 We're very happy to have today someone like Hoyser from MIT who will continue last week discussion on emergent times in holographic duality. Thanks for the introduction and for the invitation. Yeah, I'm excited to talk to you about some work on emergent times in holography that I've done with my advisor Hong Lu, who gave sort of the first part of this talk last Thursday. So first, I'll just recap a little bit of what he talked about before going into some of the things we'll discuss today. So first, let's see, there we go. So first, the main issue that we're interested in understanding is sort of these bulk time evolutions that look like this. So we're interested in understanding from a boundary perspective, how do we describe a bulk time evolution that takes some Cauchy slice like this through the bifurcation surface to another slice like this one that probes the interior of the black hole. And so the simplest sort of setup that you can study this question is the eternal ADS black hole. And it's well known that this is conjectured to be dual to the two copies of the CFT in the thermal field double states. And from the perspective of the CFT, we have this notion of time along the boundary and we have this time like killing vector outside the horizon that lets us naturally extend this asymptotic time into the boundary or into the bulk. But that time ends at the horizons. And so if we do that procedure, we're not going to be able to probe the interior of the black hole and we won't be able to understand anything that's happening in these future or past interior regions. So we're interested in understanding how do we describe infalling observers. And in particular, if we have infalling observers from the left and the right side and they can interact in this future region, how is that described from the CFT perspective? And so last time, Hong should have shown that we were able to generate, we're able to find these evolution operators in the boundary that describe these kind of evolutions in a certain limit. So they're evolutions that take observers who are living in this right exterior region across the horizon into the future and then eventually ending at the singularity. Or equivalently, we can look at the Cauchy slices. If we start with this Cauchy slice through the bifurcation surface, then evolving by positive values of S, we get Cauchy slices that look like this that are exactly this kind of evolution we were interested in that probe the interior region. And so the description of these infalling observers has to do with the fact that in the boundary theory, you can actually find these evolution operators that look like this. There's a one parameter unitary group of operators. We've got this permission generator and that generator has the following properties. It involves the left and the right degrees of freedom. This is what allows us to propagate inside the black hole. It's positive semi-definite and importantly, as I mentioned, if we take some bulk field operator in the right exterior region and we evolve with this evolution operator, we find that this operator at a later value of the parameter for large enough values of that parameter S, this capital phi is supported in the future interior. And so these conditions are interpreted in this way that this positive semi-definite condition distinguishes S as being some notion of time. It sort of distinguishes it from something like a spatial translation. And then the fact that an operator originally in the right exterior becomes supported in the future interior demonstrates that this is some sort of notion of infalling time. And so then there's a sense in which we can take those operators in the exterior regions, evolve them with U of S, and generate the interior region. So it's kind of like we're generating the interior regions from the exterior regions. Great. So that's sort of a very fast summary of Hong's talk from last Thursday. And so now I'll just quickly outline what I'll discuss today. So first I'll give an argument that there should be no notion of a sharp horizon. You can't describe the notion of a sharp black hole horizon at finite number of degrees of freedom on the boundary at finite n. And then I'll do a bit of a mathematical review on our main tool, which is half-sided modular inclusion. And I'll discuss a few of the theorems related to that. And then we'll use those properties to understand how exactly, in more detail, some of the construction of these boundary operators U of S. And so in the case of a generalized free field theory, we'll actually find that you can extend this U of S in a nice way. And it's actually completely determined the form that these operators have to take is completely determined by up to a phase. So yeah, there's a universal form for these evolution operators in that case. Great. And so yeah, as I mentioned, and as you heard last time, this is all work with Hong Lu that appeared in these two papers. So I'll just stop for a second in case there are any questions before diving into this. Great. Great. So first we'll discuss an argument that there should not be a sharp horizon at a finite number of boundary degrees of freedom. So there's a puzzle that was first pointed out by Merylton Wall in 2012. And it's related to the existence of this future region and how do we describe in following observers in ADS CFT? So first, yeah, so first, this duality of the thermo field double and this ADS black hole raises a puzzle. So first I want to imagine I'm only thinking from the boundary perspective. So I can imagine constructing this state in the two copies of the boundary theory where I've acted some unitary on the left side on the thermo field double state. And then I want to see can a measurement by an observer in the right, who would be described by these CFT degrees of freedom on the right side, can they measure the effects of that unitary insertion? And of course, the answer is no, because that measurement operator from the right side commutes with this unitary insertion. So all of these expectation values are just the same as if there was no insertion at all. And the left insertion has no effect on the measurement. So that's from the CFT side. But this appears to be in conflict with this bulk picture. And that's because from the bulk picture, we expect that the insertion of this small unitary on the left side, it should have some effects in this left region. And then those effects propagate into the future region. And if I have my observer from the right falling into that future region, they can sense those effects and they should be able to detect that this small unitary on the left was inserted. So this is what the bulk picture tells us, that they have this detection. And so this is a puzzle that was first raised by Merleau van Roel. But it doesn't directly lead to a contradiction. More so what it tells us is that if we want to describe these observers falling into the interior of the black hole, then the description of those observers in that future region necessarily needs to include degrees of freedom on the left and the right CFT. Great. And so now that we have that in mind, we'd like to understand, okay, what form should the evolution of this observer take? And what can we understand from the evolution of that observer and asking these kinds of questions? Great. So first I'll just introduce some notation. So let's imagine that we have our observer on the right side, we now understand that they're going to evolve them into the black hole and at some point, the description of that observer is going to have to have support on both the left and the right CFT degrees of freedom. So let's imagine that the probability, let's just denote the probability that this observer detects this insertion on the left by P of S. And so the bulk causal structure gives you this very clean signature. It says that P of S needs to be exactly zero for small enough values of the parameter because the observer hasn't yet crossed the horizon. But then after crossing the horizon, P of S should go to something non-zero, that insertion can be detected. So this is just the mathematical statement that that observer can't detect the left insertion until they've crossed the horizon of the black hole. So this is what we expect from bulk causal structure. And so now recall that as I mentioned a bit earlier, and as Hong mentioned last time, we found these in-falling evolution operators that take this form of a one-parameter unitary group. And so if we're describing evolution in this way, then we want to ask, okay, what is the CFT description of this observer falling past the horizon and trying to make this measurement? So let's imagine that the observer who's originally in the right region can come up with this projector that they're going to use to make the measurement. So it's a specific projector that's been designed to measure the effect of this excitation. Well, then as they fall into the black hole, they carry their projector with them and their projector evolves in this way. As the evolution is described by this U of S, so as they fall into the black hole, their projector also evolves. And of course, importantly, you will gain some support on the left theory. Great. Well, in that case, then we can write the probability in this way. This is the CFT formula for the probability. We take our state that has the excitation and we want to ask, yeah, we want to ask, okay, what's the probability that that excitation is measured after evolving by some amount S? Well, it turns out that because this projector squares to one, this probability can just be written as the norm squared of a particular state, phi of S, which we write in this way. And in particular, the positive semi-definite nature of this generator of the infalling evolution implies that this phi of S is just a vector valued function, and it can be continued to all values of S in the lower half complex S plane. So this is because of this positive semi-definite nature. And it's also it's continuous along the real S axis. But then in that case, we have Koshy's theorem. And that applies that we can't have a structure that looks like this. We can't have this bulk, what we expect from the bulk causal structure. You know, if this function vanishes on this open interval on the real axis, and it's analytic in the lower half plane, then that function has to be identically zero. So we can't have this situation where it's vanishing on this on this interval on the on the real axis. So, you know, from the quantum mechanics description from the CFT description, then we can't have this form of the probability. The probability has to either, you know, only be zero at isolated points, or it has to be identically zero. And so, yeah, and I should mention this, this is an argument. It's an adaptation of an argument due to Hager-Felt from a while ago. Great. But of course, we know that we know that, you know, there should be this bulk geometry that has that sharp horizon and so we have to ask, okay, well, how do we avoid, you know, this no-go argument for the non-existence of this sharp horizon? And a clue comes to us just from, you know, from normal quantum field theory in Minkowski spacetime. So here I've drawn a Rindler decomposition of Minkowski spacetime. I have, you know, I have a left wedge and a right wedge, a future wedge and a past wedge. And, you know, this problem is very, very similar. It's very similar. Yeah, this thermofield double case is very similar to this case of the Minkowski vacuum if we put it on a lattice. So if we imagine putting our field theory on a lattice, well, then, you know, these degrees of freedom in the left wedge and then the right wedge, this Minkowski vacuum state is just like a thermofield double state for those degrees of freedom. But we also know that when we put, you know, when we put this theory on a lattice, the light combs are actually not sharp. If I try to have any, any notion of, you know, Minkowski time evolution, I won't find sharp light combs. In fact, you know, there'll be some small tails in the commutators of evolved right operators with left operators, you know, for any value, for any value of the amount of this time that I've done. But in the continuum limits, then we really do have these sharp light combs. So somehow, you know, going to the continuum limit is showing, you know, a sharp difference between, you know, it, you know, allows the existence of these light combs. And the reason for that is essentially because the nature of the operator algebras in this discretized case and in the continuum limit is very different. You know, the operator algebras in the discretized case case are what's called type one von Neumann algebras, which are, you know, the usual von Neumann algebras that we encounter in quantum mechanics. But in the continuum limit, these left and right algebras are actually of what's called type 3-1. And it turns out that these algebras of type 3-1, they just simply don't have those projection operators that we needed to make the argument. You can't make those kind of measurements with the, you can't describe them with these projection operators in the type 3-1 algebra, they simply don't exist. And this is very intimately related to the fact that, you know, there's no Hilbert space associated to the right and to the left in the continuum limits. That only arises in the discretized case. Great. And so why does this tell us that, you know, at a finite number of degrees of freedom, we can't have this sharp black hole horizon in the description? Well, it's essentially because, you know, we have these sort of well-established dualities where, you know, the right CFT describes the right exterior of the black hole, the left CFT describes the left exterior of the black hole. But at, you know, at finite values of n, the only algebra that we have available to us on the right to describe, you know, this full exterior is the full CFT algebra. And the full algebra of operators in the CFT is of type 1, which means that those kind of projectors do exist. You know, there are, there's such a thing as local measurement and those kind of projectors exist. And that means that the no-go argument applies and we can't have this sharp horizon. You know, we can't have these commutators being exactly zero and then, and then switching on to being non-zero. Sam, does the no-go argument apply for type 2 algebras? Because, I mean, the end of December paper from Whitton, he showed that it was becoming, it was becoming type 2 infinity. So can one find some projector in such algebras? That's a good question. I, yeah, so I'd have to think a little bit more about the nature of these projectors. So in that case, in that case, my suspicion is that, is that the no-go argument does not apply for type 2 infinity. Yeah, because, because sort of to form those projectors to make that measurement, you need the notion of a local Hilbert space, which you still don't have in type 2 infinity. Thanks. Yeah, no worries. Great. So, so, so yeah, so what this means is that at finite end, you know, the only algebras we have to describe these exterior regions, the only, you know, full algebras that we have are these full CFT algebras, which are type 1, this no-go argument applies, and so we can't have an, you know, an exactly sharp horizon. And so that means that, you know, in order to describe the notion of a horizon, we have to actually sort of generalize our