 Okay, so let me review where we got to at the end of the last lecture Which was not too long ago, so hopefully you have some recollection so we started with this notion that in ADS CFT When you have a thermal state in the field theory that it's dual to a black hole and then we were Trying to be a little bit more precise. We said is that thermal state is it dual to? Just the region outside the horizon of a black hole Does it include does have information about what's behind the horizon? Is it dual to the entire? maximally extended black hole spacetime and So we had this proposal by Maldesina which said that if You think about that entire maximally extended spacetime, so this is in the sense the most complete Version of this black hole spacetime the Schrodinger black hole spacetime He said that this thing he proposed that this thing Which has two separate asymptotically ADS regions should be dual to? this natural state of Two CFTs that purifies the thermal state, so it's He says that this is dual to the thermo field double so this state if you compute the density matrix For either CFT it's thermal and that corresponds to the fact that this black hole spacetime has two equivalent Regions that look like the outside of a black hole But then it completes that spacetime into one nice Complete spacetime and this one is a pure state of the whole theory And I emphasize that This is surprising in the sense that you're starting with two completely unrelated CFTs. They don't have any interactions between them They are really two independent physical systems and the only connection between these two is that that we're putting the state into an entangled state it's a superposition of these tensor product states and if I think about the individual Tensor product states if I wrote down this state Then I would say there's two theories and Each theory is in the same state, but there's just absolutely no relation at all So if I had this two CFTs in this state I would just have to say that that is dual to two completely separate spacetimes just a two disconnected spacetimes Somehow what we're saying is that by taking a quantum superposition of these disconnected spacetimes then magically according to the proposal this amounts to actually Thinking about a new spacetime which is geometrically connected. Okay, so this is suggesting that maybe the Entanglement is responsible for this connectedness of of the spacetime Okay, and then I also mentioned that from this point of view the thermal entropy of that original thermal state of the CFT Now that we've written this purification The thermal entropy can be interpreted as an entanglement entropy or a subsystem entropy. It's the entropy of Either the first CFT or the second CFT or the entanglement entropy between the two okay, and So now from this from this picture we have Another way to think about the connection between entropy and area So the entanglement entropy between these two CFTs and that's equal to the area of the horizon But now in this maximally extended picture the horizon if I look at a spatial slice So this is supposed to be a spatial slice at some t equals zero that looks like two hyperbolic spaces connected by some wormhole and Then this horizon area is just a surface which is Dividing that space into two parts each part having one of the asymptotic regions and it turns out that this surface is I mean there would be many such surfaces Okay But this surface turns out to be the one that extra mises the area functional so it's it's It's a special surface dividing the space into these two parts Yeah, absolutely, right? So this is a this is a an interesting point one of the things that we have if so this is the causal diagram For this space time what it shows this that if I if I'm an observer Outside of the black hole I could fall into the black hole horizon Okay, but I can never come out of the other side Okay, so I can only end up at the singularity from this side I can fall in but I can't get out to the other asymptotic region. So there's no way for There's no way for me to directly influence what happens on the other side, and I think that's quite similar to To the fact that you've got the two CFTs, and they're not interacting. I can't I can't Change one and directly interact with the other what I can do is to have You know I could I could send something into the black hole and that could affect the future of someone falling into this side of the black hole And that's roughly like how when you have an entangled state you could do you could alter one side You could make a measurement on one part of the system, and then that could affect a later measurement on the other part of the system Okay, so it's a very beautiful connection between something which is entirely geometrical and something which is entirely quantum mechanical Okay All right, so I wanted to return to one of the questions that was asked in the first in the first part We wrote down this as one particular purification of the thermal state It's a special purification, which is very symmetrical on the two sides But I could I could think of many other purifications either in this same system of two CFTs or I could have one CFT and Some other CFT some other system it could be a larger a different CFT or or even some more general quantum system But I'm going to stick with this one CFT or this this two equivalent CFTs and I'll tell you what a Different purification might look like okay, so it turns out that starting from this pure starting from this purification You can show that the most general purification in this two CFT system Would be one where I take that thermo field double state and I act with some unitary some operator Which is a tensor product of a unitary operator and the identity and so if I do this then you can You find that the density matrix for the