 Testing testing that seems good. One reminder after this lecture there will be a picture taken so please don't run away so the picture will be in this area here so just wait a few minutes after the lecture ends so that the picture can be taken. So the lecture now is the first lecture of the black holes and quantum error correction is by Christopher Aikers from MIT please. Thank you and thank you to the organizers for putting this together. I'm having a lot of fun learning from the other lectures and meeting many of you and I hope that continues and please let me know if this mic isn't picking me up very well. It's kind of just hanging here. So I think my starting place for thinking about this topic is that I have many questions about quantum gravity that I would like to see answered and you know some of these questions are like what happens near singularities in black holes. Do black holes or generic state black holes have firewalls? How is space time emergent in quantum gravity because that's a thing that seems to happen etc. And one route to getting answers to these questions seems to be ASCFT, this famous duality which I'll say more about momentarily but the idea is that you have some quantum gravity theory in some D plus one-dimensional space time and it's dual to some quantum mechanical theory and one fewer dimensions and the idea is that you know in principle we understand the CFT maybe in practice it's difficult to compute things but at least because we in principle understand this side and there's a relationship between them we have a route to getting answers to the questions that we said momentarily just a bit ago and the route is this three-step process so you could say all right have some question like does this black hole have a firewall I can just formulate this question so on the on the ADS side I can use the dictionary between the ADS and the CFT to map this question to the CFT after mapping the question to the CFT I answered there so for example you might have some operator near the singularity of a black hole and you want to know what's going on there so you want to evaluate the expectation value of this operator or something so you might imagine you could map that to some operator in the CFT and evaluate the expectation value of that dual CFT operator and that would allow you to learn something so that would be the algorithm so given that you could in principle or you know we have this three-step process that we can lay out why were all of our problems about quantum gravity at least these that I mentioned not solved 25 years ago what's the bottleneck here and I would say certainly not one we have plenty of questions three isn't the only bottleneck so I'm not I'm not gonna I'm certainly not gonna say that we have the ability to answer any question we want and say n equals 4 super Yang-Mills but it's not like everyone studying quantum gravity is just studying n equals 4 super Yang-Mills you know there's other things people study and why is that and part of the answer is that 2 is a huge bottleneck so whenever we formulate this question like what what is the expectation value of this operator near the singularity of this black hole it's actually not clear to us and our current state of knowledge how to do step 2 how do you map that to something that you can evaluate in the CFT or for example you have some generic state black hole and you want to evaluate the number operator at its horizon that they could in principle tell you if it has a firewall it's not clear how to map that to something to evaluate in the CFT so this program that exists basically fixing this bottleneck understanding the map to the CFT is called the bulk reconstruction program this is the name so it goes by this name because as we'll see momentarily the ADS often is called the bulk it's like the bulk part of this of some cylinder that you might draw and reconstruction means you're looking at the dual CFT and you're trying to reconstruct the physics of the ADS out of it that's why it has this name so really just means understand CFT dictionary is in the thing that relates to languages so the point of these lectures is to tell you much of what we know about which operators in the ADS can be reconstructed in the CFT and how to do it and as far as the how to do it part I'll mention a little bit about this there actually are many reconstruction schemes but my focus will be less on particular schemes that people have constructed for doing this reconstruction and more on theorems that people have come up with about what is in general possible and not possible to reconstruct about the bulk using the CFT or different parts of the CFT that's the program let me cover some of the ADS CFT basics so first the ADS basics I'll tell you but what do I mean when I say ADS you know what I mean when I say CFT and then what's what do we know about the map relating to ADS right so really it's anti-decider anti-decider space so in gravity we care about space times that are solutions of Einstein's equations which we could write down but I'm sure you're familiar with them and ADS or anti-decider space is the maximally symmetric vacuum solution with lambda equals zero sorry lambda less than zero solution of Einstein's equations less than zero and you can write this solution down this metric in global coordinates as I called there's many different coordinates you could use of course so the hopefully this is not a sign that it's too low that I'm sitting now the metric is this one I won't use very many details of this metric or any details really show it to you for the sake of completeness so this is ADS I'll say d plus one so it has d spatial dimensions and then one time dimension so here you know this is time t this is r the radial direction and then this is the like a metric on the sphere d minus one sphere and this this L here is is called the ADS radius it's just a characteristic length scale of the spacetime sort of analogous to the Hubble radius in our universe it's related to lambda so if you plug in some lambda into I know you're trying to solve Einstein's equations with some cosmological constant it's less than zero this solves it where L is related to lambda in a way that's not important enough for me to write down but it's like d times d minus one over 2 L squared where d is the space dimensions and r runs from zero to infinity time runs from minus infinity to infinity and I might draw it this way this is a very common diagram and I will draw a number of diagrams that look like this so this is I'm really just trying to draw a cylinder where time runs up r runs outwards that's pressed the sphere I mean it would look like this so say some time slice when it hits the boundary of the cylinder would look like this circle and then time slice would fill in would be this disc here and say a circle there might look like this it's a circle in the middle of the bulk so ad yet so this cylinder is some compactified way of drawing ads where there's r goes from zero to infinity all the way out here so there's actually infinite distance in this diagram and we have not compactified time so the cylinder principle runs all the way to minus infinity and plus infinity but space is compact here or it's drawn compactified so this is the sort of conformal compactification space there's some other there's a lot of interesting physics that you can talk about that shows up nicely in this diagram won't be very important so I won't say it but I will often draw time slices so for example this time slice here I might draw just as a circle so here this is the same radio direction and in general we're not going to care just about vacuum ads this is a solution where there's this negative cosmological constant but no matter fields back reacting we will general care about asymptotically ads solutions so these are solutions of science-science equations have the same boundary conditions so they limit to this ads metric at large values of r and one particularly important asymptotically ads geometry that we will talk about is what you is ads short child and this metric this is just the metric of ads if you have a black hole so this would be an eternal black hole you can also consider solutions that come from say collapsing some ball of dust or star to form a black hole this would be the simplest black hole to talk about a metric probably very similar to the ones that you're used to say in flat space at this level it's the same an F of r looks like this there's this extra term here relative to that solution okay so this f of r has these three terms one plus r squared minus this guy which includes the mass the black hole and Newton setting various things to one like h bar and see I'm gonna talk about black holes in ads at some points and this is essentially the metric I have in mind and the pin rows diagram for it looks something like this I'm gonna sort of assume that you've seen Penner's diagrams but essentially what I mean is like over there time runs up r is outwards this these diagonal lines are the horizon the horizons of there's two black holes here connected by a wormhole so let me say this right there's two asymptotic boundaries we'll call a right boundary and a left boundary and if you take say a time slice here this time slice rather than being a disk like that it sort of looks like this this circle here is a time slice of the left boundary and this circle here is a time slice of the right boundary and the time slice that I've indicated by this line going through the ads spacetime is sort of the outside of this two so these are some examples of ads and asymptotically ads solutions and we will be thinking about states with geometries like these as we go so far I've just talked about the metric in these spacetimes but besides gravity and different metrics the theory in ads can also contain matter which for the purposes of these lectures at least until later we'll treat as so this matter we treated as quantum field theory living on say these curved backgrounds perturbatively coupled to gravity which is what we'll call semi classical gravity so I'll write it here just remind you we're gonna be working in this for now in this semi classical gravity setting where matter is essentially treated as some quantum field theory on some backgrounds such as these perturbatively coupled to the gravitation so this is what I'll call the ads basics let me tell you the CFT basics and I'll tell you how they are related so CFTs are relativistic quantum field theories that have one grain variance but also in variance under two more transformations so they're also they also have a good transformation properties under what's called dilations which transform the coordinates like this we sort of scale everything by some lambda and the generator of this D that's some operator and the CFT and then also what's called special conformal transformations which transform the coordinates like this the details of this won't matter this is another thing I'm writing essentially for completeness the generator of these transformations we'll call case of mu so these two transformations together these generators together with the generators of the Poincare group form a larger group called the conformal group SID2 and local so there's a and local operators transforming and representations of this group satisfy the following equation which will be important for one part of the talk I'll tell you right afterwards essentially this equation so here this D is the generator of dilations and this O of X is some local operator transforming and some representation of this conformal group so it satisfies this equation where this this delta here is the thing I want to emphasize so this is called the scaling dimension it's something that's assigned to each operator O that satisfies an equation like this so again there's different deltas for different O's okay and so in some sense this is these are the CFT basics I do want to emphasize that CFTs are UV completes because you have this scaling invariance so you can sort of keep zooming in as far as you want and it'll you'll have essentially the same you'll have the same CFT so they're UV complete in the sense and so there's a sense in which the CFT dual of some semi-classical gravity theory in ADS offers a UV completion of bulk semi-classical gravity so there's the sense in which you can have semi-classical gravity formulated in the ADS and we know that the CFT that we're mapping we're mapping it into with a dictionary that we'll talk about is UV complete and so in principle can give us nice UV complete answers so that's why again we care so much about this dictionary so moving on from the CFT basics let's talk about ADS CFT so fundamental to this duality is the following statement which is called the extrapolate dictionary and I believe this dictionary was first made explicit in this paper and which I'll include in some lecture notes by banks Douglas Horowitz and Martinick in 1998 but this extrapolate dictionary goes something like this so that it's in some sense the starting point for everything for everything else we're gonna talk about it's like a fundamental relationship between ADS operators and CFT operators the idea is if you have let's say some let's let Phi be some ADS operator acting at some points will call x1 so maybe I'll draw a picture here I'll go over here because I've already drawn the idea is we have some me erase this time slice I've drawn just because it's cluttered so I'm looking at this cylinder as as my picture of ADS and I want to write down some operator Phi at some point x1 comma r that's this point so it's acting somewhere in the bulk so first before I even tell you the extrapolate dictionary let me emphasize that we're thinking about this gravity theory this ADS theory living sort of in the cylinder the interior of the cylinder the CFT you should imagine is living on the boundary of this cylinder so to emphasize that I might write this way the time slices of this so the CFT is one fewer spatial dimension and some sense lacks this radial direction so a time slice of the CFT right might look like this whereas a time slice of the ADS would be the disk that fills in that circle and they both share the same time direction in the sense of forward okay so we have this operator Phi in the bulk and maybe this other Phi acting at some point x2 r so I'm isolating this r dependence just because it's gonna show up in a special way momentarily in this equation but x1 and x2 just denote all the other coordinates so their placement of time their placement of the transverse directions all that's just lumped into these other coordinates x1 x2 and so on and I might also have x3 r and so on so so I have some collection of Phi here this is some correlation function this is x1 that's xn all right a yes there or bulk and I'm gonna have them all be the same Phi I'm not not right now going to have different bulk fields in this correlation function they're all the same one because this is the context in which this extrapolate dictionary is most conveniently stated what I'm going to do is scale them in this way so there's r to the minus in Delta Delta is related to this Delta and away I'll tell you momentarily we'll have to write to the right-hand side before that makes sense this in is just the number of operators and started here and I'm going to take a limit that r goes to infinity so this is like I have my operators all placed at some radial position r and then I'm taking a limit where they're all going out to the boundary or on this time slice diagram it'll be convenient to draw them so sort of like this these are all Phi insertions and I'm gonna consider correlation functions in the limit that all of these are taken out to infinity radially and the idea the statements of this dictionary is that this correlation function in this limit scaled this way equals a certain CFT correlation function and that is see this O of x1 so this x1 is the same one here so it's the same place in the in the transverse directions there's no r in that O position of course because we've taken the limit that r goes to infinity so all CF that makes sense all CFT operators live on the boundary of this cylinder so they shouldn't have real radial position and so on for all of them and this Delta here let me see if I can use a different color this Delta here is the scaling dimension this okay and part of the statement of the dictionary is that for each bulk Phi there exists some CFT O that satisfies this question and this O is called the CFT dual or you might call Phi the ADS dual good so this is one nice relationship between ADS quantities and CFT quantities and it in principle specifies the duality but at least in practice it has a major limitation and because of this limitation we're gonna have to consider we're gonna have to think about the dictionary you know the map between operators in the bulk and boundary a little harder we're not done just because we know that so the limitation is that okay this is a time slice of ADS again in the circle on the boundary you might imagine as a time slice of the CFT if I wanted to say evaluate the correlation function of Phi's you know these bulk operators Phi sitting at the boundary using the CFT I could do that this tells me how to do that so that's great but what if I want to do something a little different and I want to consider the four point function of operators Phi that are not at the boundary but are still sitting at some finite radial distance this doesn't manifestly tell me how to do that using this CFT so we're gonna have to work a little harder and people did this and this limitation to some extent was mitigated or figured out you know people figured out a way around it in 2006 so all of this remember was around 1998 shortly after ADS CFT was written down people really understood the mapping between certain operators on both sides at the level of say this equation took until about 2006 for people to start writing down things like the following which will be called hkl reconstruction this is the name that it goes by I said 2006 by these authors I won't write their their names down here but I'll put them in the lecture notes Hamilton kabot Lipschitz and low they wrote a series of good papers about this that I'm about to tell you but you can also if you want a pedagogical introduction that goes into more details than I'm about to you can see Daniel Harlow has some lectures that you can find online from 2018 they're his TASI lectures so if you look up Harlow TASI TI TASI sorry it's small you can find these they go into great detail so the idea behind hkl reconstruction is that in certain cases we will be able to evaluate using the CFT and say four point function of operators deeper into the bulk and the idea in words is essentially that operators have to solve the equations of motion which are some partial differential equation and you can solve these using Green's function methods that have boundary conditions at spatial infinity given by the extrapolate dictionary so the spirit is you sort of start with your phi inserted at some finite radial distance and ADS and you sort of solve the equations of motion out radially to the ADS boundary and there you know it'll be some mess of operators some integral of many operators but each of those you can relate to boundary operators the CFT operators using this extrapolate dictionary so this will give you some expression involving Green's functions for these bulk operators in terms of some integral over boundary operators I have a question yeah yes perhaps these are naive questions but since in a theory of dynamical gravity the film morphism can be thought as a gauge symmetry I mean is there some subtlety in defining these operators where the insertions are in the bulk yes I consider them as gauging variant observable so yes thank you for that question yeah so I am I was being a little fast and loose by what I mean by local operator here to really define these in a gauging variant way I might want to say an operator inserted at some renormalized distance L starting at this point on the boundary and going going in this renormalized distance L so each of these might be defined in a gauging variant way like that and it's these dressed operators that I might try and reconstruct and it's it's something even here I can't do this using just the extrapolate dictionary itself does that answer okay does it matter how you dress yes each of these dress a different way would be a different operator but there is a for certain questions how you dress will be a subleading effect but yeah so like this is a different operator from say this but for many questions that say leading order one over G that difference won't be important sorry can we think this dressing as like in usual gauge theories like electrons attached to wheels online yes it's the gravitational version of that yeah so the gravitational version is there at the end of some geodesic say that's one way to do it there's many ways to do it but it's exactly more this may be a follow-up of the Francesco's question but usually the statement in gravity is that since local operators essentially does not really make sense or actual degrees of freedom live on the boundary typically all the question that you can ask are answered by by the regulator of the CFT which live on the boundary so which question do you need to ask in order to be interested really in correlators where the points are in the bulk good example is when say you can construct operators that are anchored to points in the boundary and therefore gauge invariant in this sense but aren't only supported on the boundary they come in some finite radial distance and so like these guys here that I've drawn will be an example so even though yeah so this these fies now you know they're all different by 1 by 2 by 3 by 4 because they're all they dressed in their own independent way so I want to compute this correlation function the gauge theory analog would be like some these would all be operators that insert some charge dressed with some Wilson line to the boundary and this will be non-trivial to answer and certainly not possible just using the extrapolate dictionary because they're not completely supported on the boundary they have some point I sent inside there's other questions that we might ask which is I'm gonna be very interested in which I could use this black hole diagram to answer so before I talk about the black hole actually let me mention some other situation so let's say sorry I'm gonna erase this we just have vacuum ADS and I want to compute some scattering process so I maybe insert at the boundary at some early time two particles like A and B and they in the bulk fall in and then they interact here and then they come out sorry I drew my cylinder too short so let me erase it again and then draw it really tall so that these two guys start at the boundary fall in interact and then they come out maybe in some scattering process ultimately hitting the boundary perhaps in different places and I can ask about correlation functions of all the operators of operators living just in the boundary and this would be something I could do using the CFT even though this process is happening in the bulk I could ask about the outputs given these inputs using the bulk and that's all fine and this might be something I'm interested in doing and then I don't really have to do anything more complicated than using the extrapolate dictionary however there are other processes which I am personally interested in where you can't just it doesn't the problem doesn't just reduce to this even though it's something as simple as scouting so I could for example here take some operators inserted now there's a black hole and these two particles inserted at the boundary will fall in and they'll interact here which is inside the horizon of these black holes and then they'll have outputs which hit the singularity don't come back out and I might want to understand what the outputs are here so how the scattering works in this highly this perhaps highly curved regime and this is not something that I can do using just simple the simple extrapolate dictionary because these guys never come back close to the boundary so this is another example of the type of thing I would like to be able to answer but we have to work more on the dictionary so sure just to make sure I'm understanding it is still true that all the all the information of gravity is on the boundary but there are some processes which are not captured by simple correlation function of local operator on the boundary that appears to be the case okay thank you these are great questions so as you were saying this idea behind hkl reconstruction is that you you write these fies perhaps what they're dressing as some smeared out operator on the boundary by using these greens function methods combined with the extrapolate dictionary and the answer you get looks something like this so so some five X this is an operator in the CFT that's supposed to you act like five X so so on the right hand side here this is this is the CFT by as an operator so you know the the oh that shows up on the right hand side when phi is on the left hand side of the extrapolate dictionary so those are the ohs that go here and this K is a greens function where capital X you should regard as a boundary point like a point in the CFT and then little X is like a bulk point so that's you know phi is living at some point little X oh is living at some point capital X so we're integrating over a set of boundary points that I'm calling S of X what are those boundary points I need to draw a picture for you to explain it so here again I'm drawing this ADS cylinder let's say we had inserted phi here so this is the point little X the support so this region S sub X is some patch of the boundary cylinder that it's a patch that is space like or null so not time like to little X so if you drew the light cone in the bulk from this point it would hit the boundary somewhere and so this is supposed to be forward and backwards light cone hitting the boundary at these circles that's what these circles I drew we're trying to be and this region S of X I'm gonna write it in blue is this region I'm trying to shade in blue it's like space like or light like related to this point X in the bulk this is the this is one formula that hkl has so I'm gonna write this so this is the boundary points space like to X so by this equation I mean this is some CFT operator and that it's the CFT operator that's supposed to reconstruct this bulk operator no matter all X is belonging to the left conform of it or your right good yes so this good this is a great question so I'm gonna get to the black hole geometry momentarily there's actually a subtlety that shows up there and I that's your pre you're already thinking of the right question for now imagine this formula in the context of this cylinder where there's one CFT to talk about and then this O is the some operator in that CFT a version of this will work also for the black hole case but it gets complicated because there's a black hole and there's a singularity and I want to tell you momentarily how to what's your green function it's a I don't remember the form of it it's it's not the bulk boundary propagator I remember that it's something else also if you compute the Laplacian of this green function do you get a so you don't get just a delta function at the boundary yeah if you compute the Laplacian right like plug it into the equations of motion you get you're saying because if it was a delta function of the boundary then that would be the bulk to boundary propagator no I don't know what this K is and I think it is the one that's it yeah it solves the equations of motion in that sense I don't remember off top my head many properties of this greens function I remember it's not the one it's not the same K that shows up in the in written diagrams it's kind of ugly yeah yeah the explicit form that was a lot of the work that hkl did but it won't it won't be important for where I go should I think about this identity like an operator equation I mean that every correlator of the files is equal to is obtained by the coordinator of the CFT using this transformer yes exactly oh good yeah let me mention a few caveats about when it's formidable that's exactly what you're gonna get so that that is the idea is this is imagining this is an operator equation but I wrote this in the pre-field setting so there there's some closed form K that you can write down when there's interactions in the bulk so if I have an interacting field you can include those sort of perturbatively so this formula you have to add to this infinite series of terms not in all cases does it converge this is one of the problems with this that I'll get to yeah so this is a free field guy and it also depends on the particular background we're talking about so this would so this is best understood in vacuum ADS so this would be an exact operator equation for the free field case in a limit where where Phi you can ignore the back reaction to Phi so if you could if you could exactly ignore the back reaction to Phi then this would be an exact operator equation but if you try to insert too many then all these excitations will back react the geometry will change and then for very different geometries you're supposed to use a different K so so then this would no longer hold so it's sort of in an approximation where you're you're really talking about this semi-classical limit where this is an operator what will fail will be like the linearity of this because this is like a general answer if you change the green function here right what will fail is that yeah you might be able to word it like the linearity this would definitely it's this guy should change and he definitely does change on different backgrounds yeah so in the limit that Phi doesn't back react then this would say the same and then it would be there's some linearity issue so one thing I want to mention about this guy is that even though we've written in this equation Phi in terms of some big integral over CFT operators that don't live on just one time slice of the CFT but are actually you know in some time band you actually only need a time slice and the reason is that this is not special to ADSE of T it's just a you know statement in any QFT in quantum mechanical system but so let's say this is some time slice down here this is some operator oh inserted at some point X it's on future time T so let's say this is T equals 0 you can represent you can write this operator is some fairly complicated operator acting at this earlier time and it'll have support some non-trivial fairly complicated support at the intersection of this time slice and the past light cone and it's essentially this operator if you're acting it here but this O of X 0 is some operator acting on this time slice and this H is some is the Hamiltonian governing your quantum system so it's this relatively complicated operator that you can act at T equals 0 which will implement this operator at a later time so conversely if I have this operator here say in this formula so that some of these O of X's are acting not at T equals 0 but say it's some later earlier time I can replace them with operators acting just at T equals 0 be it slightly more complicated looking operators so instead of being local like these guys there'll be some sort of smeared out mess but it's something you can in principle do so the idea is if you just evolve each of those O's to T equals 0 using the CFT Hamiltonian we have this bulk field phi at point X oops no it's future light cone hit the boundary here so this region S of X is you know this big time band but we can represent this phi using this method on just this T equals 0 slice which I'll draw in green or any other slice but you know we can in particular represent this operator as some operator on this time slice the reason I'm saying this is that that's just a convenient way to think about you know we're mapping some operator that's living roughly on one time slice of the bulk to some operator living roughly on one time slice of the CFT that's sort of a nicer relationship to think about then mapping it to something that's more smeared out in time these are the pictures I'm going to draw later it's this mapping from one time one time to an operator at another time and let me now ask the question great now that we can do this we can talk about reconstructing at least some operators deep in the bulk as some operator in the CFT with a fairly explicit formula at least in some cases is this everything we want I'm going to tell you not really and let me argue why so the reason it's not really everything we want is that it doesn't appear that such green funk greens function solutions generally exist for example for geometries that are far from vacuum ADS so HKL really figured out how to do it around vacuum ADS but when there's a lot of back reaction or you're even just doing it around some different backgrounds like when there's a black hole they don't always exist and there's a nice picture for this is related to the question we're asking earlier so I'm gonna draw the black hole pinners diagram I don't know if I ever made it explicit but these curvy lines and the top and bottom are the singularities of the black hole and inside these diagonal lines is the interior the diagonal lines of the horizon and so if you try and do this greens function method for some operator here we run into trouble the reason is that essentially you just followed the same algorithm you would try and evolve you you'd evolve this operator phi out radially you would hit the singularity so yeah there's some some data that you need to reconstruct to write this phi at some further out point but that data is data that's supposed to live at the singularity which is a problem this doesn't allow us to use CFT data to write this phi because some of the data we need isn't in the CFT it's whatever is at the singularity it's just a problem we just can't in this situation write this phi using this method as a boundary operator solving the equations radially hits the singularity no good so and this is sort of exactly the situation that I am most interested in is understanding what's going on inside black holes so the fact that this method doesn't allow us to do that I view as a big short come let me now directly address the question that you're asking before I move on from hkl reconstruction to what is more powerful in general type let me mention one thing which is called rendler reconstruction I think that was the name so it's a different type of you know it's the same papers by hkl and it's a it just uses the fact that this kernel here is k wasn't unique there is there were different ones you could choose and some of them allowed you to reconstruct phi in the CFT not using say an entire spatial slice which is using a sub region of the CFT and this is very important because it's gonna be like the types of reconstruction we talked about afterwards so let me emphasize this here again I'm drawing the same sort of CFT ADS diagram or have the same 5x that I'm trying to write as a boundary operator and before we argued that there was some formula like this that allowed you to reconstruct this phi using the whole time some some like t equals 0 slice so some operator in with support of the entire time slice of the boundary turns out when you're solving the equations radially this is the the sort of slogany way of talking about using this greens function solution to write phi as a boundary operator you don't have to solve it radially in all directions you can sort of try and concentrate in one direction and this sort of works so you can just sort of solve it this way and what you get is that you need not land with this integral supported or with this k say having support on all boundary x it can have just supports on sub region let's see how do I draw this it'll be easier if I say it this way so let's say let's say I pick some sub region B of the boundary so I there's some time slice this is some circle on the on the boundary of the cylinder I'm picking a sub region of that circle and I'm calling B it's this this solid sub region and now if you find the domain of dependence of this B in the CFT so that this is like you you consider it's the space time region that it knows all the data about so you know this is like if you found it's it would look like a diamond it's like something like this and this is like if you just consider it's called the domain of dependence where you basically chop out the causal future and past of the complementary region so this is some patch of the boundary cylinder you take this whole patch of the boundary and now you look at its bulk past and future this creates what's called the causal wedge not super important so it like create the bulk past and future intersect at some line which I'm drawing here is a screen line that defines something called the causal wedge the details won't be super important and I'm happy to go into more detail about how this thing is defined the statement of Rindler reconstruction is that any any point or any local operator in this causal wedge this is a bulk region admits a reconstruction just just on B but there's some formula here where the support is not on the whole region S sub x that I defined but instead just on this patch of the boundary here and you know just like before you can take any operator in this patch and then evolve it to be some fairly complicated operator that lives just on this one time slice that we were calling B so 5x admits reconstruction on just a sub region of the CFT this is a an example of sub region I think I maybe went fast on that explanation let me give you an example that maybe makes it clear so if you consider this black hole case I gave I said there was a problem trying to reconstruct this operator phi and started here inside the black hole because when you solve the equations radially you hit the singularity and so you can't define all the boundary data you need to reconstruct you might complain that if I just to find some operator here which is outside the black hole so this will be phi acting at some point x prime this is outside the black hole and it would seem to have the same problem as this guy if I use that formula right because his his past and future looks something like this and so while his he has support on some finite time band of the right CFT part of the data we need the part to the left seems seems to be at the singularity on the left side that's his past and future hitting the singularity on this side that's so that would seem to say you couldn't reconstruct this operator in this background using hkl reconstruction but actually you can the way out is you have to just remember that you can do Rindler reconstruction with a different kernel not that one but the one that just has support on a patch of the boundary in this case it only you only need for anything in the right exterior so outside the black hole here to the right you can use just this right CFT to reconstruct it so you can just solve the equations for phi radially outwards this way it has support on just this finite time band here you don't need this left side so maybe that's an illustration to you of the power and importance of this idea of Rindler reconstruction so you can just use a patch of the boundary with these methods so does this does this a reconstruction procedure give some problems with a non-uniqueness of the region that you can pick yes absolutely absolutely does which is exactly what I'm going to talk about next lecture so to tell everyone what was just pointed out here I said you can sort of choose to solve like represent phi as some operator on some boundary sub region B but it's not unique what that B is this is actually not just a little toy curiosity it's very deep the deep statement about the ADS CFT dictionary which we will discuss in detail next time but for now this is what I wanted to say about Rindler reconstruction and before I close let me give you a taste of where we're going we introduced the extrapolate dictionary and we said this is great this is the fundamental in some sense fundamentally defining the dictionary but it's not telling us everything we want to know like about the inside of black holes HKLL does a little better it allows you in some cases to reconstruct operators further in the bulk but still doesn't do everything we want because it doesn't reach all the way in to the interior you could ask is there a better scheme that does allow you to say reach further in or reconstruct operators behind black hole horizons and we would like to do better than just guess at some reconstruction scheme and then check how good it is we would like to be systematic about it and prove some theorems about what can in principle be reconstructed using what data of the CFT we want to know like prove a theorem is there any way of writing a CFT operator that reconstructs this guy and maybe if so what is it and in the last few years that's exactly the sort of theorem we've been able to write down so we can prove statements that sort of systematically tell us what is and isn't possible it will be possible in case in some cases to reconstruct operators like this and when this isn't possible is very deep and important related to the information problem I want to talk about all of this good so we want to turn from this which is sort of a pre-2015 to the post-2015 paradigm of reconstruction schemes that work as generally as possible with certain theoretical guarantees and what I will tell you about next time is going to start with the so-called quantum extremal surface prescription quantum extremal surface formula and this will often I'll just call it the QES formula the QES prescription and I'll tell you what it is it's a prescription that was written down not originally with the goal in mind of understanding bulk reconstruction it was written down originally as a tool for computing the von Neumann entropy of CFT subregions and it just turns out people figured out that it is perhaps the most insightful formula that exists for telling you what you can and can't reconstruct and explaining it and explaining how people started to see that it was very useful for understanding bulk reconstruction will be the topic of next lecture you mean for questions before the discussion I mean the discussion will be after the second lecture of the afternoon I was wondering is there also some Euclidean version of this reconstruction problem or maybe it's not interesting or it's too trivial yeah definitely it's not one that I've thought about as much there should be a way to yeah I'm sure a lot of similar things can be said like you could certainly do a version of hkl reconstruction and actually I guess one thing that I would wonder is in the Euclidean setting what are the questions we're trying to answer there I'm sure there are some I guess it's in this Lorentzian setting where we can ask things like what's going on behind this horizon which is why I have mostly yeah maybe even as a comment on the question I mean I think you already said this but of course in the Euclidean there will be no interior and I think at least at the classical level which probably extends the problem is essentially sold by Fepham and Graham in the mathematical sense right but but that was that was just a quick comment my question was actually with this time-slice thing vague question sorry is there some inter interplay with this idea of complexity in the sense that time evolution generates complexity and somehow there is the lore that very complex operators are hard to reconstruct so maybe if you wait too long in the future you still run into trouble yeah great yeah so that's certainly true yeah so at some level this operator I was saying yeah as you're remarking on is nasty when you time all the back and try to represent here but there's a sort of formal another definition a more formal definition of complexity that you could give where if this T was too long like this this local operator with say some so far in the future it was exponential in some parameter then it would literally be probably formally complex to write it in the operator you're out here would be formally complex to act have high complexity in computer science sense this is true yeah it is related to some things that people are talking about these days with like pipe on lunches and so on with how hard it is to reconstruct operators behind the horizon so let me hijack the answer to this question to also emphasize the point that I didn't say which is one thing I'm focusing on any lectures is this question of is it possible to reconstruct some given operator in some say portion of the CFD and I am not going to really discuss the question of how hard is it to reconstruct what you've given that you can reconstruct it how complicated in the computer science sense is that reconstruction all of that is a very interesting question and it seems to be the case that often times operators behind the horizon can be very complicated to reconstruct say using some patch of the CFD or some patch of the hockey radiation that is deeply related to this yeah okay I think we should yeah just super short you say it's like can be complicated in a computer theoretic sense but is it do we know if it's complicated in a physics sense like is there been any progress on someone writing down something like hkl that you can actually use practically yeah good correlators in the interior or something yeah so yes I will mention at some points the formulas we have that do the explicit reconstruction so so one formula that essentially should work anytime anything could possibly work it's called the pets reconstruction formula and this is a fairly explicit formula and in certain examples like this so-called West Coast replica wormholes paper that some of you might be familiar with they actually you know explicitly talk about they're able to like you do this calculation do the reconstruction explicitly they can they can compute exactly how well it reconstructs the operator because there's always some error I haven't mentioned that yet but when you reconstruct the operator might do a great job but not exact not not exactly reconstructed and you can compute how well these formulas and so in this physics sense of reconstruction there is some understanding of what the formulas look like for reconstructing these things that have even been implemented in certain cases in general is very difficult and looks very ugly and in general it won't look like some nice integral with some kernel times some local operator there's actually a good argument why the more powerful types of reconstruction that we're going to talk about given to you by the pets map guaranteed to work by this don't look like that and they can't and I would be happy to mention why that is they're generally more complicated okay let's postpone the question to the discussion and don't leave we have the picture now so just come all in front and look at the back let's thank Chris for the lecture after