WEBVTT 00:00:30.000 --> 00:00:32.000 Okay, yes, all right, great. 00:00:32.000 --> 00:00:34.000 Um, 00:00:34.000 --> 00:00:40.000 Yeah, so I'm gonna talk about a number of related topics in the context of 00:00:40.000 --> 00:00:45.000 quantum cosmology from the point of view of path integral and holography. 00:00:45.000 --> 00:00:51.000 And I don't have that many sides, so I'm happy to discuss things as we go. It's more fun that way, anyway. 00:00:51.000 --> 00:00:53.000 Um, and I guess this stuff is… 00:00:53.000 --> 00:00:55.000 confusing enough that… 00:00:55.000 --> 00:00:59.000 questions and discussion are good. 00:00:59.000 --> 00:01:04.000 So, by way of introduction, let me begin by saying that 00:01:04.000 --> 00:01:11.000 you know, not everyone maybe agrees, but from my point of view, I would say we now have a fairly robust 00:01:11.000 --> 00:01:16.000 framework for thinking about the black hole interior and the information problem. 00:01:16.000 --> 00:01:19.000 Don't know the answer to every question, but um… 00:01:19.000 --> 00:01:28.000 We know the answers, at least plausibly, to quite a few questions that we didn't before. Um, and so the way that I like to summarize the situation is I say that 00:01:28.000 --> 00:01:42.000 The effective field theory of the black hole interior is non-isometrically encoded into the microstate degrees of freedom, leading to a page curve that's consistent with unitarity via the quantum extremal surface formula. 00:01:42.000 --> 00:01:45.000 So, if it was… 00:01:45.000 --> 00:01:47.000 Four years ago, I would have been… 00:01:47.000 --> 00:01:58.000 giving this talk, trying to defend this statement here and this bullet point. This is my attempt to summarize what we learned in, uh, in, uh, one sentence. 00:01:58.000 --> 00:02:00.000 But today, I'll just… 00:02:00.000 --> 00:02:03.000 say that this is true, and um… 00:02:03.000 --> 00:02:18.000 go on from it. And so one thing that's natural to do, you know, whenever you think you learn something about black holes, so, well, quantum gravity is really only important for two things. It's important for black holes and cosmology, and so if we think we learned something about black holes, we should 00:02:18.000 --> 00:02:21.000 see if we can use it to learn something about cosmology. 00:02:21.000 --> 00:02:26.000 So, unfortunately, 00:02:26.000 --> 00:02:28.000 Uh, when we take this slogan here, and we try to… 00:02:28.000 --> 00:02:34.000 generalize it or apply it to cosmology, we're led to a rather disturbing conclusion. 00:02:34.000 --> 00:02:44.000 Which is that the Hilbert space of quantum gravity in a closed universe has dimension 1. So, in other words, quantum gravity in a closed universe has zero degrees of freedom. 00:02:44.000 --> 00:02:48.000 Um, and… 00:02:48.000 --> 00:02:53.000 you know, then you immediately, of course, ask, you know, what are we to make of this, right? You know, how can you account 00:02:53.000 --> 00:02:58.000 For the richness of human experience in a one-dimensional Hilbert space, or 00:02:58.000 --> 00:03:01.000 Alternatively, is this an argument that we don't live in a… 00:03:01.000 --> 00:03:08.000 in a one-dimensional Hilbert space. You know, sorry, that we don't live in a closed universe. 00:03:08.000 --> 00:03:12.000 Daniel, isn't quantum mechanics projective? 00:03:12.000 --> 00:03:17.000 Isn't quantum mechanics projective? Yeah, so it means that it's very boring in a one-dimensional Hilbert space, right? 00:03:17.000 --> 00:03:23.000 So the projective Hilbert space. If the Hilbert space is one dimensional, projective space is a point. 00:03:23.000 --> 00:03:28.000 Yeah, that's correct, yes. So that's why I call it zero degrees of freedom. 00:03:28.000 --> 00:03:31.000 Yeah. Yeah, sounds bad, I agree. 00:03:31.000 --> 00:03:34.000 So, um… 00:03:34.000 --> 00:03:40.000 So last year, um, several of us proposed a way to try to make sense of this problem, 00:03:40.000 --> 00:03:54.000 Um, where we suggested something that on, you know, on some level is crazy, it's somewhat extreme, you know, the defense for that is that this problem of zero degrees of freedom is a very serious one, and so we're going to have to do something crazy to… 00:03:54.000 --> 00:03:57.000 To deal with it, okay, whatever we do. 00:03:57.000 --> 00:04:08.000 And so we proposed that the laws of physics, the fundamental laws, somehow should treat observers differently than they treat the rest of the things in the universe. 00:04:08.000 --> 00:04:14.000 Now, you know, this is something that actually, for example, Bohr would have been fine with, but uh… 00:04:14.000 --> 00:04:18.000 you know, there's a… 00:04:18.000 --> 00:04:24.000 in the years since the 1920s, probably there's been a pushback against that point of view, that the observer is just… 00:04:24.000 --> 00:04:30.000 Made out of a bunch of quantum particles, just like everything else, and there's decoherence and so on. Now… 00:04:30.000 --> 00:04:39.000 Now, I agree with some of that. On the other hand, I'm not really a many-worldser. I don't think it's actually science until you add the board rule, which brings the observer back in some special way. 00:04:39.000 --> 00:04:42.000 Um, but still, you know, trying to… 00:04:42.000 --> 00:04:44.000 it's always a bit concerning when you try to… 00:04:44.000 --> 00:04:47.000 give yourself special treatment in the laws of physics. 00:04:47.000 --> 00:04:51.000 Um, so… so maybe this is all wrong for that reason, I don't know. 00:04:51.000 --> 00:05:02.000 Now, in, uh, so there were two versions of this proposal, so the one that I'll focus on in this talk is, uh, although I won't focus on it, but I'll use it occasionally in this talk. 00:05:02.000 --> 00:05:06.000 Um, the idea was that what you're supposed to do is… 00:05:06.000 --> 00:05:15.000 is essentially include into the laws of physics, you know, when you want to compute the predictions that are seen by some observer, you have to first 00:05:15.000 --> 00:05:27.000 use a quantum channel, really called a quantum to classical channel, that decoheres the observer in the pointer basis. And only after you've decohered the observer are you allowed to ask about what they see. 00:05:27.000 --> 00:05:30.000 And what we argue in this paper is that, um, 00:05:30.000 --> 00:05:39.000 If you use this decoherent channel, then it allows for a semi-classical physics to be recovered up to errors which of order e to the minus the entropy of the observer. 00:05:39.000 --> 00:05:43.000 Um, and we had some philosophy about how that's the only… that's a… 00:05:43.000 --> 00:05:49.000 good enough. You know, if you're an observer, you don't need semi-classical physics better than e to the minus your entropy. 00:05:49.000 --> 00:05:58.000 And so that was our proposal. This other proposal of the West Coast proposal was, uh, is kind of the S Observer going to infinity. 00:05:58.000 --> 00:06:03.000 limit of this one, to the extent that that makes sense. 