first copy of the CFT is changed But the density matrix for The second copy is the same okay, so it's still Okay, so if I think of the the thermal density matrix This state would be a purification of that Which is not symmetrical So the the one side is still a thermal density matrix. The other side is now some other density matrix And so the question would be so the question could be well, what would that be dual to? How could we understand that on the gravity side? So a simple way to think about it is to take this you to be Some local operator some local operator in In the CFT one okay, or a slightly smeared local operator if we want to make everything really Well-defined So what does that do so I take this I take this state of the two CFTs And now you just act with a local operator in one of them So in field theory that does something like creating a for example creating a particle it creates some excitation a local excitation in one of the in one of the field theories and In the gravity picture Okay, what does what does create acting with some local operator do? So that local excitation in the field theory corresponds to something where say I act with the operator at this t equals zero That might That would correspond to something where I perturb the gravity solution out close to the boundary Okay, so an ADS CFT doing things locally or changing the UV physics Corresponds to changes which are out near the boundary So I so I perturbed the CFT Corresponds to a local perturbation near the boundary in the gravity system, but then that propagates inward or actually generally speaking that will have effects Going forward and going backwards So that simple kind of Modification where I start with that purification and I change it to a different one You see it might correspond to a different geometry. It's mostly the same as the first geometry, but now there's a little bit of Maybe there's a few gravity waves Propagating in and bouncing off there and Propagating back through the horizon. Okay, so a different pure purification Might in this case correspond to a slightly different geometry If I Acted with a local operator down here That could change That could change the space time in this region if I acted over here It changed it in this region. So generally if I think about acting with You maybe you as a combination. Maybe it's a collection of local things So these would make changes And you notice that in general I could change any place in the geometry accept the region Here except the region outside that horizon on this side. I can't bite by what we just said I can't perturb the boundary on this side and change anything outside here Okay, so we can change geometry They're outside. I'll call this wedge two. So let's say this is wedge one and one two So what does this adjust? So we have these different purifications They have different Density matrix for CFT one, but the same density matrix for CFT two What we see on the gravity side is that you've got space times which are generally The same as that space time in this region here Wedge number two, but different everywhere else Okay, and so what this suggests is that the density matrix row two that we started with the thermal density matrix If we want to say what is that really dual to? It's kind of dual. It's it's dual to the part outside the horizon so it suggests that Row thermal is dual to Just the outside Horizon region and then different purifications would be different space times with that same exterior But some different geometry past the horizon. So it's not you there's not a unique way to extend this geometry We just usually talk about the symmetrical way. Yes. Yes. Yeah, so it's just that if I right if I consider A field theory state say I say of the vacuum of the field theory and Let's say I'm talking about a scalar field and I I act with this operator psi It's a local operator at time zero and position zero So roughly speaking. I mean this state compared to the vacuum it now has Roughly speaking I've created a particle at this place. Okay, but then if I think of the state as a whole I mean this particle didn't just appear In this state, there's a history and so it must have it must have come in From some past and maybe it it spread out and like for some reason because of the way I've constructed It's localized at that particular time, but then in the future in the past It would be it would be delocalized and so so it's only kind of at the one time when it when it's localized Okay, and and it it there's changes in the past and changes in the future and then on the gravity side that that would be indicated by this Yes, I'm sorry, what did you do in one side? Okay, good. Yeah, so so right. I mean if I If I have this and I measure the energy at once I mean this this I'd have to I'd have to have this whole CFT set up in my lab somehow But yeah, if you were able if you were some kind of Experimenter that had control over this and you could measure the energy of the one side precisely then you could put it into a you could put the whole state projected into this and You know presumably this would I argued already that that would correspond to just two disconnected space times So that would just destroy the entire Connection between the two tops so far. I'm just talking about So far, I'm just imagining pure ADS where So so if I have the it's not sorry not pure ADS, but Just the like this this geometry and and maybe small perturbations of it. So so far. I'm just considering a case where Where it would not be unambiguous and later we'll talk about how to How to understand what to say In more general space times Just one other point. So so row thermal Suggestions dual to the outside the horizon region Another piece of evidence for that is that if I know the density matrix for the one CFT Then of course, I can I can calculate lots of observables like correlation functions as long as they're localized to one side and Using these kinds of correlation functions, I could for example probe anywhere in this region by say calculating a response function if I calculate something like this oh of x1 oh X2 Correlation functions like this in the CFT where maybe maybe this one is in the future of this one Then you can think of that in the gravity side as Sending in making a perturbation here and then measuring how it affects the state here okay These kinds of observables would be sensitive to any small Local change within this wedge. Okay, so let's say I suppose I put a little mirror So so I have I have the unperturbed black hole state But now I change it by putting a little object in here could be a mirror or another object, okay Well, then I could I could detect that object. So from the gravity point of view I could detect that object by sending in a signal and Measuring but making a perturbation here and measuring What comes out here Okay, so I could it could be a light it could be a laser beam I could if I sent in a laser beam here, I would I would find the laser beam out here Okay, and that's the kind of thing in the CFT point of view. That's the kind of thing I would I would be sensitive to if If I calculated one of these Response functions, I make a perturbation in one place calculate the effect in another place so with these kind of simple observables that I were at where I Where I could calculate them directly from row thermal from from row on on the right-hand side You could see that I can probe basically anywhere in that wedge and any place in this wedge is is kind of accessible by these by these observations which map over to Response functions, okay, so you can actually see more or less directly how you could go about Figuring out the geometry in that region using this density matrix, okay So I just calculate all these correlation functions and and that will tell me about that region But they don't tell me about the other regions and then we have this other argument that They can't really tell this this density matrix can't really tell me about the other regions because there are lots of Purifications and the other regions are different in different purifications or questions about that. Okay, so for this black hole spacetime Then we have That the thermo field double seems to correspond to the whole thing according to Maldesina The density matrix for the left half seems to correspond to this wedge the density matrix for the right half seems to correspond to this wedge and apparently so So if I know this density matrix and I know this density matrix That actually doesn't tell me the whole state. That's that's just a limited amount of information What's left is the details of how the two sides are entangled with each other Okay, so if I want to ask how is the information about these regions three and four behind the horizon, how is that information contained in Here it seems that it should be contained in the entanglement information About how the first CFT is entangled with the second CFT details So that's interesting in the sense that The the physics of this region, I mean that they'll say the local fields in this region I mean according to this They don't really correspond to Knowing what those are doing. It's not about some Additional degrees of freedom in the field theory We've already got if we wanted to just describe regions one and region two you've already got the Two CFTs and region three is really just about Understanding that is about understanding how those are entangled with each other Yeah, so I think if I if I chose a relatively smooth operation here Then then I could expect a smooth solution here if I did something more singular It could be singular so Schenker and Stanford have talked about these shockwave solutions recently that are are Basically like this if I do something if I do something sharply localized on the boundary then then I can expect some shockwave type solution and Stanford and Schenker discuss these things so those are those are not Completely smooth and then I can imagine smoothing it out by smoothing out the operator that I act with. Oh I'm not sure if I want to do that. Let's Okay Yeah Yeah, I mean yeah, I would I would Think that Yeah, okay, so here's okay. Let's just as a diversion your You know, I think I think the question could be if I consider this is sort of an open question But if I consider a More or less generic state of these two CFTs Okay, so so of that combined system So I have I have the two CFTs Choose some very generic say high-energy state Maybe I fixed the energy and I choose a generic state of the two CFTs. What is that dual to? and What you would realize is that the density matrix for each of these two sides is very close in in in general in a generic state, it's very close to being thermal, okay and so you would you would conclude that Whatever geometry is dual to this kind of generic state. I think would have two Schwarzschild black hole exteriors But then it's a it's kind of an open question. What what is on the inside and I think Right, I think that literature starting from Stanford and Schenker and then later Suskind and others I'm not sure what the current status is I mean, I think they're suggesting that maybe maybe these generic states are dual to something with a long and complicated wormhole But I don't know if there's evidence Whether it should be purely geometrical or not. I mean, I would have guessed that maybe it's not geometrical And let me so let me just mention one thing which is again a little bit of an aside But it connects with some things that people talked a lot about in recent years So there's an interesting consequence of this observation that you know We're saying the thermal state in a sense it's dual to the black hole exterior Okay So remember this thermal density matrix. It's just an ensemble of High-energy it's an ensemble of states of the CFT and generally if I took a typical state If I took a typical state from that ensemble This is what we would call a black hole microstate. So this is a typical state in this ensemble is a pure state of one CFT And this is one of the things that the entropy is counting. So the question is well, what does that correspond to? This is actually closely related to a teacher's last question What does that correspond to if the thermo field double is dual to this and the density matrix is dual to this What is what is the micro? What are the microstates dual to and I mean naively because because this one is Some ensemble of all these microstates So it's confusing because you might think that the microstate is dual to a black hole with some Geometry behind the horizon So you might think Yeah, here's my here's my microstate. Here's the black hole horizon and And then there's some geometry behind that I'm not exactly sure but but at least up until a few years ago people were largely convinced that if you fell into a black hole there would be smooth geometry and and They probably would have said that it doesn't really matter which microstate you're looking at that geometry should be similar Okay, so so if that's true if all of these microstates have some smooth geometry behind the horizon Then You would probably conclude that the ensemble of microstates also has that same smooth geometry behind the horizon Okay, but we've argued that that can't be the right interpretation For this ensemble because different purifications of this have different geometries behind the horizon So one way out of this there's two ways out of this one way out of this is to say These microstates actually have no geometry behind the horizon. This is the firewall answer You could say all of these microstates are you get to the horizon and that's it and then it would be kind of consistent to say that the Ensemble of all of those is this region outside the horizon the other way around this is what's known as state dependence and That is to say that Okay, so so if I wanted to argue precisely that The ensemble would have some smooth that the ensemble would have some smooth geometry behind the horizon Given that all the microstates do I might try to find an operator Where if I take the expectation value of that operator in the microstate This would this would be how I would tell that the geometry behind the horizon is smooth Okay, so supposing that I could find some operator like that that would be Answering the question is the geometry behind the horizon smooth and suppose that there exists such an operator Which is state independent which where I could use the same operator for all of the different states This is normally how it works in quantum mechanics if I want to ask the same question. I use the same operator Well, if there's such a state independent operator Which has the same answer for all the microstates? It's going to have the same answer for this density matrix and it would tell me that the geometry behind the horizon, you know exists and is smooth and And that that's sort of a contradiction because we already know that there are Purifications of this where you don't have a smooth geometry behind the horizon Okay, so the the way around it is to suggest that if I want to learn that if there is a geometry behind the horizon of Black hole microstates and I want to learn about it. I need to conjecture that the operator I need to use for the different microstates is different and then I can't give this averaging argument I can't use the argument for the ensemble so This connects with you know, there's a big recent discussion of whether black holes if you're taking into account quantum mechanics whether they have firewalls at the horizon and to me it seemed that the output of that discussion or one of the outputs was that That that many people agree on was that you either have firewalls or you have state dependence And this is I think a simple way to to argue for that okay, so we've talked a lot about black holes and how how this Decomposition into density matrix for the one CFT and the other CFT Works and wet what the different density matrices are dual to and how the black hole and entropy can be interpreted as a subsystem entropy What we're working towards is the a generalization of this Where instead of this kind of artificial system of two CFTs I just consider say a single CFT and then I look at a region of that CFT and its complementary region So what we're going to get to is is the claim that many of the things that I just said for this Black hole the two-sided black holes are also true if I just think of a single CFT and I kind of arbitrarily divide it up into two subsets and So on the way there It's interesting to Recall a fact about Okay, so we're just going to now consider the case of just a single field theory And I want to recall a fact about actually arbitrary quantum field theories local quantum field theories So I'm going to start on Minkowski space remind you that actually this thermo field double state Makes an appearance Even if I'm just considering a single quantum field theory and even if I'm just considering the vacuum state of a single quantum field theory Okay So Okay, so this doesn't have to be a holographic field theory or a conformal field theory. This is this is general so far As a result from the 1970s as well Okay, so so we have our field theory in Minkowski space and what I want to do is consider a half space so For example x greater than zero and so I want to ask what is The density matrix Okay, now I have to be careful. What do I mean by the density matrix for half of a quantum field theory? Okay, so local quantum field theory it has degrees of freedom over here and it has degrees of freedom over here If I imagine Regulating this by some some lattice for example, then I would have specific degrees of freedom on one side and and degrees of freedom on the other side And I could say I could I could then Definitely say okay. What is the quantum state of this subsystem? At least in the regularized theory It's clear that I could talk about the subsystem and I could talk about the density matrix And then maybe takes take some limit I want to ask the question What is the what is the density matrix for the half space and the answer and I won't have time to Review the derivation of this but I'd be happy to do that Outside of the regular lectures So the answer you may have heard of is that it's a thermal state but not with the regular with respect to the regular Hamiltonian with respect to What's called the Rindler Hamiltonian? It's respect with respect to Hamiltonian which generates Basically, it's actually it's actually it generates boosts in the original. It's one of the boosts generators It generates a flow in this right-hand wedge Which people would call Rindler time Okay, so I'll tell you exactly what I mean So you can consider this vector field. Okay, so that's the that's the time like unit vector and the vector field in the x direction This is x and t so this is this is some particular vector field which I'm considering just in this right hand Actually this right hand wedge okay, so if I if I follow the flow of this vector field it looks like this and And this is a symmetry of Minkowski space. It's the it's the boost symmetry and so I can use the know-their procedure to associate a conserved quantity to that symmetry and That conserved quantity is I'll call it h eta. I'll call it eta the time eta is like the time coordinate that is Moving along those that flow h eta is the generator of that which I said was the boost generator And so the density matrix it turns out is just the thermal state If I if I wrote down the thermal state with respect to that Hamiltonian Which which I could do by okay, so this would be the usual way e to the minus h that That's the thermal density matrix for that Hamiltonian the claim is that is the Density matrix for the half space and there's a there's a simple path integral derivation of that I can either give you references for that or explain it But for now, we'll just take that as as a true calculation Okay This is related to the statement an accelerated observer We'll see a thermal bath of particles because this is this is the time if I if I'm an accelerated observer I would move along one of these trajectories and so it's like my time as an My time coordinate as an accelerated observer and the statement that the the density matrix is thermal is related to the statement at all I will experience thermal physics as one of these accelerated observers Okay, so that's and so this is just a statement about the vacuum of a of a quantum field theory And we have a similar statement about the the other half Okay, so it's symmetrical So similarly row left is equal to there's there's some other time And if I want to think of the entire state the entire Minkowski space vacuum Okay, so we now have a state where there are two equivalent sides and each half is described by a density matrix which is thermal and So you might guess that the full state if I wanted to I could represent it as a thermo field double state Okay, and that turns out to be true so even if I'm thinking about a single quantum field theory and I'm thinking about the vacuum state and I just arbitrarily divide the space into two halves Then this state has a description as some entangled state between the two halves There's a there's entanglement even in the vacuum state And it takes exactly the form of the thermo field double state that we looked at before If you look at a free field theory you can you can be even more specific You can define a set of field theory modes in this wedge and a set of corresponding field theory modes in this wedge and then that basically each each mode in this wedge is Entangled with the corresponding mode on the other side. Okay, so so the vacuum state of a field theory Has this characteristic entanglement which has the structure of the Thermo field double state Incidentally, if I remove that entanglement if I try to if I try to consider a state where this side is not entangled with this side What happens is that the stress energy tensor becomes singular all along this region, okay, so states so this is tangled and States with no entanglement for example Taking a typical state on either side you get a singular Yes, I'm not sure so I guess they're dependent on your definition The argument I know is for a local quantum field theory. Yeah. Yeah. Why does it become singular? I mean, it's like So I've changed the state somehow In the free field case what I can do is simply so so there I define I have the usual modes of my fields on all of Minkowski space and And then I define these other modes On the left and right and and I just I just calculate I can I can calculate That the usual Minkowski space vacuum in terms of these left and right modes. It is a state that looks like this I can then Calculate what happens. I could basically start with a state instead like this which which is not entangled between the two sides if I try to translate that back to To Something involving the modes of the whole theory I Guess I just find that the I mean I just calculate the stress energy tensor and I find that it has a singularity there So I'm not sure. I mean, I think it's basically that By by disentangling them at that one point You've you've made a large modification to like the UV physics so somehow if if you want a non-singular stress tensor then the the the UV physics should should be the same as the vacuum state basically If I zoom in to just very small distances Then everything should including the entanglement between local degrees of freedom should look like the vacuum state But if I completely disentangle it then even if I zoom in You know a very large amount then I can still detect differences Compared with the vacuum state and so it corresponds to a very singular kind of modification of my field theory state Okay, great. So that's that's a statement about any field theory now what I want to do is Go back to CFT I want to think about I want to think about the CFT On a sphere But now not in the thermal state just in the vacuum state. Okay, so this is dual to What's called pure global ADS space time? Hey, this is a space time where where the time slices are hyperbolic space Negatively curved space. It's some warped product of hyperbolic space and and time and what I want to do is Think about a similar question. Okay, so instead of dividing the plane up into two regions. I Want to divide the ball up the sphere up into two regions The Bar okay, so CFT again in the vacuum state on a sphere and I want to ask well, what is the Density matrix for this half and what is the density matrix for this half and you know, how can I write the full state as some entangled state? Okay, so the good news is that I can map this there's some conformal transformation that maps this problem to this problem So precisely Okay in in order to Be able to draw this. Okay, so so so this ball is that's that's like the boundary Okay, so that's that's the geometry. It's the boundary geometry at some particular time This is the geometry my CFT is living on so that split into two halves Corresponds to like this half of the ball and this half of the ball B bar So let's let's draw In order to be able to draw this I'm going to flatten it out a little bit Because I want to draw a time as well So the statement is that this there's a there's a conformal transformation of the metric of this of Minkowski space That maps this Rindler wedge into What we call the causal diamond associated with that ball Okay, so so think about all of the points P where every causal curve through P passes through B So it defines a diamond On this diagram here Okay, so this is the ball the diamond that I'm talking about it's it's a dime It's something on the boundary going into the past and future Okay, and so the claim is that by by making a conformal transformation from this picture I can map this Rindler wedge to this diamond-shaped region associated with this ball and this region here maps to a diamond-shaped region associated with the complimentary ball So it's it's important to understand this picture. So if you have questions It's a good idea to ask them So we so we've got we've got a sphere The inside of this cylinder we're not worried about yet That's that would be the inside of ADS, but I'm still thinking about the field theory. So I'm thinking about the sphere I've just drawn an S1, but in general, it's an SD and then there's time and then I've drawn a Ball on this I've divided the sphere into two parts. Okay, so that's like on this S1 I've divided it into this part in this part and then for each part. I've considered the causal diamond on the boundary and and so the claim is by some Informal transformation. Okay, so now the CFT in Kowski space can be related by the symmetry to a CFT on a part of the sphere times time, okay and and The two Rindler wedges of Minkowski space map into these two wedges over here So the bottom line is by knowing what the density matrix is in this simple case And I know this for all quantum field theories and then I'm going to apply a conformal transformation Then I I'm going to know immediately what the density matrix is For this half of the ball and for this part of the ball And I also know I can also I can also say that the full state Of my CFT on the ball is like a thermo field double state On the between the left half and the right half of the ball So let me write down that conclusion. So the vacuum state on the ball is equal to What we learn is that it's equal to Something that looks just like the thermo field double state be Bar Where these energies are so so that Rindler Hamiltonian it maps into some There's a there's a Symmetry generator that lives inside this ball. There's a conformal killing vector that lives inside the ball So in this that lives inside this particular region, okay? So there's a particular definition of time in this region of us for a CFT. There's a particular time like vector field That defines this a symmetry. Okay, I'll call that H data to pick another Greek letter and so there's there's So there's a particular natural Hamiltonian that defines a time inside that ball and there's one that defines a time inside that one and Now my vacuum state of the field theory is equal to this thermo field double state with respect to those two Hamiltonians Yeah, yeah, so the same one. Okay, let Okay, let's just be a little bit more so so the entirety of Minkowski space the entirety of Minkowski space it gets mapped to one particular Wedge of this global ADS Okay, so I've the site okay the side view is this okay, so the entirety of Minkowski space Gets mapped into this diamond on the boundary of that cylinder and then These two wedges I'll use the colored chalk Get mapped into This wedge and and some other wedge over here Okay, so it gets mapped into wedges. Okay, so yeah, it's one conformal transformation Okay, so if I know this data Minkowski space that I just do the mapping and I get this Okay, so what's the point of all this? We're at we're at a good spot now because what I've done is I Just started with a plain old CFT in the vacuum state Nothing to do with thermal physics or black holes or anything and what we realized is that The formal structure of that state the entanglement structure of that state If I make some arbitrary decomposition into some ball And some other complement of that ball then the entanglement structure looks exactly like what we had in the black hole case it is a thermal field double state and so we we can kind of use the same arguments as When I was talking about the black hole, so if we ask what is row left and row right dual to and You know you you can so so I made all these arguments in the black hole case, but they're exactly the same So it suggests that the the density matrix for this ball Would be dual there's some there's some now bulk wedge and a bulk wedge there Okay So I'm just I'm just going to repeat all the things I said for the black hole, but now in this case So Left dual to region one row right dual to region two and It suggests regions three and four one two three four I related to How that exactly how the two sides are entangled we can even ask about the what about the What about the actual entanglement entropy between the two sides? Does the entanglement entropy between that ball and the other ball? Match with the area of With the area of the surface Okay, so that was that was the in some sense the starting point in the black hole thing that we had the connection between Thermal entropy an area we later understood that as the a measure of the entanglement between the two CFTs in the thermo field double Now we've shown that just in a regular single CFT you can actually view it as a thermo field double We can make similar arguments that the left and right density matrices map into these two wedges that end at this particular surface in the bulk Defined by you know which By this causal construction which rich regions can we communicate with And so if it's like the black hole case then we would say that the Entropy the entanglement entropy between the two halves should match with the area of that surface okay, and The answer that is that it actually does match They're both infinite but so that's That's good But there's a more detailed matching even if even if you regulate the thing Okay, so in ADS CFT we kind of understand that if you put a UV cutoff on the field theory This corresponds to having some IR cutoff for example just considering the region of ADS out to a certain Distance but then not going all the way not going the infinite distance to the boundary okay So for a conformal field theory for one of these ball-shaped regions you can actually compute the entanglement entropy and It matches exactly with the area of This surface that I've drawn in the bulk Geometry, so that's the first statement where we've generalized the Beckenstein Hawking calculation to the sorry the Beckenstein Area entropy connection to a case where now we're not talking about a thermal state at all We're just talking about say the vacuum state and we're talking about some subsystem in the vacuum state and we find that the Entropy of that subsystem matches with the area of some naturally associated. I mean basically basically what we've done is we've made this Pure ADS look a bit like a black hole by making some arbitrary choices And then we carried over the entropy area connection and we found that it worked and so to I Guess I'll finish up So to conclude I mean basically then I think this is a way of motivating What Ryu and Taki and agi did it's not really how they motivated it They basically now said what about More general states and more general regions okay, so what What if I instead of looking the vacuum state I consider some excited state, which is dual to some other geometry Okay, and what if instead of considering this region of the ball I consider some other Some other region of the ball, okay Is there still this connection between entanglement entropy and The area of some surface in between Okay, and their conjecture basically says yes that that there's that there's this can that there's this connection in general Okay, so if I have a state psi and whatever geometry is dual to it And I divide up the boundary into a region a and some region a complement Okay, so in all the examples so far the subsystem entropy of a Was equal to the area of some surface that divided the bulk space into two parts Where one part had the boundary a and the other part had the boundary a bar Okay, so in the general conjecture, it's the same thing we think about we think about the two Regions we think about some spatial Slice and we think about some surface that divides that into two parts Okay, but then the question is well which exactly which surface Should we be talking about what is the natural surface and this is this is kind of the meat of the conjecture You're even tacky nagi say that the entanglement entropy is equal to the area Remember when we talked about the black hole case There were lots of surfaces that separated in the two halves The horizon turned out to be the one with extremal area it extremizes the area functional And this is the this is the conjecture in general that it's the area of a tilde Which is the which extremizes so it divides space into two halves where you know a a prime with boundary a and a bar prime with boundary a bar and and It extremizes the area Functional and this is actually the covariant version of the conjecture By hubini and ranga mani and tacky nagi Ryu and tacky nagi originally just considered spatial Spatial surfaces and this this is covariant so it's supposed to apply okay So next time so it's kind of a motivation of how to go from black hole Entropy area connections to entropy area connections in general This is the Ryu tacky nagi conjecture and the the last two lectures I'm going to talk about first a little bit of evidence for that more evidence for that and then Consequences there are lots of amazing consequences of that and so that's all for today