description of, you know, this infalling observer and the measurements that they'll make. And so here I'll discuss this general horizon criterion. So again, yeah, I'll remind you that we can't use, you know, this idea of projectors, because we might not have those in the algebra that we're using to describe these observers. And so we need a new criterion. And so, and so, you know, adapting, there's a formula for measuring, you know, local differences of states given by Buchholz and Ingveson. And so adapting that to this scenario, we should consider this function. And so what this function basically does is it, you know, it scans over all possible operators I can find in the right theory, not now we're not assuming that we have projectors that have these properties that we want. Instead, we scan over all operators in the right theory, and we ask are any of the expectation values of those operators different with the insertion of this left, you know, this left unitary or, yeah, compared to when the left unitary is not inserted. And so this criterion for a sharp horizon that also applies, you know, in this type 3 case is that, you know, instead of asking about that function p of s that we discussed previously, we should ask about this function f of s, where we're scanning over all the possible operators in this right theory. And so the sharp horizon signature is that this f of s is identically zero before we cross the horizon, and then it can become nonzero after we cross the horizon after we've evolved by some parameter s, which is larger than, you know, this distance to the horizon. Great. So that's sort of the story about no sharp horizons at finite end. And I'll just stop for a couple of seconds in case there are any questions before going on to talk about half-sided modular inclusion. Great. So now I'll just review a little bit about half-sided modular inclusion. And some of this stuff hopefully is familiar from Hong's talk last Thursday. But I'll just remind everybody of this story. So this slide is a bit abstract, but essentially, what is a half-sided modular inclusion? Well, it's a structure that you may or may not have in your operator algebra. So the way that it works is you start with a von Neumann algebra and a state. And so this here is just sort of setting up some jargon. If we have a von Neumann algebra and a state, and then if we act, you know, all operators in that von Neumann algebra on the state, and then we take the set of all those states, if that set is dense in the Hilbert space, then we say that omega is cyclic for the algebra m. And then similarly, another thing we can do is we can again act all those operators in the algebra on this state. And if it turns out that the only operator in that algebra that can annihilate the state is the zero operator, then we say omega is separating for m. And these cyclic and separating properties are essentially the algebraic way to say that this state omega is in a highly entangled state on this algebra m. And so there's a very familiar example of a cyclic and separating state, which is just the Minkowski vacuum. You know, if I take a Rindler algebra, then the Minkowski vacuum is cyclic and separating for that algebra. It's like highly entangled on the two sides of the Rindler decomposition. So then what's a half-sided modular inclusion? Well, in order to define this structure, we need to choose a subalgebra. And you choose a subalgebra n of m. And you have to choose it to satisfy two important properties. So the first is that this state omega has to still be cyclic. So yes, the state omega, I'm assuming, is cyclic and separating for m. And now I have to assume that for the subalgebra n, the state is still cyclic for n. So it's sort of trivially already still separating because n is a subalgebra, but the cyclicity is a bit more non-trivial. And then moreover, it has to have this following property under modular flow with respect to m in the state omega. So if I do the modular flow of all, you know, these operators in this algebra n, I have to be contained within n for negative values of the parameter. And so if I can find a subalgebra that has this special structure, then we have what's called a half-sided modular inclusion. And then there's these theorems of Borchers and Wiesbrock that guarantee that this unitary group exists. There's this one-parameter unitary group with a positive semi-definite generator. And moreover, that one-parameter unitary group actually preserves the state omega for all values, for all real values of the parameter. And it was later shown by Borchers that this kind of structure can only happen if this algebra m that I started with is a type 3-1 von Neumann algebra. And so that's why it was so important, as Hong mentioned last time, for us to find a type 3-1 von Neumann algebra, for us to find that algebra coming out of the CFT in this emergent way. Can I ask you a question? Yes. Let's say we are at finite and it's very large but finite. So the algebra is type 1. Do you know in what sense which part of this theorem breaks down or in what sense it breaks down? Right. Yeah, that's a good question. Right. So essentially, I think essentially, yeah, so maybe I can say just a few words about this, the proof of this theorem. So the proof of the theorem essentially shows that having this kind of structure requires the modular, requires essentially requires the modular operator to have this continuous spectrum going along, yeah, I guess for the modular operator along the entire positive real axis. And so I think it's that spectral condition that breaks down. I wouldn't, yeah. Yeah, but I mean, presumably this breaks down in a mild way in the sense that the spectrum will be almost dense. It will be like the gap will be exponentially small between an energy eigenstate. So there should be some sense in which some aspects of the theorem continue to hold. Right. Yeah. So I guess, yeah, I think maybe at least my expectation is that you should be able to construct objects like U of S that sort of approximately have these properties. Yeah, I guess, yeah, all of these theorems are stated for the exact case. And that's why they, and that's why this type three one exists. And so, yeah, it's a very interesting question of, you know, what exactly, what do you have to impose, you know, if I have the type one case, is there some way I can choose that type one algebra such that I get an approximate version of this structure? Yeah. Great. Yeah, yeah. So yeah, so all the statements basically that I'm going to make today are assuming, you know, you have these exact properties. And it would be very interesting to understand what happens in the approximate case. Yeah. Great. Okay. So now, I'll talk a little bit more about the modular properties of this group U of S. So there's more theorems by Borchers and Wiesbrock. And so here, this is the other important theorem that I'll show is now, you know, now suppose that I have some unitary group U of S. And it obeys this property that, you know, if I do evolution by a negative value of the parameter, I preserve this big algebra M. Well, in that case, you can show that any two of the following three properties will imply the third. So the first property is that the generator is positive semi definite. The second property is that the vacuum, you know, or the state omega is preserved for all values of the parameter S. And then the third is that I have this sort of algebra with the modular operators. So, you know, for the modular operator, if I do the modular flow of U of S, I just get U at some new value of S. And if I do the modular conjugation of U of S, I just get U at the negative value of S. And so from the theorem I've just shown you, we know that the U of S that we get from modular conjugation satisfies these first two properties, and therefore it must satisfy this third property as well. And so this theorem sort of fixes the algebra that this group of operators must satisfy with the modular operator and modular conjugation. Great. And so these properties of U of S are all important because they're going to allow us to really constrain U of S in the case that our algebra is a generalized free field algebra. Great. So here I'll just summarize all these properties of U of S that are going to go into this next sort of more technical part of the talk. So the first thing is that the generator of U of S is Hermitian. So that's, you know, like this unitary group, if I take the Hermitian conjugate, I just get it, you know, I get the same unitary group but at the negative value of the parameter. And all the applications that I'll discuss in the future, I'm going to be taking that state omega, that cyclic and separating state, to be the thermo field double. And so those, you know, the preservation of the state omega, it now turns into this preservation of the state, the thermo field double for all values of the parameter. As we just mentioned, we have this modular conjugation property that slips the sign of S. The modular algebras tells us that the modular flow of U of S is just U at some new value of S. And then finally, U of S was this one parameter group, you know, it's a unitary presentation of the real line. So if we multiply these operators, that's the same as taking, you know, the operator at the sum of those two values of the parameter. So these are these five properties that will all go into constraining U of S. Great. And so, you know, so now I want to talk about computing U of S in general. And so the first thing that I'll emphasize is that those theorems of mortars and Vs rocks, they're existence proof. They tell you that, you know, if I can find this subalgebra n that has those special properties, then this U of S exists, but they don't really tell you anything about how to construct it. And in the completely general case, if I wanted to compute U of S in general, it's a very hard problem, because I have to know the modular operator for both n and for n. And you know, there are not very many cases where we know the modular operator, you know, very precisely. But what we find is that in the case that this algebra m, this larger algebra that I start with, is generated by generalized free fields, then we can actually completely, we can show that this U of S has this universal form and it's completely fixed up to a phase factor. And in particular, that that phase encodes the choice of n. So you know, so any one of these structures, because U of S has all those nice properties that we talked about on the last slide, that completely fixes its form in this generalized free field case, up to some phase and that and yeah, and in fact, in fix, in fixing that form, you don't even need to mention that the subalgebra n just those properties of U of S. And so that phase sort of encodes the choice of n. Great. So now I'll go over what's the calculation strategy that we're going to use. And so the first thing and the thing that really makes this possible for generalized free fields, is that we'll see that this this these unitary operated U of S can be described by matrices or really they're really infinite dimensional matrices and we're like linear maps. But I might go back and forth saying matrices and linear maps. And so we're going to take two steps to compute U of S. The first step is we'll show that that permissivity of the generator, the modular conjugation property and the invariance of the thermo field double under U of S. So those three properties together, they uniquely determine an extension of U of S from negative values of the parameter to positive values of the parameter. And so this is important because U of S for negative values of the parameter is a little bit simpler to understand because it actually preserves this algebra n. It doesn't mix this algebra m with anything else for negative values of the parameter, whereas for positive values, we're going to get mixed with the commutant of n. And so and so yeah, so this this extension sort of allows us to focus only on the negative values of the parameter, because we know how to extend it to positive values. And then we'll sort of zoom in on this case where the parameter is negative and we'll see that that group property, the fact that we have this representation of the group of numbers on the real line, that modular algebra property, permissivity of the generator and invariance of the thermo field double again. So all those properties are coming into it again. They completely fix the form of the matrix that describes U of S for negative values of S up to this phase. Great. So this is the strategy that we're going to take. And before diving into that, I'll just I'll just briefly, you know, sort of fix some notation on generalized free fields and what do we mean by that. So in this discussion, I'll take this algebra m, I'm going to write it as m sub r. So you can imagine that, you know, like this is the algebra of operators on the right side of the thermo field double, and it's commutant is the algebra on the left. And so then, so then a generalized free field algebra is generated by operators that look like this. So they look a lot like a free field. So I have, you know, here's my operator and then I have some coordinates on some space time manifold that I'm imagining studying my generalized free field on that manifold. We have these mode functions U. And then we have basically these creation annihilation operators that I'll denote by A. And so one important, yeah, so just, yeah, to mention notation, I'm not going to write, I won't write a dagger anywhere in these definitions of A. I guess, yeah, first I'll say the sum over momenta, what's different about a generalized free field from a free field in this mode expansion is that we also have to sum over independent values of the energy or frequency. You know, we don't have the energy or frequency determined in terms of the spatial momenta by some dispersion relation because we don't have an equation of motion. These generalized free fields don't satisfy that equation of motion. And so actually, you know, you have these independent degrees of freedom at different values of the frequency. And there's a similar mode expansion in this left side, which is I'll take to be related by the CPT conjugation. So, you know, these sort of left mode functions that would appear, they're just given by these complex conjugates of those mode functions. And oh, yeah, one thing I meant to mention is that here, you know, in this mode expansion, I didn't write A and A dagger, you know, instead I just wrote it with a single K. So, you know, the Hermitian conjugation will take an operator of negative frequency to an operator of positive frequency and vice versa. So that's sort of fixing the notation. And then, okay, the very important part of generalized free fields is that the commutators are all complex numbers. If I look at the commutators of these operators, it's just a complex number. There's not a non-trivial operator on this side. And this is sort of the hallmark of generalized free fields. And so, again, fixing notation a little bit, I'll use these indices from the start of the Greek alphabet, alpha and beta. Those take on the values like R and L right and left. And so, the operators on the two sides commute with each other that's reflected by this delta, alpha, beta. And then we have this sort of, you know, the usual commutation relations for creation and annihilation operators here. As I mentioned, Hermitian conjugation is just going to flip the sign of the momenta. And then modular conjugation acts, you know, in this well-known way. It exchanges this algebra M with its commutant. And so, in particular, yeah, the modular conjugation acts in this way on this position space operator, you know, it takes an operator, for example, on the right side to the same operator on the left. Similarly, a left operator to a right operator. So, this sort of alpha bar notation is just to be interpreted as, you know, you take the opposite side. If I have R bar, that's equal to L, L bar is equal to R, you know. And so, this is sort of just fixing the notation that we'll use to describe these generalized free fields. And so, now we have to define the thermo field double state. And it's defined in this way. It's defined by, oh, and one thing I should mention is I will always work in units where the inverse temperature is 2 pi. And so, that's why we'll have this e to the minus pi omega here. So, the thermo field double states is the state that, you know, if I act with, you know, say a right, you know, a right annihilation operator, then that's the same as acting with a left creation operator up to some, up to some pre factor here that depends on the temperature. But in this case, I'll just work in units where the inverse temperature is set to 2 pi. And the reason this is important is it allows us to define operators that annihilate the thermo field double. So, none of these a-alphas, so none of these a-rights or a-lefts annihilate the thermo field double. But, you know, we have to do this Bogolubov transform into operators that do annihilate the thermo field double. And so, we define these operators in this way. And then, for positive values of the frequency, those operators will annihilate the thermo field double. And, you know, these operators are sort of, it's, you know, like the more familiar case where you're just quantizing about the Minkowski vacuum or something. You have annihilation operators that annihilate the state and creation operators that don't. So, now that all of this notation is fixed, we're going to, we're going to describe the evidence. Can I ask you? Yes. What was the function epsilon of k that you had in the previous line? Yes. Good point. Sorry. Yeah. Yeah. I guess this is a non-standard. This is just the sine function. So, if omega is positive, it should give plus one. If omega is negative, it should give minus one. Yeah. These things just sort of appear because I've chosen not to write a and a dagger, but instead label by positive and negative frequency. But, sorry. In the generalized free field, don't you have something like the dimension of the operator? It will be the mass of the free field in the bulk. So, for this description, yeah, for this description, the mass, yeah, I guess maybe it's a point of normalization, but the mass or the dimension, yeah, so the mass in the bulk or the dimension of the operator is completely encoded in this function u k. Let me see. Yeah. Yeah. Okay. Great. Thanks. Yeah. Good. Great. So, with these operators that, you know, so, yeah, one important thing is that, you know, these C operators and A operators, they're just a change of basis from each other. And so, you know, if we can describe the evolution of the C operators, then we can do a basis transformation and we can describe the evolution of the A operators. And so, we'll want to describe the evolution of, you know, one set of these oscillator operators using, and we want to understand the evolution under this U of S, this U of S that we get from the half-sided modular inclusion. And in particular, because we get U of S from the half-sided modular inclusion, as I emphasized earlier, it has those five very important properties. Great. And so, here's an important part about generalized free fields. The fact that all those commutators were complex numbers means that when I do the evolution of this operator under U of S, well, that just has to give me some new operator. But I actually have, you know, a basis for those operators here. These A k's or C k's form a basis for these operators. And this is a very special property of generalized free fields. It's reflecting the nature that, you know, they're free fields. And so, you know, basically all this statement is is that the evolution of some operator is some other operator. And at this point, we don't know anything about the form of lambda or sigma. These are these matrices that I discussed earlier that describe the evolution. And we'll see that we can really constrain these matrices all the way down to being fixed all the way down to a phase. Great. So, yeah. So, yeah, as I mentioned here, yeah, you know, lambda and sigma, they're related by some basis transformation. They're just describing this transformation in different bases. And they're completely fixed up to a phase by those conditions that U of S is known to satisfy. So, the first condition is the hermitsity of G. So, you know, if I take this evolution, and I do the and I do the Hermitian conjugation, I can either first expand, you know, in terms of sigma and C and do the Hermitian conjugation, or I can take, you know, I can use that U of S dagger is equal to U of minus S. And, and, you know, using and then do the expansion afterwards. And then you'll get this equation, you'll find that sigma has to satisfy this property, you know, it's complex conjugate, is the same as if I just reverse the signs of the momentum. Great. Then there's the modular conjugation property. And that says that, you know, if, you know, if I do the modular conjugation, and if, you know, if I first do the expansion in terms of sigma versus, you know, using the fact that J, you know, revert J squared is equal to one, that it reverses the sign of S, then I can I can deduce from that this this following condition that sigma has to satisfy. So if I take a complex conjugate, this essentially is coming from the fact that J is anti unitary. Then, then that has to be the same as if I, you know, reverse the right and left sides, and then also take this parameter to be negative. So this is just these are just some properties that these matrices have to satisfy. And then finally, I'll talk about this thermo field double invariance. So that was the idea that U of S for any real value of the parameter S has to preserve this thermo field double state. And yeah, in particular, what that means is that now if I act with the involved one of these CK oscillators, while U of S preserves the thermo field double state, that's what's appearing on the right hand side. So the C alpha K still annihilates the thermo field double from positive frequency. And so I still have to get zero when I evolve this operator. And so yeah, so in particular, the fact that the evolution preserves the thermo field double means that it cannot mix the positive and negative frequency C type oscillators, those oscillators that are like the annihilation creation oscillators for the thermo field double. So this allows us to, you know, decompose this matrix sigma into a very specific form. There's one matrix A, which parameterizes, you know, how if I evolve an operator positive frequency, how does it mix with other operators of positive frequency? And similarly, another matrix B, that tells us if I evolve an operator of negative frequency, how does that mix with other negative frequency operators? And so these sort of important properties of sigma, it turns out that you can look at all those properties and put them together and find that sigma for S positive can be completely determined in terms of sigma for negative values of S. And so then the last thing that I'll recall about U is that it's preserving this algebra M for negative values of the parameter. So if I do this evolution by negative value of the parameter, then I have to stay within the algebra. And so what this means, what this means, thinking about those A oscillators again, which lived either in the right or the left algebra, what this means is that if I take a right oscillator of some, of some, you know, quantum numbers K, if I evolve it by negative value of the parameter, then it's overlapped with the left oscillator of any, you know, any quantum numbers K prime for negative values of the parameter, that has to be exactly zero. Otherwise I wouldn't have this property up here. I would have mixed the right algebra with the left algebra for negative values of the parameter. And so what we'll do is we'll just define, you know, this matrix C now without any of these right or left labels to be the matrix that describes the evolution of these right oscillators for S, for S negative, because we know that they're not going to mix. We know that they're not going to mix with any of the left oscillators for negative values of the parameter. And so with, with all of these properties that U of S has to satisfy and taking those conditions on the matrices that I just showed on the last two slides, that Hermit, Hermit city of G modular conjugation and invariance of the thermal field double will determine sigma in terms of this C K, K prime of S, which is only defined for negative values of S. And in this way we extend U S from its, from its, its negative value of S, you know, description to all values of S just from those properties that we know U of S must satisfy. And so here I'll just I won't go into any detail about this, but this is just, you know, a picture of that full extension of U of S. So here, so here I've described in terms of these lambda matrices, but, but you know, as you can see, you know, if I take this lambda right, right, well that was by definition this C of K, K prime for negative values of S, but you know, any other possible values I could have chosen for alpha and beta or for K and K prime all all and for different values of S, those are all determined in terms of this, you know, this single, the single matrix C K, K prime in this way. And so this is sort of the solution of all those properties that extends U of S from negative values of the parameter to positive values of the parameter. Great. So that's the first, the first step in, in, you know, constraining what, what form this U of S has to take. And so the second step is to determine what can we say about this C K, K prime matrix that described the, you know, the evolution of these operators for negative values of the parameter. So we have this group property that tells us, you know, that if we take this product, it's the same as the sum. And we have this modular algebra property that tells us the modular flow of U of S is you at some other, at some other value of S. And this actually will completely fix C K, K prime up to a phase. So the first thing that we have to note is that the modular flow in this thermo field double states, you know, the modular flow for these left or right oscillators is actually just a time translation. It acts as a time translation in the following way. So if I do the modular flow of this, of this oscillator here, I pick up this phase in front. And it just like it just like you would expect for a time translation. And in particular here, there's a bit of an abuse of notation. This alpha, if it's right, should be interpreted as a plus one. If it's left, it should be interpreted as a negative one. That's just reflecting the fact that the left theory is a time reversal of the right theory. So the time's running in opposite directions. And so then we can play the same game. We can play the same game with with this modular property. So we take, you know, the modular, the modular flow of this operator, and either we can do our expansion in terms of the matrix, the matrix first and take the modular flow after using this equation, or we can use this modular property and first, you know, do the modular flow of U of s, and then do the expansion afterwards. And all that boils down to this equation here at the bottom, which is a which very, you know, it completely fixes the s dependence of this matrix lambda. So right, so this matrix lambda, you know, the only difference in terms of these two matrices is the, you know, the value of this s parameter that we're at. And that, you know, changing that s parameter has to change the phase in this way. It has to, you know, change us from this alpha omega on this side to this beta omega prime on this way side. So this is very constraining property on the s dependence of this matrix lambda. And in fact, it completely fixes it to take the following form. So it has to take, it has to be of the form s to some complex power, you know, involving the various sides. And then, you know, the frequency, the two frequencies in this way. And then if you trace that back to the mapping to C k k prime, this tells us that the s dependence of C has to be of this form. So that's the s dependence of this matrix C k k prime of s is completely fixed to be of this form. Can I just, just to fix it. So all this you could just state in Minkowski Rindler for a free field, right, more or less? Right. So, so, so yeah, so if you, if you look at like the Rindler decomposition of Minkowski for a free field, then, yeah, then you'll find the evolution matrices look exactly like this. Yeah, if that was just to understand in that context. So, sorry, so k and k prime, you're in how many dimensions? So, so this is, it's, yeah, I guess it's general at this point. I haven't said anything about the manifold that I'm working on. So, so k and k prime are vectors? Sorry. Yeah, so they could be vectors, but it also could be, like, you know, if I have, if I have like a boundary cylinder, then, you know, I can have angular momentum and then maybe a frequency associated with the time direction. Okay. Yeah, yeah, so they're, they're just sort of general quantum numbers that are characterizing the the manifold you're working on. Yeah. So Samuel, can I also ask a question? You started this discussion by writing down the evolution of mode. And on the right hand side, you had the linear relationship with A. So there was sigma times A. So the fact that we do not have any additional higher order terms, I guess can be understood by the intuition that the model Hamiltonians for m and m are effectively quadratic in the, in the operators, right? Yes, yeah, that's yes, yeah. Okay, yeah, yeah, that's exactly right. Yeah, so that yeah, that will guarantee that we don't generate those terms. And then that's guaranteed by the sort of free field nature of, yeah, of this problem. Yeah, great. Okay, but can you please translate everything? Let's say for Rindler and Minkowski in two dimensions, let's say it's a key and any right, right, right. So u of s is like some deformation of the Minkowski evolution, time evolution, whereas the modular flow. So m is like the right Rindler wedge. Yeah, so yeah, you can imagine. Yeah. So an example of this is to choose m as the right Rindler. And then the modular flow is like the Rindler evolution. With the t, t being the Rindler time. Okay, okay. And s is like the Minkowski time or but some deformation of, yeah. Yeah, so s is like the, yeah, s is like the Minkowski time. The simplest example of this is to, yeah, I should have mentioned that. The simplest example of this is to take the Rindler wedge, take m to be that algebra, then just shift the Rindler wedge by a null translation in like a u direction. Just shift that down and then take the algebra in that region. And that's n. So under the modular