00:06:03.000 --> 00:06:06.000 So, um… 00:06:06.000 --> 00:06:10.000 So that was the talk I would have given last year. 00:06:10.000 --> 00:06:15.000 But it's this year, so I won't give that talk either. Instead, what I'll… 00:06:15.000 --> 00:06:23.000 talk about is some potential weak points, or at least questions that I felt were left over by this proposal, and uh… 00:06:23.000 --> 00:06:25.000 In this talk, I'll try to… 00:06:25.000 --> 00:06:30.000 address those questions that I felt were left over. And so let me say a bit about that, right? So… 00:06:30.000 --> 00:06:37.000 So, I'll have some more concrete questions on the next slide, but first let me say the rough idea. So, um… 00:06:37.000 --> 00:06:42.000 you know, the way these papers try to phrase the emergence of quantum mechanics in a closed universe is… 00:06:42.000 --> 00:06:47.000 In terms of transition amplitudes between arbitrary closed universe states, 00:06:47.000 --> 00:06:50.000 Um, and… 00:06:50.000 --> 00:06:59.000 In the discussion, um, we both ignored the possibility of disconnected contributions to transition amplitudes between 00:06:59.000 --> 00:07:06.000 closed universe states, so those are contributions where the universe, together with the observer, is sort of fluctuated out of nothing in the Harle-Hawking state. 00:07:06.000 --> 00:07:11.000 So we just essentially ignored that 00:07:11.000 --> 00:07:14.000 possibility. For example, you could say we impose some sort of 00:07:14.000 --> 00:07:17.000 approximate global symmetry that suppressed the… 00:07:17.000 --> 00:07:20.000 production of the observer with the universe in it out of nothing. 00:07:20.000 --> 00:07:31.000 But, you know, okay, so we did that, we had some story, but, you know, you could say, well, if the theory allows for this, then our rules should also allow for it, and we need to 00:07:31.000 --> 00:07:34.000 figure out what to say about it. And so that was… that was the sort of… 00:07:34.000 --> 00:07:42.000 on-ramp for me. Um, but actually, along the way, as you'll see, I ran into a whole mess of issues about how this approach 00:07:42.000 --> 00:07:52.000 relates to the third quantized approach involving alpha parameters. Um, and so actually, this talk is mostly going to be about that. So this talk is actually just mostly going to be about 00:07:52.000 --> 00:07:56.000 How do we understand the gravitational path integral? And then the observer will sort of… 00:07:56.000 --> 00:08:00.000 appear towards the end, uh, in… to help us… 00:08:00.000 --> 00:08:02.000 resolve one of the issues that comes up. 00:08:02.000 --> 00:08:08.000 And so I'll also, at the very end, I'll point out a puzzle that relates these ideas to the string landscape. 00:08:08.000 --> 00:08:15.000 Um, which, again, is sort of very, uh, yeah, we'll see when we get to it. It's puzzling. 00:08:15.000 --> 00:08:20.000 And so this talk is based on a paper I put out, um, maybe 2 months ago, 00:08:20.000 --> 00:08:26.000 Uh, the same week, there were two other papers, one by Ying Zhao, my collaborator on this work, and then another one by… 00:08:26.000 --> 00:08:30.000 some superset of the Berkeley people from the other… 00:08:30.000 --> 00:08:32.000 Berkeley paper. 00:08:32.000 --> 00:08:40.000 Yeah, I wouldn't say these three papers are all totally aligned. There are various points of agreement and disagreement between all of them. 00:08:40.000 --> 00:08:42.000 I think you should view all of this as, you know, 00:08:42.000 --> 00:08:50.000 up for debate and, uh, not settled. I'll tell you my perspective on all these issues. 00:08:50.000 --> 00:08:57.000 So, to motivate this a little more, let me try to ask some more concrete questions before we get into the more technical part of this. 00:08:57.000 --> 00:09:00.000 Um, so here, here are four questions that I would… 00:09:00.000 --> 00:09:07.000 want to understand. So… so first of all, how does, you know, if we say the Hilbert space is one-dimensional for holography in a closed universe, then… 00:09:07.000 --> 00:09:17.000 How does that one state relate to the Hartle-Hawking state, right? Do they live in the same Hilbert space? Do they have anything to do with each other? People sometimes say they're the same. We'll see that that's not actually true. 00:09:17.000 --> 00:09:20.000 Um, but then we do want to understand how they're related. 00:09:20.000 --> 00:09:30.000 Secondly, in the third quantized approach to quantum cosmology, you know, which Duda Coleman Giddings, Strominger, 00:09:30.000 --> 00:09:32.000 et cetera. Um… 00:09:32.000 --> 00:09:38.000 There's a large closed-universe Hilbert space that's spanned by alpha states, which I'll explain that in a little bit. 00:09:38.000 --> 00:09:46.000 Uh, and then you can naturally ask, you know, if you have this big Hilbert space spanned by alpha states, how's that related to this idea that there's only one state? 00:09:46.000 --> 00:09:50.000 In a closed universe. 00:09:50.000 --> 00:09:56.000 Also, uh, you know, in perturbative string theory, right, we sum over topologies that connect 00:09:56.000 --> 00:10:10.000 you know, ingoing and outgoing string states, which looks like the same kind of sum over topology as we do in the arguments that the Hilbert space is one-dimensional, but the string… this, you know, the Hilbert space of string scattering space is certainly not one-dimensional, and so how is that, 00:10:10.000 --> 00:10:12.000 compatible with the one-state… 00:10:12.000 --> 00:10:14.000 Claim. Okay. 00:10:14.000 --> 00:10:17.000 Um, and then finally, 00:10:17.000 --> 00:10:25.000 you know, what's the relationship, if there is any, between this observer proposal I mentioned and the third quantized approach? You know, are there two solutions to… 00:10:25.000 --> 00:10:31.000 The one-dimensional Hilbert space problem, are they really the same? You know, what's going on? Okay, and so, um… 00:10:31.000 --> 00:10:36.000 And what we'll see is that the answers to all these questions depend on how we interpret 00:10:36.000 --> 00:10:39.000 the gravitational path integral, that's kind of going to be the… 00:10:39.000 --> 00:10:42.000 the common thread going through this talk, um, 00:10:42.000 --> 00:10:48.000 And so we'll now talk about that. But so first, any questions about the introduction before we get into it? 00:10:48.000 --> 00:10:53.000 Everyone's happy. 00:10:53.000 --> 00:10:56.000 Okay. Um… 00:10:56.000 --> 00:10:59.000 So let's start with path interval. 00:10:59.000 --> 00:11:05.000 Now, I'm going to first make an assumption that some people like and some people hate. I actually more hate it, but… but… 00:11:05.000 --> 00:11:11.000 I'll make it anyway, because I have to start somewhere. So I'm going to assume that the gravitational path integral is well-defined. 00:11:11.000 --> 00:11:16.000 Okay. Now, as far as we know, that's only really true in 1 plus 1 dimensions. 00:11:16.000 --> 00:11:21.000 you know, maybe in 2 plus 1, but even that's kind of borderline, because of the sum over topologies. 00:11:21.000 --> 00:11:30.000 Um, and so we're gonna have to walk back this assumption later, but for now, let's make it, because it's instructive to see where it leads to, and then we can… 00:11:30.000 --> 00:11:33.000 try to think about how much of what we learn will… 00:11:33.000 --> 00:11:37.000 go through in a higher dimensional context where it's not well-defined. 00:11:37.000 --> 00:11:43.000 Why do you need it? Why can't you just assume it's an effective theory, like in string theory? 00:11:43.000 --> 00:11:48.000 Um, well, I think you have to see what I do with it before you see the answer to that question. 00:11:48.000 --> 00:11:52.000 Yeah. But I'm going to do… like, I'm gonna do some… 00:11:52.000 --> 00:11:56.000 Mathematical arguments here, which are really going to need it to be well-defined. 00:11:56.000 --> 00:11:59.000 Uh, and uh… 00:11:59.000 --> 00:12:06.000 Yeah, part of that is because I want to make contact with things that other people do, um, in the context where it is well-defined, but I… 00:12:06.000 --> 00:12:10.000 Yeah, yeah, probably that's the best thing I can say. 00:12:10.000 --> 00:12:18.000 And it sounds from your what your comment about three-dimensional gravity that you're taking it for granted that we have to sum over topologies. 00:12:18.000 --> 00:12:26.000 Uh, no, I'm not. Actually, we're gonna discuss that here, but indeed, that, indeed, in 3D, that's the issue, is does this sum over topologies make sense? 00:12:26.000 --> 00:12:27.000 Right. 00:12:27.000 --> 00:12:32.000 In 2D, it makes sense, probably, but in 3D, I don't know how to sum over 3 manifolds, I don't know if it makes sense. 00:12:32.000 --> 00:12:33.000 Um, 00:12:33.000 --> 00:12:38.000 that that's difficult, despite a lot of claims in the literature and and. 00:12:38.000 --> 00:12:44.000 Well, yeah, as I said, I don't… I don't know if it makes sense. 00:12:44.000 --> 00:12:45.000 Yeah, well… 00:12:45.000 --> 00:12:49.000 4 4 manifolds. Well, any finitely generated group is pi one of some 4 manifold. So good luck. 00:12:49.000 --> 00:13:01.000 Yeah, right. I mean, so, well, maybe I should say why it's not well-defined, right? There's two reasons. One is be… in higher dimensions, it's… there are two reasons. One is that gravity is not renormalizable, and the other is that the sum over topology is not under control. 00:13:01.000 --> 00:13:09.000 In 3D, you have the sum over topology problem, but not the renunalizability problem, and somehow in 2D, you can do both. 00:13:09.000 --> 00:13:11.000 Okay. 00:13:11.000 --> 00:13:21.000 Now, even in situat… so I want to start by emphasizing that even in situations where the path integral is well-defined, there are actually 3 different ways 00:13:21.000 --> 00:13:26.000 And they all appear in the literature for how to relate the path integral to quantum mechanics. 00:13:26.000 --> 00:13:44.000 Uh, and the three rules differ exactly in this question Greg just brought up of which are the topologies that we sum over, okay? And so I want to… I want to be above board about all three, because, you know, I think a lot of the confusion in the literature and in discussing with people comes from people conflating various of these 00:13:44.000 --> 00:13:47.000 options without being careful about it. 00:13:47.000 --> 00:13:53.000 So, the most straightforward thing we can do is canonical quantum gravity. 00:13:53.000 --> 00:13:57.000 Okay? So, in canonical quantum gravity, 00:13:57.000 --> 00:14:01.000 We want to compute… so first, look at the diagram on the left here. So we have some field… 00:14:01.000 --> 00:14:06.000 eigenstate in the past, some field eigenstate in the future. 00:14:06.000 --> 00:14:10.000 And we want to compute a transition amplitude between them. Now, 00:14:10.000 --> 00:14:24.000 Because we're doing gravity, the Hamiltonian is 0, so that doesn't appear. And also, because we're doing gravity there, diffeomorphism constraints, so what we're really computing is a projection onto gauge invariant states. Uh, the matrix elements of that, okay, but that's what… 00:14:24.000 --> 00:14:31.000 canonical gravity computes. Um, and in particular, it computes it by only summing over the cylinder topology here. 00:14:31.000 --> 00:14:42.000 Okay, that's what canonical quantum gravity does. So if this spatial manifold is sigma here, then this spatial manifold up here is also sigma, and the spacetime in between is sigma times an interval. 00:14:42.000 --> 00:14:45.000 Okay, that's what comes out of canonical quantization. 00:14:45.000 --> 00:14:49.000 And so that's our first approach to the path interval. 00:14:49.000 --> 00:14:52.000 Now, what is this picture on the right? Well… 00:14:52.000 --> 00:14:58.000 Yeah, so this is something that's a bit… it's one of those things that it's a bit annoying to explain, but it's too convenient to… 00:14:58.000 --> 00:15:05.000 to not do it. So this is some notation of Meryl and Maxfield. So basically, this issue of the diff constraint is annoying. 00:15:05.000 --> 00:15:15.000 you know, then you have to worry about how do you take the inner product between Wheeler-DeWitt wavefunctions and group averaging and all this stuff, which, from my point of view today, is a distraction. 00:15:15.000 --> 00:15:24.000 And so what Meryl and Maxfeld proposed is that rather than preparing, you know, computing overlaps of states that are prepared on a finite 00:15:24.000 --> 00:15:34.000 Koshi Slice. You take the cosmological constant to be negative, and then you prepare states using asymptotic Euclidean ADS boundaries in the future and past. 00:15:34.000 --> 00:15:42.000 So when I write a J, I mean the sources for some fields on an asymptotic Euclidean ADS boundary. 00:15:42.000 --> 00:15:51.000 Right? So, so the cost of doing this is that you have to take lambda to be negative, which of course is not something that you want to do in quantum cosmology. 00:15:51.000 --> 00:15:57.000 Um, I think everything I say in this language, I can also say in this language, it's just more annoying. 00:15:57.000 --> 00:16:00.000 And so, for this talk, we'll just pay the price. 00:16:00.000 --> 00:16:04.000 of having the negative lambda, but then in return, 00:16:04.000 --> 00:16:17.000 are the states that are defined by these boundaries with sources J are gauge invariant, so we don't need to worry about this projection. We just have a gauge invariant inner product summing over whatever you can put in between these two Euclidean ADS boundaries. 00:16:17.000 --> 00:16:19.000 Okay? And that defines… 00:16:19.000 --> 00:16:23.000 So you're summing over metrics on this shaded object? 00:16:23.000 --> 00:16:27.000 Yeah, that's right. Metrics and field configurations, if there are matter fields and so on, yes. 00:16:27.000 --> 00:16:30.000 What's the signature? I think you're doing Euclidean quantum gravity? 00:16:30.000 --> 00:16:46.000 Yeah, this is… this is Euclidean here, yeah, yeah. So, I'm defining some set of states by a Euclidean ADS boundary, and then I sum over Euclidean things here. Now, strictly speaking, there's some issue of the sign of the conformal factor and so on, so there's some contour issue, which I don't want to get into, but uh… 00:16:46.000 --> 00:16:47.000 Yeah, up to that point. 00:16:47.000 --> 00:16:50.000 Right. Right? So I mean, that immediately raises all sorts of contour questions, as you just said. 00:16:50.000 --> 00:16:56.000 Yeah, yeah, so indeed, there are contour issues, and I think the right way to say it is that, um, 00:16:56.000 --> 00:16:58.000 I think you can… 00:16:58.000 --> 00:17:00.000 Yeah, yeah. 00:17:00.000 --> 00:17:08.000 Yeah, I mean, somehow, I mean, in the end, I want the inner product here to be the same. So this one on the left is Lorentzian, and so you don't have that issue. 00:17:08.000 --> 00:17:09.000 Hmm. 00:17:09.000 --> 00:17:14.000 Right? Somehow, I want to use… ask with the Hartle-Hawking state, I want to have a sort of Euclidean preparation of the Lorenzian state. 00:17:14.000 --> 00:17:15.000 Um, yeah. But I won't claim that I have a complete understanding of the contour in this. 00:17:15.000 --> 00:17:21.000 Mm-hmm. 00:17:21.000 --> 00:17:30.000 So, is there something that guarantees that this Scalar product satisfies the rules of a Scalar product? 00:17:30.000 --> 00:17:35.000 Um, yeah, yeah, because it coincides with canonical quantization. 00:17:35.000 --> 00:17:44.000 Yeah, if you ask that question again on the next slide, you'll be absolutely right to ask it, but in canonical quantization, the inner product is a good inner product. 00:17:44.000 --> 00:17:47.000 Well, actually, I mean, is there some notion that. 00:17:47.000 --> 00:17:52.000 What some components of the boundary are ingo and some are outgoing. 00:17:52.000 --> 00:17:59.000 Um, I was gonna talk about that later. Um, that has to do with whether you gauge CRT or not. 00:17:59.000 --> 00:18:00.000 Um… 00:18:00.000 --> 00:18:02.000 Okay. 00:18:02.000 --> 00:18:03.000 Uh-huh. 00:18:03.000 --> 00:18:10.000 I would say, yeah, I would say there's a CRT operation that… well, I think… let's ask that in the next… in the next slide, I think that's the right place to ask that question. 00:18:10.000 --> 00:18:13.000 Okay, well, in Bordism theory, you have to say. 00:18:13.000 --> 00:18:14.000 Yeah. 00:18:14.000 --> 00:18:21.000 you know, some some components are ingoing, and some components are outgoing, and that's true in both unoriented and oriented boardism. 00:18:21.000 --> 00:18:26.000 Yeah, so that's this issue of the bras and ketz that Edward and I were discussing, so, uh… 00:18:26.000 --> 00:18:30.000 So, I would say that there's a map where you can move 00:18:30.000 --> 00:18:38.000 final states… well, not actually not in this canonical context, but in the more general context that we'll do in a sec, where we sum over all topologies. 00:18:38.000 --> 00:18:47.000 You do have the ability to move final to initial at the cost of a CRT transformation of the source. 00:18:47.000 --> 00:18:55.000 Okay, well, a topological filter, and these pictures are starting to look like topological field theory. But one has to do is choose what's called a co-orientation. 00:18:55.000 --> 00:18:56.000 Um, yeah, yeah. 00:18:56.000 --> 00:19:03.000 even in unoriented Buddhism theory, you choose a co-orientation, which is an orientation of the normal bundle. Again. 00:19:03.000 --> 00:19:14.000 Yeah, so it… well, I mean, okay, I wasn't… I wasn't planning to make this a talk about gauging CRT, but if I did, I would say the following thing, which is that when you restrict to real wave functions, 00:19:14.000 --> 00:19:17.000 You no longer need the co-orientation. 00:19:17.000 --> 00:19:25.000 So, for CRT invariant states, you drop this extra label that, um, Edward talks about in his paper. 00:19:25.000 --> 00:19:28.000 Yeah. 00:19:28.000 --> 00:19:38.000 Okay, so anyway, so this is definition number one of the gravitational path integral, is you only sum over the cylinder topology, and that's the canonical approach, okay? 00:19:38.000 --> 00:19:48.000 You know, in some sense, this is the oldest approach, right? This is the approach of DeWitt and, you know, whoever from the 50s and 60s, Dirac, etc. 00:19:48.000 --> 00:19:52.000 So, the second approach, which I think was really most 00:19:52.000 --> 00:20:00.000 Clearly articulated by Coleman Giddings, and Strominger, although certainly it had antecedents in the string world sheet and so on. 00:20:00.000 --> 00:20:03.000 Um, is what I'll call baby universe field theory. 00:20:03.000 --> 00:20:07.000 Um, which is, you know, this third quantized approach. 00:20:07.000 --> 00:20:12.000 Um, yeah, and then so recently, Mel… 00:20:12.000 --> 00:20:13.000 Okay. 00:20:13.000 --> 00:20:17.000 Sorry, sorry, sorry, Daniel, I'm still thinking about what you, your previous slide. You could have taken disjoint unions of cylinders. 00:20:17.000 --> 00:20:29.000 Um, yeah, that's correct. So, so here I told you the rule, if there's one boundary. If there's more than one boundary, then you allow cylinders connecting the different copies, but they can only connect the identical sigmas. 00:20:29.000 --> 00:20:32.000 When you say one boundary, you mean one connected component of the bound. 00:20:32.000 --> 00:20:43.000 Um, right here, I mean one connected component. If there's more than one connected component, then you can have cylinders that connect, and then there's a rule that the universes are bosons. 00:20:43.000 --> 00:20:44.000 Yeah. 00:20:44.000 --> 00:20:51.000 Let's see the initial and final slices in your picture, at least, those are supposed to be closed manifolds. 00:20:51.000 --> 00:20:54.000 Yes, that's correct. So that's why I drew them that way. 00:20:54.000 --> 00:20:56.000 Yes, absolutely. 00:20:56.000 --> 00:20:57.000 Yeah, that's what I mean by closed universe. 00:20:57.000 --> 00:21:01.000 Compact metapholes without boundary. 00:21:01.000 --> 00:21:10.000 Okay, so… so let's now do the second approach to the path integral, which is the third quantized approach that I'm calling baby universe field theory. 00:21:10.000 --> 00:21:19.000 So here's what baby universe field theory looks like. So, sorry, I tried to draw sufficiently representative example so that you… I just have to explain it once. 00:21:19.000 --> 00:21:23.000 So, the idea is that now you allow arbitrary numbers, 00:21:23.000 --> 00:21:28.000 of these connected components that Greg was just asking about in the future, in the past. 00:21:28.000 --> 00:21:32.000 So here I've drawn an example where there's two in the future and one in the past. 00:21:32.000 --> 00:21:39.000 Okay? And then you sum over all the ways of connecting them, including both connected and disconnected contributions. 00:21:39.000 --> 00:21:49.000 So, so this works like Feynman diagrams, where there's a vacuum bubble piece of completely disconnected things that exponentiates and factors out, so that's here, the things with no boundaries. 00:21:49.000 --> 00:21:56.000 And then you sum over things where each component of the bulk spacetime is connected to at least one of the boundaries. 00:21:56.000 --> 00:21:59.000 So that's the usual, sort of, Feynman diagram type. 00:21:59.000 --> 00:22:02.000 combinatorics, um… 00:22:02.000 --> 00:22:04.000 And so then the idea is you, um… 00:22:04.000 --> 00:22:11.000 you use this path integral, which we're assuming is well-defined, including the sum over topology, 00:22:11.000 --> 00:22:16.000 Um, to define, I guess what I should call a sesquilinear form. 00:22:16.000 --> 00:22:25.000 on the linear span of, uh, states, you know, of these objects that are labeled by sources for some definite number. 00:22:25.000 --> 00:22:27.000 of boundary, uh, boundary components. 00:22:27.000 --> 00:22:34.000 And so this, uh, yeah, this vacuum bubble thing here, we refer to as 00:22:34.000 --> 00:22:38.000 following Meryl Finn Maxfield. Okay, so that's the definition. 00:22:38.000 --> 00:22:41.000 Um, so, um… 00:22:41.000 --> 00:22:46.000 Now, here's where Tom's question comes, which is, how do you know that this… 00:22:46.000 --> 00:22:53.000 Sesquilinear form is actually an inner product, and in fact, in general, it's not, because, um, 00:22:53.000 --> 00:22:58.000 It can have zero modes, so there's two comments to make about that, so what… 00:22:58.000 --> 00:23:04.000 So, so that issue was ignored in the old literature. What Marilyn Maxfield, the whole paper, is about that issue. 00:23:04.000 --> 00:23:20.000 So the way they phrase it is that you just have to view it as an assumption that this is positive semi-definite. If it's positive definite, even better. If it's positive semi-definite, then you're supposed to, uh, quotient the linear span of these J states by, um, by the set of null states. 00:23:20.000 --> 00:23:30.000 to define a Hilbert, an actual Hilbert space, uh, that lives on the… whose elements are these equivalence classes? 00:23:30.000 --> 00:23:35.000 So that's the construction of the baby universe field theory Hilbert space. 00:23:35.000 --> 00:23:42.000 So, Daniel, if you're doing anything real as opposed to… 00:23:42.000 --> 00:23:47.000 Well, even for Maroff and Maxfield, I… I forget… 00:23:47.000 --> 00:23:51.000 Do they integrate over the moduli? 00:23:51.000 --> 00:23:57.000 Yeah, well, I mean, in principle, yes. I mean, in the models where they actually did calculations, no? 00:23:57.000 --> 00:24:00.000 Yeah, so… 00:24:00.000 --> 00:24:01.000 Okay. 00:24:01.000 --> 00:24:03.000 They just, uh, had a topological model. I think… but I think… I think we could probably… 00:24:03.000 --> 00:24:08.000 In some sense, put Sodschenk or Stanford in this category, and they summed over the moduli. 00:24:08.000 --> 00:24:15.000 Yeah, so if you're… if you're summing over the moduli, this sum is not Burrell summable. 00:24:15.000 --> 00:24:20.000 So, there's… there's another mathematical question. 00:24:20.000 --> 00:24:24.000 of, you know, what this scalar product means. 00:24:24.000 --> 00:24:25.000 Yeah, yeah, so, I mean, that's right, so our… 00:24:25.000 --> 00:24:27.000 after you… I mean… 00:24:27.000 --> 00:24:33.000 Yeah, so already in that case, there's a convergence issue with the sum, right? So, in the Merrilliff-Maxfield model, the sum is convergent. 00:24:33.000 --> 00:24:34.000 Right. 00:24:34.000 --> 00:24:42.000 And so there's at least one case where we can actually do this algorithm I'm telling you. I mean, but don't worry, I'm gonna complain about this algorithm later, okay? I promise. 00:24:42.000 --> 00:24:43.000 Okay. 00:24:43.000 --> 00:24:53.000 But I first want to present it, because, like, half the conversations I hear about assume this algorithm makes sense, and so we… I want to present it before… before we… I complain about it. 00:24:53.000 --> 00:24:56.000 Um… 00:24:56.000 --> 00:25:01.000 Okay, so in Baby Universe field theory, um, there's a particular state that is… 00:25:01.000 --> 00:25:07.000 nice that we like to think about, which is the Harle-Hawking state, which is the state where there's no past boundaries. 00:25:07.000 --> 00:25:10.000 So here's… here is the sum… 00:25:10.000 --> 00:25:14.000 Contributions to the two-boundary Hartle-Hawking estate. 00:25:14.000 --> 00:25:18.000 Okay, and then there's more, right? You can put a handle, etc., you know. 00:25:18.000 --> 00:25:22.000 So, the norm of this state is the sum over vacuum bubbles, this allott. 00:25:22.000 --> 00:25:29.000 Which I'll remind you was the exponential of the sum over connected, uh, spacetime, uh, closed connected space-times. 00:25:29.000 --> 00:25:33.000 really compact connected space-times. 00:25:33.000 --> 00:25:43.000 So, okay, so fine. So far, we didn't really do much. Okay, so now, so now Greg, pay attention. Here's where we're gonna assume that everything is well-defined. 00:25:43.000 --> 00:25:47.000 So here's the key step of Merrillfin Maxfeld. 00:25:47.000 --> 00:25:50.000 is we introduce an operator Z-hat, 00:25:50.000 --> 00:25:56.000 that acts on this pre-Hilbert space by creating an extra universe with a 00:25:56.000 --> 00:25:58.000 source configuration J. 00:25:58.000 --> 00:26:01.000 Okay? And, um… 00:26:01.000 --> 00:26:10.000 So, you know, it acts on the state, you know, J1 through JN, and then it gives you J, J1 through JN. I should have written that equation, sorry, I forgot to write that equation. 00:26:10.000 --> 00:26:13.000 So, if you think about it, you can… 00:26:13.000 --> 00:26:17.000 quickly convince yourself that these C hat operators all commute with each other. 00:26:17.000 --> 00:26:23.000 Essentially, that's because we treat the closed universe as bosons, so when you create a J, it doesn't matter which slot 00:26:23.000 --> 00:26:29.000 You created in, and so you can act with these Zhats on the Harle-Hawking state in whatever order you want, and you get the same 00:26:29.000 --> 00:26:33.000 you know, state with the product of J's afterwards. 00:26:33.000 --> 00:26:39.000 Oh, by the way, sir, I wanted to explain this notation. So you see I'm doing square brackets for the baby universe field theory inner product? 00:26:39.000 --> 00:27:01.000 That's to be distinguished from angle brackets, which I did for the canonical gravity inner product. So they're not the same, because, for example, here, if I put the square brackets, there would be a disconnected contribution also. There would be other topologies, not just the cylinder, okay? So with the angle, it's the canonical inner product with the square bracket, it's the third quantized bit of a universe field theory in a product. 00:27:01.000 --> 00:27:09.000 I strongly advocate that everyone who works in this business adopts this notation, because otherwise people get horribly confused. 00:27:09.000 --> 00:27:11.000 Okay, uh… 00:27:11.000 --> 00:27:12.000 Back to where we were, oh yeah? 00:27:12.000 --> 00:27:19.000 Oh. Yeah. Well, as you know, I wrote a paper with Benergy. 00:27:19.000 --> 00:27:26.000 because I found the mayor of Maxfield paper. to my mind, extremely confused. 00:27:26.000 --> 00:27:28.000 Mm-hmm. 00:27:28.000 --> 00:27:39.000 where we. in that language we coupled a topological an arbitrary topological field theory to. 00:27:39.000 --> 00:27:40.000 Yeah. 00:27:40.000 --> 00:27:46.000 topological gravity by topological gravity. You just. Wait, wait topologies by some. 00:27:46.000 --> 00:27:47.000 Yeah. 00:27:47.000 --> 00:27:57.000 some factor to the Euler character. And you know we defined a harder locking state and a dual harder locking state. 00:27:57.000 --> 00:28:07.000 You don't need to assume any kind of inner product, and the inner product is not Aleph. 00:28:07.000 --> 00:28:08.000 Is it what you don't like this question? 00:28:08.000 --> 00:28:11.000 the the the I mean, that really surprised. Yeah, that equation right there. That really, really, really, really surprised me. 00:28:11.000 --> 00:28:15.000 But I think all our formulas are well defined. 00:28:15.000 --> 00:28:16.000 Well, I think… I think this follows. 00:28:16.000 --> 00:28:18.000 And that. Not really true. 00:28:18.000 --> 00:28:20.000 I mean, I think this follows from the definitions, right? I defined the… 00:28:20.000 --> 00:28:37.000 It… well, yeah, yeah, yeah, yeah, if you write the pictures, it looks like it's obvious, but it's if you… if you follow our rules, and we were. 00:28:37.000 --> 00:28:38.000 Yeah. 00:28:38.000 --> 00:28:40.000 Our rules were just intended to make precise mathematical sense of what Merolf and Mackfield were trying to say. And it just wasn't true. 00:28:40.000 --> 00:28:47.000 Well, I'm not sure what to… I mean, I'm hesitant to try to… because I don't remember the details of what you did, so I'm hesitant to try to get into the details of… 00:28:47.000 --> 00:28:50.000 sorting out the difference between the two proposals. 00:28:50.000 --> 00:28:53.000 Well, we could talk about it. But. You know, I think. 00:28:53.000 --> 00:29:00.000 I mean, I think the way I define this here, this is true by definition, right? Because I just defined it to be this, 00:29:00.000 --> 00:29:05.000 And so if I don't put any Js, then this first factor is not there, and then I have the second factor. So what more do you want? 00:29:05.000 --> 00:29:08.000 Well, look, if you write, if you write precise formulas. 00:29:08.000 --> 00:29:17.000 Using the formalism of topological field theory and summing over the topologies, then you can write precise formulas for the Hartle-Hawking state. 00:29:17.000 --> 00:29:18.000 But I think it… the only thing I assumed here is that… so when I said I assumed the path integral is well-defined, 00:29:18.000 --> 00:29:23.000 And just not. 00:29:23.000 --> 00:29:28.000 Uh, the weak version of that is that I meant that the terms in this sum 00:29:28.000 --> 00:29:34.000 are in these two sums are well-defined, right? And then, of course, the strong version is that the sums converge. 00:29:34.000 --> 00:29:35.000 Um, 00:29:35.000 --> 00:29:47.000 Well, the sums do converge for general. General, semi-simple general semi. Well. 00:29:47.000 --> 00:29:48.000 Yeah, but then since I defined this… 00:29:48.000 --> 00:29:50.000 For generic topological field theories. Where the string coverage is. 00:29:50.000 --> 00:29:51.000 But I defined this using this picture, right? So it's just a fact about this picture. 00:29:51.000 --> 00:29:57.000 Anyway. Look, what do you mean? Daniel, what do you mean by the Jays? 00:29:57.000 --> 00:30:02.000 Uh, yeah, so for me, the Jays are sources at these Euclidean ADS boundaries. 00:30:02.000 --> 00:30:06.000 How are you? 00:30:06.000 --> 00:30:16.000 I think it's the… the Euclidean is not… in a topological context, it doesn't really matter that it's ADS, though, right? We just need some boundaries. 00:30:16.000 --> 00:30:17.000 Well, you need sources on the boundary? 00:30:17.000 --> 00:30:18.000 Well, you need a little more than that. Any Frobenius algebra associated with a circle. 00:30:18.000 --> 00:30:27.000 You need boundary… you need boundary conditions for the fields, right? So yeah, if you like, these are boundary conditions for the fields that you sum over. That's what these are. 00:30:27.000 --> 00:30:32.000 I'm labeling a Hilbert space that's spanned by, uh… 00:30:32.000 --> 00:30:34.000 lists of boundary conditions. 00:30:34.000 --> 00:30:42.000 for the bulk fields, modulo null states, and I had to assume to make a Hilbert space, that it was positive, that this 00:30:42.000 --> 00:30:48.000 Sesquilinear form was positive semi-definite. 00:30:48.000 --> 00:30:49.000 Yeah, I mean, I… yeah, why don't we go in for now? I mean, I'm telling you what the assumptions are. So, you know, we can certainly… 00:30:49.000 --> 00:30:53.000 Okay. 00:30:53.000 --> 00:30:57.000 discuss whether the assumptions are true. 00:30:57.000 --> 00:30:58.000 Yeah. 00:30:58.000 --> 00:31:04.000 Okay, I insist that the formulas that I wrote with Banerjee are mathematically precisely well-defined, and are the closest. 00:31:04.000 --> 00:31:11.000 thing I could make use I could find that could make some kind of sense of what Marolf and Maxfield were saying. 00:31:11.000 --> 00:31:16.000 Well, I think in their topological model, this all works, right? Like, it's, you know, it's a pretty boring model, but… 00:31:16.000 --> 00:31:22.000 There was so much even in their topological model, there was a great deal of. 00:31:22.000 --> 00:31:25.000 There was a lot of confused statements. I'm going to be blunt. 00:31:25.000 --> 00:31:26.000 I don't disagree, I mean… 00:31:26.000 --> 00:31:28.000 There were a lot of really deeply confused segments. 00:31:28.000 --> 00:31:31.000 I mean, Greg, there's a reason I'm reviewing this myself. 00:31:31.000 --> 00:31:43.000 is because I also found some of the presentation confusing, but I think the logic for the version I'm telling you, I think, is fairly clear. 00:31:43.000 --> 00:31:48.000 Yeah, so, okay, so let me go on. So now you define an operator Z hat, 00:31:48.000 --> 00:31:52.000 that acts on this Hilbert space? 00:31:52.000 --> 00:31:58.000 are well related to the pre-Hilbert space, but then it survives the quotient, because it's by definition orthogonal to null states. 00:31:58.000 --> 00:32:03.000 Um, uh, well, no, what you have to show is it maps null states to null states. 00:32:03.000 --> 00:32:05.000 Um, 00:32:05.000 --> 00:32:11.000 So this operator, uh, what it does is it creates an additional universe with this boundary condition J. 00:32:11.000 --> 00:32:18.000 Um, and so these operators commute with each other because, um, because closed universes are bosons. 00:32:18.000 --> 00:32:23.000 And so now here's where Greg's other question about CRT comes in, okay? 00:32:23.000 --> 00:32:28.000 is… I assume that for every future boundary J, 00:32:28.000 --> 00:32:32.000 there's a past boundary, J star, which gives the same path integral. 00:32:32.000 --> 00:32:39.000 So I can move a J on one side to a J on the other side at the cost of a star, uh, without changing the value of the path integral. 00:32:39.000 --> 00:32:45.000 And so that's the… that's the CRT operation. 00:32:45.000 --> 00:32:51.000 So, if that's true, which I assume it is, then you further show that the dagger of a Z 00:32:51.000 --> 00:32:56.000 is also a Z, which means that the Zs are normal operators. 00:32:56.000 --> 00:32:58.000 They commute with their own daggers. 00:32:58.000 --> 00:33:06.000 So they can be diagonalized, and in fact, since they're all mutually commuting normal operators, they can be simultaneously diagonalized. 00:33:06.000 --> 00:33:11.000 Um, leading to a set of eigenstates, alpha, these are the alpha states, 00:33:11.000 --> 00:33:17.000 that are eigenstates of all the z-hat operators at once, with eigenvalues Z alpha of j. 00:33:17.000 --> 00:33:32.000 Okay? So… so this step right here, Greg, I want to highlight, is the one where I really had to make a fairly strong assumption about how well-defined this formalism is, because this statement is some kind of application of the spectral theorem. 00:33:32.000 --> 00:33:38.000 Um, and, you know, you don't get to use the spectral theorem if everything has giant error bars on it. 00:33:38.000 --> 00:33:41.000 Well, you have a community of C star algebra. 00:33:41.000 --> 00:33:47.000 Well, you didn't show a C star, actually. You have a commutative algebra with a star. 00:33:47.000 --> 00:33:50.000 Uh, yeah, yeah. Um, it would… 00:33:50.000 --> 00:33:52.000 That's not it. That's not quite a sea star algebra. 00:33:52.000 --> 00:33:57.000 Yeah, there's something about the norm or something, you mean? 00:33:57.000 --> 00:33:59.000 Yeah. 00:33:59.000 --> 00:34:03.000 Yeah. 00:34:03.000 --> 00:34:06.000 Mm-hmm. 00:34:06.000 --> 00:34:07.000 you know. 00:34:07.000 --> 00:34:10.000 You need a norm square. You need a norm. You need a norm which, moreover satisfies the norm of a star is the norm of a squared, and you need completeness in that norm. So you you're really and the hardest thing to show. 00:34:10.000 --> 00:34:13.000 is that the norm of AA star is the norm of A squared. 00:34:13.000 --> 00:34:17.000 Uh, yeah, yeah. Yeah, but I mean, in higher dimensions, like… 00:34:17.000 --> 00:34:26.000 we're in trouble on a much more basic level, right, because we just can't even define this Gescal linear form to get started, right? It's, um… 00:34:26.000 --> 00:34:33.000 So, you know, so that… when I say it's well-defined, I mean, I even worrying about this sort of technical C-star issues. It's just we can't even get off the ground. 00:34:33.000 --> 00:34:36.000 if the path integral is not well-defined. 00:34:36.000 --> 00:34:41.000 And, you know, and this is not like, you know, 00:34:41.000 --> 00:34:53.000 you know, if you make small mistakes, right, like this argument that the Zs all commute and you can simultaneously diagonalize them is not feel like an argument that's stable under small errors. You know, it's something where it really needs to be true. 00:34:53.000 --> 00:34:56.000 Um, for this conclusion to follow up. 00:34:56.000 --> 00:35:04.000 Um, but for now, following the majority view in the field, I'll assume that it is true for a little bit, and then I'll complain about it later in the talk. 00:35:04.000 --> 00:35:09.000 Well, I've already done some of the complaining now, but I'll do more of it later in the talk. 00:35:09.000 --> 00:35:16.000 Okay. So, why is, why are these alpha states so interesting? Why are certain… 00:35:16.000 --> 00:35:22.000 people jumping up and down, excited about them, you know, going back to Coleman and continuing 00:35:22.000 --> 00:35:24.000 To the present. 00:35:24.000 --> 00:35:29.000 It's because by inserting a complete set of these alpha states into one of these 00:35:29.000 --> 00:35:31.000 Um, overlaps. 00:35:31.000 --> 00:35:36.000 We can give it an average interpretation, right? So we take these… 00:35:36.000 --> 00:35:41.000 this J state, and we view it as a product of a bunch of Z hats acting on the Hartel-Hawking state. 00:35:41.000 --> 00:35:47.000 We take the J prime bra, and we view that as a bunch of Z daggers acting to the left on the Hartle-Hawking state. 00:35:47.000 --> 00:35:58.000 And then we insert one complete set of alpha states, and then since all the Z hats are eigenstates of the alpha, you only need to insert one. You don't need to insert many. So you just have one sum, 00:35:58.000 --> 00:36:01.000 Over alpha, and then a product. 00:36:01.000 --> 00:36:08.000 of these functions, these eigenvalues, that just depend on the boundaries separately. 00:36:08.000 --> 00:36:11.000 Each boundary gets its own factor. 00:36:11.000 --> 00:36:19.000 And then this probability distribution, P alpha, that you average over comes from the overlap between the Harle-Hawkins state and the alpha states. 00:36:19.000 --> 00:36:25.000 When you say they're complete, could they be overcomplete or are they orthonormal, the alpha states? 00:36:25.000 --> 00:36:31.000 Um, I was assuming orthonormal, because I'm really using the spectral theorem. 00:36:31.000 --> 00:36:37.000 Yeah. 00:36:37.000 --> 00:36:38.000 Uh, unclear, right? 00:36:38.000 --> 00:36:41.000 I say. Well. What is what is alpha running over? I mean, you're sort of assuming that's a discrete spectrum. If it's a. 00:36:41.000 --> 00:36:45.000 No, I'm actually… no, well, I wrote a sum, but it could be an integral, actually. I don't want to commit to that. 00:36:45.000 --> 00:36:48.000 Okay. And then there would be a density of states. 00:36:48.000 --> 00:36:50.000 That's correct, then it could be an integral, that's right, yeah. 00:36:50.000 --> 00:36:53.000 Yeah, I don't want to commit on that one. 00:36:53.000 --> 00:36:54.000 Actually, I'll comment a bit more on that later. 00:36:54.000 --> 00:36:58.000 Okay. 00:36:58.000 --> 00:37:12.000 So, um, now this equation actually is one of the reasons why there's been a lot of confusion about this subject, right, is because it suggests that there's some kind of averaging interpretation to the inner product in baby universe field theory. 00:37:12.000 --> 00:37:18.000 Um, but I really want to emphasize that this is not an average over theories. This is a conventional, 00:37:18.000 --> 00:37:26.000 quantum theory. It has a Hilbert space that's spanned by the alpha states. You can have superpositions of states with different alpha. 00:37:26.000 --> 00:37:31.000 And, uh, you have operations that take you from one alpha to the other, acting on this… 00:37:31.000 --> 00:37:33.000 Hilbert Space, um… 00:37:33.000 --> 00:37:39.000 And in particular, if you multiply two transition amplitudes in baby universe field theory, 00:37:39.000 --> 00:37:43.000 Um, you don't include… 00:37:43.000 --> 00:37:45.000 Topologies that connect them. 00:37:45.000 --> 00:37:52.000 So, like, say here I have two one-universe overlaps, J1 prime to J1 and J2 prime to J2. 00:37:52.000 --> 00:37:55.000 Um, and I, uh, I multiply them. 00:37:55.000 --> 00:38:00.000 Um, alright, this should have been Aleph-2 minus 2 here, but anyway, it doesn't. Sorry about that. 00:38:00.000 --> 00:38:07.000 Um, so this is, uh, you know, this is… this one is a product of things like… sum over things like this. 00:38:07.000 --> 00:38:10.000 And this one is a sum over things like that. 00:38:10.000 --> 00:38:16.000 But then you don't include, you know, geometries that connect to J1 prime here to J2 over there. 00:38:16.000 --> 00:38:19.000 or J2 prime here to J1. 00:38:19.000 --> 00:38:26.000 Over there. Because, you know, these overlaps are just numbers, and you're multiplying them, right? So this is a factorized 00:38:26.000 --> 00:38:29.000 quantity. 00:38:29.000 --> 00:38:37.000 So, um… yeah, and in particular, it's not equal to the transition amplitude between two universe states, right? So, like, we could have also computed 00:38:37.000 --> 00:38:49.000 The overlap from J12 to J1'2 prime, and that certainly would have included geometries that connect between 1 and 2, but this product of 1 to 1 and 2 to 2 does not, uh, does not include that. 00:38:49.000 --> 00:38:52.000 Okay. 00:38:52.000 --> 00:39:00.000 People fight with me about this, but I don't know why, because it's just obvious. This is the definition of this quantity. 00:39:00.000 --> 00:39:01.000 So, Daniel? 00:39:01.000 --> 00:39:02.000 Yeah. 00:39:02.000 --> 00:39:12.000 In the original Coleman paper, and the ones that followed it, and also in the Saudshanker-Stanford paper, 00:39:12.000 --> 00:39:13.000 Yeah. 00:39:13.000 --> 00:39:20.000 Um, it's… it's kind of important that the average that you're doing is an average over… 00:39:20.000 --> 00:39:25.000 Um, local couplings in a conventional 00:39:25.000 --> 00:39:29.000 field theoretic treatment of a single universe. 00:39:29.000 --> 00:39:34.000 And I don't see anything like that in this description here. 00:39:34.000 --> 00:39:38.000 Um… 00:39:38.000 --> 00:39:44.000 Well, yeah, so in com… because common was… in common's thing, you're not in the closed universe. 00:39:44.000 --> 00:39:50.000 In common Sting, you're living in an asymptotic region, and you're summing over microscopic wormholes? 00:39:50.000 --> 00:39:58.000 So, does alphas have to do with what's going on in the baby universe sector, but you're not living in the baby universe sector, so for you, it's just some coupling constant. 00:39:58.000 --> 00:40:06.000 But that's not… here, I'm imagining that we're in the baby universe, and so then the things we measure depend on alpha. Like, it's not a… 00:40:06.000 --> 00:40:11.000 Yeah, it's a bit different than the context that Coleman was thinking about. 00:40:11.000 --> 00:40:16.000 Well, I certainly understand that Coleman made an extra assumption. 00:40:16.000 --> 00:40:21.000 about the wormholes being microscopically small, that's not being made here. 00:40:21.000 --> 00:40:22.000 Yeah. 00:40:22.000 --> 00:40:28.000 But, um, it… it also seems to me that you're… 00:40:28.000 --> 00:40:31.000 Um, what you're getting here is… is… 00:40:31.000 --> 00:40:39.000 quite different than has… than occurred in either of those two examples. 00:40:39.000 --> 00:40:46.000 Well, so Thad Chenker and Stanford, you're about to see what that is. So I told you I have 3 interpretations of the path integral, and we're currently on the second one, okay? 00:40:46.000 --> 00:40:47.000 Okay. 00:40:47.000 --> 00:40:49.000 So that one's gonna be the third one. 00:40:49.000 --> 00:40:50.000 Okay, good. 00:40:50.000 --> 00:40:52.000 So just hold off on that for a sec. 00:40:52.000 --> 00:41:08.000 Yeah. Yeah, let's go to that now. I don't want to get too far behind here. So… so my… so the third approach to the path integral is what I'll call average holography. So I want to be clear, it's the same path integral either way? It's the quantum mechanical interpretation that we assign to it that's going to be 00:41:08.000 --> 00:41:11.000 be different between this and the previous one. 00:41:11.000 --> 00:41:13.000 Okay? Um… 00:41:13.000 --> 00:41:18.000 So this average here, we said that all… we said that we compute an overlap 00:41:18.000 --> 00:41:25.000 You know, each overlap has an average interpretation, but we're not averaging over overlaps, okay, right? Like, if we square 00:41:25.000 --> 00:41:31.000 this two overlaps is just the product, okay? There's no average over the products, right? 00:41:31.000 --> 00:41:33.000 So, in this third approach, 00:41:33.000 --> 00:41:40.000 We decide to switch, instead of viewing the Hilbert space spanned by alpha states as fundamental, 00:41:40.000 --> 00:41:43.000 We say that we view alpha as 00:41:43.000 --> 00:41:46.000 a parameter that labels… 00:41:46.000 --> 00:41:48.000 fundamental theories. 00:41:48.000 --> 00:41:51.000 And, um, so we… 00:41:51.000 --> 00:41:59.000 we define some states, so now the double angle bracket are these states that live in the holographic dual at fixed alpha. 00:41:59.000 --> 00:42:11.000 Um, and we just defined the states to be these functions, Z alpha of J1 to Z alpha of jn. And so this is indeed a one-dimensional Hilbert space, because the state is just a number, right? It's a product of these… 00:42:11.000 --> 00:42:16.000 numbers. So each state with some number of j's, it just gets mapped to this number. 00:42:16.000 --> 00:42:18.000 Okay? Um… 00:42:18.000 --> 00:42:19.000 And so then, in this language, this… 00:42:19.000 --> 00:42:26.000 would have made it a very large dimensional space, but I think, saying its functions of J's. 00:42:26.000 --> 00:42:28.000 Um, well… 00:42:28.000 --> 00:42:29.000 Yeah, it's a bunch… it's a bunch of states. I mean, we… 00:42:29.000 --> 00:42:35.000 I mean, you know, any. Any, any Hilbert space could be written as a sum over one-dimensional Hilbert spaces, right? 00:42:35.000 --> 00:42:43.000 No, no, but this is not a sum, right? This is saying that every state… so this is a basis if… you could try to say it's a basis, it's a very over-complete basis, right? 00:42:43.000 --> 00:42:46.000 So, I diff… I… I take… 00:42:46.000 --> 00:43:02.000 for any set of the values of the Js, right, that defines a number, but it always defines a number. And so… so… so that number… the numbers all live in the same one-dimensional Hilbert space called the complex numbers. 00:43:02.000 --> 00:43:08.000 Well, I mean, if I if I consider the set of complex valued functions on a manifold. 00:43:08.000 --> 00:43:10.000 I could say that, you know, for each point I just number. 00:43:10.000 --> 00:43:16.000 No, no, but then you allow a superposition, then you allow super… right, I mean, 00:43:16.000 --> 00:43:20.000 The thing is, Jay is not, like, uh… 00:43:20.000 --> 00:43:24.000 Yeah, these are not wave functions, right? These are the states themselves. I didn't compute the overlap with anything. 00:43:24.000 --> 00:43:25.000 Right? 00:43:25.000 --> 00:43:31.000 Our Jays were like boundary values of fields. Aren't they points in some infinite dimensional manifold? 00:43:31.000 --> 00:43:34.000 Um, yeah, yeah, but I'm just, uh… 00:43:34.000 --> 00:43:39.000 See, if I put a psi here, overlapped with J, then what you're saying is correct? 00:43:39.000 --> 00:43:44.000 But I'm not doing that. I'm saying the state itself is a number. 00:43:44.000 --> 00:43:50.000 Not the wave function. The wave function is always the number, but the state itself is a number. 00:43:50.000 --> 00:43:52.000 Okay? 00:43:52.000 --> 00:43:58.000 Um, and then if you look at this, so this averaging equation that I wrote here, 00:43:58.000 --> 00:44:04.000 In this language, what we say is that this path-integral overlap with the square brackets 00:44:04.000 --> 00:44:11.000 is equal to an average over an overlap in a one-dimensional Hilbert space with the double angle brackets. 00:44:11.000 --> 00:44:16.000 Okay, this is the key equation relating average holography, and uh… 00:44:16.000 --> 00:44:18.000 And the Baby Universe Field Theory. 00:44:18.000 --> 00:44:25.000 Okay? Um, oh, sorry, one time my phone is ringing, let me see what that is. 00:44:25.000 --> 00:44:28.000 Um, so, um… 00:44:28.000 --> 00:44:32.000 So this is a key equation, right? We say that instead of interpreting this… 00:44:32.000 --> 00:44:37.000 thing as a sesquilinear form on a Hilbert space of large dimension. 00:44:37.000 --> 00:44:42.000 We say it's an average over inner product in a one-dimensional Hilbert space. 00:44:42.000 --> 00:44:45.000 Okay, that's what this equation says. That's this average tallography. 00:44:45.000 --> 00:44:47.000 The average is the average over alpha. 00:44:47.000 --> 00:44:57.000 Yes, the average over alpha. This sum alpha, P-alpha here, yeah. So, it's the same equation, it's this equation, I'm just saying now, without the sum, if I look at these things by themselves… 00:44:57.000 --> 00:45:10.000 I interpret them as an overlap in this one-dimensional Hilbert space, where the overlap is really just you multiply the numbers, right? