is sometimes called a nil cut. The picture is as follows. So in Rindler space, you just take space time and you divide it on a plane. So in this picture, that would be like if space was this way and time is going up, then we would just divide it along this plane here. That would be Rindler space. The generalized Rindler space is like a wiggly, it's like a wiggly Rindler horizon. So the idea is that you give the Rindler horizon some curvature, but the boundary of the, is not just totally arbitrary, the boundary of the Rindler horizon lives on a nil surface. So that's what I've drawn here. I've drawn the, this is the nil surface in the U direction. So space goes out this way and out that way. So you pick a curve. These are traced over A prime and these are traced over A. Yeah. They have changed, they have changed. So row A prime, that's why, so this is, yeah, let me explain that. So A prime, so if A is A prime B, we've traced over B. So to get from here to here, we've traced each density matrix over B and then recalculated the relative entropy. We've done a partial trace of both of those density matrices. Okay, so we pick this nil cut, which I'm gonna call lambda, so I'm gonna call this curve here is at U equals lambda gamma of X perp. So perp is the direction into the board and then the this, so once you've picked a curve like this, you've divided space time into two halves, sort of wiggly halves. One of them is over here, region A, and it sort of just curves, okay, so there's a sort of lifting, I think we're just sort of lifting up the corner of the rug and then region B is everything else, okay? So the strategy now is to study entanglement or these quantum information inequalities as you deform the cut. For example, you can start with just ordinary Rindler space where it's completely straight and then you can ask what happens to all these entanglement quantities when you start pushing the, when you start adding some wiggles to that cut and the first order wiggle is actually gonna give the anac. So with this setup, we're gonna choose sigma, so sigma is called the reference state. We're measuring how far we are, we're sort of measuring how far we are away from the reference state. So choose the reference state sigma to be the vacuum, to be any pure state. Monetonicity under partial trace says that the derivative with respect to lambda by sigma A, negative. So gamma is the shape of the wiggle, lambda is the size of the wiggle, that's what this picture says. If we increase lambda, we shrink region A. If you're used to thinking about entanglement at a fixed time, then you're used to just drawing a plane and choosing region A and B on that plane. But in Lorentz and Varian theories, you can just as well, if you have a causal diamond like this, if this was your original region A, a tilted region like this is a subregion of region A and all the usual entanglement logic applies to that subregion. The reason, I mean, it's clear that it's a subregion because you can think of region A as being this slice plus this null thing. Okay, so you just deform your time slice until it looks like this and then you trace out the null piece. Okay, so this is a very common trick in deriving things in relativistic theories using entanglement is to not just think about regions in the sense of space but regions in the sense of space time. That's right, yeah. It really is just a subregion because we could just, instead of calling it A, we could call it B A prime and then it's really just a subregion. Okay, so with our conventions here that our cut is being deformed up that way, the increasing lambda is shrinking region A so that's why that was a less than zero. Similarly, D lambda S of rho B, sigma B can only increase. So adding these two inequalities together, we have the relation zero is less than D lambda S rho B sigma B minus S of rho A sigma A. I'm just gonna plug in the definition of the relative entropy and reorganize the terms a little bit. So there's the definition is over there on the first board. So there's the modular Hamiltonian terms and then there's the entropy terms. So we have H B, H A in state row and then we have the entropy terms of sigma B minus S rho B minus S of sigma A plus S. So just copy that definition onto this board twice. Now the entanglement term is all just canceled with each other and the reason for that is that the entanglement entropy of a region is the same as the entanglement entropy of its complement because we're working in a pure state. So when you have a balance between what's happening to region A and what's happening to region B and all the entanglement term is just canceled. So in detail like S of sigma A is always equal to S of sigma B no matter how you do the cut. Okay, so these terms cancel. That's the vacuum term, the reference state but the same is true in any state. So these terms cancel. So here I've used S of rho A is equal to S of rho A complement. So the last thing we need to do is understand these modular Hamiltonians. So we've reduced this inequality to this statement about modular Hamiltonians. Now this statement is gonna be that this is the anac but that's what we have to get to. So I'm gonna just give you the formula for these modular Hamiltonians. Before I do the wiggly cut, I'm gonna remind you the modular Hamiltonian for Rindler space, which is just one where you set lambda x, lambda gamma up to zero, modular Hamiltonian for Rindler. Well we know that Rindler space is thermal in the Rindler sense. There's thermal with respect is thermal at temperature inverse temperature two pi with respect to the Rindler Hamiltonian which is the generator of Lorentz boosts. Okay, so if we take the log of the Rindler thermal density matrix, then the modular Hamiltonian HA just two pi times the boost generator which is the integral for the perpendicular directions and integral dy from zero to infinity of y p zero zero. This is the generator of boosts. I've just used the fact that the normal is time direction and then we get the boost vector is y in the t direction. Using conservation you can rewrite this. The expression that I just wrote since it's integrated over Rindler space like that. It's an integral over this region. But using conservation you can deform this and write it as an integral over the Rindler horizon. This is a conserved current and by pushing this integral up to the horizon you can write it as two pi real du from zero to infinity is over this Rindler horizon. So this part has been well known for a very long time but what's been understood only recently is how to do this for wiggly for wiggly generalized Rindler horizon. So general null cuts. I'll write the answer and then say some things about it. So the answer for a general, oh and I should say also that this is pretty easy to derive. In modern language, with modern techniques and thinking about this as a Euclidean path integral you could derive this in about five minutes. So this is fairly easy. For the general null cut the formula is almost the same. So the formula for ha is two pi times integral dx perp, integral du. So we're gonna write it as an integral over the future horizon here which means that u runs from, so it's gonna be an integral over this region. Okay, so the limits of integration on u are that it starts at the wiggly cut and goes to infinity. So it's from lambda gamma x perp infinity and then it's just the same expression but now we have to shift by the starting point. u minus lambda gamma x perp times tuu. That's the whole answer. Okay, so this was first derived, I think by Aaron Wall in free field theory. Then it was derived in this paper of Faulkner and collaborators for a slightly wiggly, so at first order in the wiggle and they had some reason to conjecture that it might be true in general and then it was derived in general by Cassini, Teste and Tarova just last year. So I'm not gonna give the derivation. It's pretty hard. It's pretty hard to derive this. It would take a long time. I'll just say that I was really surprised to see this formula. So in general, a modular Hamiltonian does not have to be nice. Remember the definition of the modular Hamiltonian is that you take any density matrix and you just take the log. There's no reason in general for that to be a nice expression. And prior to this one, there were maybe two or three known cases where we could actually calculate a modular Hamiltonian. There was Rindler space. You can do a little bit better in two dimensions by doing conformal mappings or free fields. I think that was, those are all the ones I can think of. Oh, a sphere, you can do a spherical region in conformal field theory. I think that's it. So this was a new one. In general, modular Hamiltonians could be totally crazy, totally non-local, impossible to even reasonably write down expressions. But for the null cut, the statement is that there's this very simple looking modular Hamiltonian, which is just basically a sum of Rindler Hamiltonian, a Rindler modular Hamiltonians along the transverse direction. So there's like, if you just cut it into transverse planes, you just write down the Rindler Hamiltonian for each of those and that gives you the right answer. So that can be derived a couple different ways. I think the most direct derivation is that you apply perturbation theory to the density matrix. You can think of this deformation perturbatively, so you apply perturbation theory. You work very hard to calculate the first order term and then you can write down a differential equation that describes the flow as you increase this deformation and then you can solve that differential equation. Okay, so we're just gonna take this equation and plug it into our inequality there. If we take the derivative of the modular Hamiltonian and then we're just gonna look at the first order case. So we're gonna take the derivative, the first order deformation, so we take the derivative and evaluate it zero, then the derivative acts on this equation in two places. It acts on the limit of integration and it acts on the integrand. But this one does nothing at first order because that will just pick up for you the integrand at the endpoint, which is zero. Okay, so we can just ignore the action of the derivative here and we easily get the answer, which is two pi, sorry, minus two pi times integral x perp gamma x perp integral du from zero to infinity duu. Did I get the sign wrong? I think I got the sign wrong. Okay, I hope that doesn't mess up my enic but it works out if you do all the signs correctly. But the question is whether I'm gonna get the right sign when I do the enic. I might have, the minus was right? Okay, good, good. Okay, so now this integral is from zero to infinity. Okay, so this is what some people call this the half enic quantity. It's like it's the half null energy. You don't integrate over a complete null line. Actually, this thing, we didn't actually have to do this thing where we subtracted two terms. We could have just studied the modular Hamiltonian of one region and then we would get an inequality relating this to the entanglement entropy. But since we're talking about the anac, I subtracted the two sides to get rid of all the entanglement terms and then if we do the same thing for region b and add these up, we get d lambda hb minus h a is the integral two pi integral dx perp gamma x perp times the anac thing, du minus infinity to infinity, duu. We said that monosynicity of relative entropy imposes positivity on this combination. Now this is true for any gamma. This is true for any positive gamma because we were assuming that gamma was going in that direction. So this is true for any positive gamma. Then it must be true for the integrand itself, integral du from minus infinity to infinity, du, positive. And that's our third and final derivation of the anac, now from a completely different point of view. This I think is the most general of the derivations that we talked about. I didn't have to say anything about whether it was a free theory or an interacting theory. I didn't have to use a biography. There may be some assumptions about the continuum limit of entanglement entropy and what exactly it means to split the Hilbert space into regions A and B. I think that it would be nice to understand some of those issues a little bit better, but there are subtleties that I think will work out. I think so, although I haven't really thought about it. So I think the answer is yes, but I could be missing something. Okay, so I'm gonna close with just some, discussing some things until I run out of time. Okay, so we've talked about, we've sort of used the anac to talk about various relationships between correlators, holography, quantum information, and shown how we can do various calculations that relate these. They're sort of tied together by this idea of working on the light cone. Here we were doing correlators on the light cone. Here we were doing deformations along the light cone. So this is a limit where we can do lots of calculations and match these things together. There are a few, okay, so there are a few things that I didn't really discuss that I just wanna mention in the last five minutes in words. So one is, these have to do with sort of extensions and applications of these ideas. The first is the conformal collider bounds. So this is work of Hoffman and Maldesena. Oh, eight. I just wanna tell you how these fit in. So this came prior to all the more recent work on the anac. In that work, what they did is they studied conformal field theories and they just assumed the anac. Okay, so they assumed the anac as input. They had some physical arguments for it based on the idea that a calorimeter in a collider should not give you a negative number at the end of the experiment. That was their argument for it. But they just assumed that the anac was true and used it to constrain OPE coefficients. The way they did that is by studying states like this. Okay, so the anac, this is a positive operator, it has to be positive in any state. Once you have that fact, you can use it to constrain the data of the theory. You can constrain conformal dimensions, you can constrain OPE coefficients. In particular, so if you're familiar with CFT, then you know that the three-point function of scalar operators is uniquely fixed up to a coefficient. That's almost true for spinning operators. The functional form is completely fixed, but there are different tensor structures that can show up. So like TTT, for example, in a conformal field theory is completely fixed up to three numbers. People often call NS, NF, and NV. So these are just the coupling constants. Instead of having one OPE coefficient in this three-point function, there are three OPE coefficients, and those are their names. So Huffin and Maldesena derived bounds, used the anac to derive inequalities on these coupling constants. And you can do much more elaborate things, you can study more complicated states. Recently there was a paper by Cordova, Maldesena, and Teriachi where they looked at similar effects in superposition states of T and other operators and they derived some new set of bounds. So that's one application of the anac. Second thing that I didn't really get into is the question of large N. The anac is a statement about any conformal field theory and we derived it using the OPE. We expect that holographic theories, I should say not just large N, but also a large gap, just a way of saying that the theory has to be so strongly coupled that higher spin particles are lifted up out of the spectrum. So the anac is true in any theory, but we expect special things to happen in holographic theories, okay? So you might imagine that if we repeated, say the OPE analysis in a holographic theory, or not using holography, but just purely from quantum field theory, if we repeated the analysis in theory with large N and a large gap, we might hope to get a stronger bound and that's exactly what happens. So the way that works in practice is that now we have a one over N. In our OPE analysis in these lectures, we've been using the leg cone limit as a small parameter, but now we don't have to do that anymore. Now we have a one over N as a small parameter, which means that the kinds of things we've been doing, you can redo, but in a much broader range of kinematics. So you can sort of trade V goes to zero for one over N goes to zero, and then you can apply all the same techniques, but not even being in the leg cone limit. You can apply the same methods in a non leg cone limit. And if you do that, you get a stronger set of constraints. Instead of getting the ANNIC, instead of getting the ANNIC, you get a more elaborate operator obeys a positivity condition. From the field theory point of view, that operator is just very complicated looking and terrible, but from the, if you rephrase it holographically, it's the length operator in ADS. So instead of getting the ANNIC, you get a constraint on the emergent length operator. You can use that length operator to derive constraints. So just like Hoffman and Maldesena did for TTT here, you can now evaluate things like T length operator T and impose positivity on things like this. When you do that, actually something sort of magical happens, which is that you don't just get inequalities anymore. This looks like an inequality. But what happens is that you get upper constraints on the ends and you get lower constraints. And then in the large N theory, the two just collapse on top of each other. So you get upper bounds and lower bounds. And as you take N bigger, you just get, they just are the same. So instead of getting inequalities, you fix the couplings. They're just set to certain values up to one remaining coefficient. And the values that you get are exactly, as they must be, are exactly the prediction of Einstein gravity. So the conclusion is that taking this ANNIC, so if you take what we've done for the ANNIC and you kind of follow your nose and extend it in the natural way to large N theories, that you can derive Einstein gravity at the level of the Graviton three point function, the Graviton three point couplings. And this was actually, at least our initial motivation for thinking about these causality constraints was that there was a paper by Kaman Ho Adelstein, Maldesena and Jabayev a few years ago that used, that studied the relationship between causality and the Graviton three point function from the gravity side and understood that the two were closely related. So the idea was to understand that from the CFT. I guess I'll stop there. Okay, thanks.