flow, which was the boost about the origin, then that shifted Rindler wedge is preserved under the boost about the origin by negative values of the boost of the Rindler time. And then what you find, the U of s that you get from that procedure is actually just a pure null translation along the Minkowski u direction. That's right. Yeah, that picture I remembered. I'm not trying to understand why these matrices are universal in that picture. I mean, there must be some easy way to see this. Yeah, why, okay. So that is, why is it so uniquely determined up to a phase? Right. I guess, I think because the form of this U of s has to satisfy all these properties. So it's a very special form of evolution operator. Yeah. So in this example, you're saying that the U of s can only be this null translation? No. So yeah, so yeah, I guess, yeah, maybe what I was saying is that in that example, for those choices, U of s is a null translation, but I might be able to choose some different choice of sub algebra. And then I might, and then of sub algebra that I'm using to construct this half sided modular inclusion. And if I choose a different choice, then I might not get a U translation of U of s. But the matrix description of the form will look exactly like this, except the phase will be different. So even though the new U is not really a null translation, or is it a null translation up to a phase? So yeah, so this, it doesn't have to be a null translation. Because you're looking at some different basis of key, I mean, of these A and A dagger operators. Yeah. Yeah. I think, yeah, maybe the simplest examples are these null translations, but it doesn't have to be a null translation. Yeah. But nevertheless, this matrices are determined up to a phase. Yes. Yeah. Yeah. Yeah. So I mean, these A operators, yeah, these A operators might be some very strange object that we don't usually talk about. Yeah. Great. Great. So now that we fix this s dependence, we'll assume that this evolution preserves the angular momenta or the transverse momenta. So yeah, whatever are these sort of spatial momenta on the manifold, we'll assume that it preserves those transverse momenta, and then see k, k prime the s dependence is fixed. So all that's left is some function g of k and k prime. Well, then you can go through, you can go through, you know, plugging this into into the fact that, you know, u of s times u of s prime is the same as u of s plus s prime. So into that group property, and then you get some equation that looks like this. And this is actually a very stringent constraint on g of k and k prime. And it's solved by this taking g of k and k prime to have this form. So we choose some function lambda, now just of a single momentum, you know, we choose the same function and we put it in the denominator for k prime. And then there's this gamma function that appears. And if we take this form, then we're guaranteed to satisfy that group property. And then the hermicity of the generator g, well, if you track, you know, those properties through all these definitions, what you'll find is that, you know, taking the complex conjugate, I have to get the negative values of the momenta. And so now translating into, you know, this new parameterization of the function, then we have to get this property of the lambda function. And then finally, the thermo field double invariance. Well, what this says is that, you know, if I evolve these oscillators for some value of s, that's negative, which is what we're considering here. Well, if I evolve them by the same value of s, the thermo field double is invariant. So that had better not change this two-point function. And if you, you know, you go through plugging in this parameterization of u of s, what you'll find is that this implies that lambda k times lambda minus k has to be equal to this one over two cinch factor. So after all of that, after plugging in all those properties and looking at those equations, you find that lambda k has to take this form. And this is this phase that I was talking about, the fact that we've only determined the evolution up to a phase, we're still allowed to put an arbitrary phase here in that form. And then that's reflected in this C kk prime, this matrix that's describing the evolution for negative values of s, in this way. It's reflected in this sort of arbitrary phase that can appear in front. And that phase sort of encodes the choice of the subalgebra n. So just sort of going through this, you know, from the right to the left, we assumed that we were going to be diagonal in the transverse momenta. So we have this, we're conserving those transverse momenta. This gamma function factor comes from the fact that we want to satisfy this group property. We want, you know, we want this, we want the product of the two evolutions, give us the evolution of the sum of the two parameters. This minus s is fixed by the modular properties that you have s had to satisfy this s dependence. And then this square root of the cinch factors that, that's determined by this, this invariance of the thermal field double under this evolution. And so putting that all together, putting the two parts together, we have, we have, you know, this form of lambda, which is describing this, this evolution u of s. So lambda is completely determined from C kk prime of s. And then this is the general form that C kk prime has to take, which is completely fixed up to this phase. Great. So are there any questions about that? That's, that's sort of the roadmap of this calculation. Great. So now I'll just quickly go over some, some applications before concluding. So, so the first application is as Hong mentioned last time. So now if, if we take, you know, these generalized free fields on the boundary, remember that these generalized free fields don't satisfy an equation of motion. They're independent operators and independent at different times here. So this n is a genuine sub-algebra. And so now we know that we have a generalized free field theory on the boundary. We have a half-sided modular inclusion. So that u of s, by the calculation we've just done, is completely constrained up to this phase. But computing that phase in general can be difficult. So the way that we compute it is, is we conjecture that, you know, this algebra of operators below this fixed time slice is, is described by the algebra of operators at some, you know, cutoff at some crucible u coordinate. And then you can actually compute that that phase has to be given by the phase shift of these bulk fields at the horizon. And so now that we've computed the phase, we've completely fixed u of s. And it's that choice of u of s that gives rides to this crucible u like translation taking us from the right into the future. And our, or, you know, alternatively you can draw it in terms of Cauchy slices over here. And with that u of s, we can generate those interior regions from the exterior regions in this way, as I mentioned earlier. And then one thing that that's important in all of this is that this, remember that that u of s just was required us to choose a nice choice of sub algebra n. It turns out that that, you know, in this black hole case, there's actually an infinite number of choices that we can make for that subalgebra n. So there's an infinite number of those evolution operators that we can find. And so I'll just conclude with some future directions related to this work. So the first thing is that one crucial feature of our work was the emergence of this type three one algebra. And we'd like to better understand the emergence of that type three one structure at large n. And specifically, we'd like to understand this, you know, in some more concrete models, like maybe the syk model, how exactly does that does that transition happen from this type one algebra to the type three one. We'd like to better understand the role of bulk singularities. Now that we have this evolution of operators that that that can go in and hopefully probe the singularity, we'd like to better understand how do we see the singularity from the boundary theory. We'd like to apply this framework to emergent summaries, symmetries in the SYK model, and also and also, you know, near horizon symmetries. And finally, we'd like to study, you know, the emergence of the black hole interior from these single sided black holes using this kind of structure. So thanks very much for your attention. And I'm happy to answer any more questions. Thank you, sir. And thanks a lot for your talk. There are any more questions? Sorry, can I ask a question? So if instead of the thermophilic state, we start with another state where there is a unitary on the left side, some particle. I guess this formalism will give you a different G. Mm hmm. Which seems to sort of contradict what you would calculate from effective theory in the bulk where the evolution operator for a from the right waves into the future waves is is a fixed operator independent of the state that you're considering. And like, for example, in the Koski space, maybe this light translation is the correct generator that, you know, moves the observer through the real horizon. But if you try to reproduce it via this formalism, you would get a different result depending on what unit areas you have worked with on the left side. Right. We have any way of, let's say, removing this or defining some criteria under which you use this to see as the physical evolution? So, yeah, so I think, yeah, we don't, yeah, I guess I would say, yeah, we don't have a criterion for choosing a particular U as a physical evolution. So as I mentioned at the end, there's an infinite number of choices for this U. And one way is exactly as you've mentioned. If I act, if I act some small unitary on the left, then then I get a value, yeah, I get a new, I get a new generator G, which is related basically by conjugation of that unitary. And so, right, as long as that unitary is small enough that it doesn't destroy the bulk geometry. But yeah, yeah, but my comment here is that if you look at the time evolution from the bulk point of view, from effective theory in the bulk, you would expect that the operator, the unitary that will transform a local operator on the right side to an operator behind the horizon, the operator itself should be independent of the presence of where you left or not, whether you have a U left on the other side or not. While this mathematical formalism predicts that J transforms by this conjugation by the unitary on the left. Right. So, okay, so, okay, so are you imagining we have, we have a preferred evolution. So, for example, in the bulk, we can say we just want to evolve by crucial time, some crucial time and evolve. Right. Okay. Right. Yeah. So, yeah, so in this case, I think, yeah, you will still, or maybe you say it will be differently. This formalism, for example, predicts that the state itself, the state you start with is invariant under evolution with you. We sort of make sense, maybe for the thermophilic state, which is, well, there are no excitations on the exterior, but if you take an excitation with, a state with excitations, also the black hole that will fall in, why would we expect this to be invariant under you, from a physical point of view? Right. I guess, I guess maybe, I think maybe, I guess, yes, from our perspective, these different choices of, yeah, these different possible evolutions are describing different possible families of observers falling into the bulk. And so, I think you might not, you might not, I guess, expect them, yeah, they won't all have the same evolution operator, I think. Yeah. Yeah. Okay. Yeah. Okay. I would say, yeah, you annihilating, yeah, leaving the state invariant, if I just think in Minkowski space, sounds like a strong requirement, right? I mean, it's not generically, for a, I mean, except for the vacuum, it's not true, for any other state. So how come this requirement is so central to your construction? Right. Yeah, I guess, yeah, that's a good question. Yeah, so I guess it's, I guess, from our perspective, it's more of a, it was a requirement that came out of, it's a property that came out of these theorems of Borchers and Wiesbrock, that you do preserve the state with the evolution. You could apply it, I mean, before going to holography, right? Suppose you just try to apply it in Minkowski space. I'm just trying to imagine, if you take instead of the Minkowski vacuum, if you take a slightly excited state, can you find, how can you, how can the time evolution operator of any kind annihilate the state? I mean, no, not annihilate it, leave it invariant. Right. So I think in that case, you could run this Borchers theorem on a simple try, I mean, simple example of like this Bison, you know, I mean, for Miller observer, right? Just do it for Miller observer. Then how should I think about, I mean, there must be a way to think about it. I'm just trying to understand in the context of Miller, yeah. Right. So I guess, so maybe, yeah, so in the context of Rindler, I mean, I guess, right, yeah, one crucial input, yeah, so yeah, one crucial input, I guess, was that we were in a thermal state as seen by the Rindler observers. And, right, and using, right, and so that's why when we, I guess, if we purify that state to the thermal field double, then we end up with that. If you were slightly excited over the thermal state, then that's not. So, yeah, so if you're, if you're sort of, I mean, yeah, so if you're, so yeah, so say you can apply some, maybe some small unitary to give you these excitations. Then, yeah, then this construction still goes through, but you'll get a new generator. And I mean, essentially, the generator transforms simply. So maybe this is the answer. If I apply some small unitary, so imagine I apply some small unitary to excite the Minkowski vacuum a bit, and then I run through this construction, what I find is that this, this new generator G is basically related to the old generator G by conjugation by that unitary. So essentially, the preservation of that state is just the preservation of the vacuum again, but I have to take off the unitary, you know, annihilate the vacuum and then apply the unitary again. You're only imagining unitary, but suppose I just take Minkowski and act with a dagger, yeah, one particle state, right, right, that's the unitary, right. Yeah, in that case, I guess I would have to think more carefully about the properties of that state. Yeah, yeah, yeah. Yeah, yeah. Can I ask one more technical thing? You mentioned that this phase that you found for the matrix that you had C or sigma was the same as the phase of the phase shift of the fields near the horizon. Is there a, is there any way to understand that without going through the calculation? Right, yeah, that's, is there any intuitive explanation of this? Yeah, yeah, I guess, yeah, that's a good question. Yeah, is there an intuitive way to understand that? So I think, I think essentially, so an important feature, I guess, of this U of S, right, is that there of G maybe is that it's S independent. So we have the same U of S everywhere in the bulk. And basically what that means is we can study what these null evolutions look like in the near horizon limits. And then in that case, you get the phase shift showing up. So, yeah, yeah. I see, okay. Yeah, yeah, thanks a lot, Sam, for a very nice talk. Great, thanks so much for all the questions. Thanks. Thanks. Thanks, yeah. Thank you again. Thanks.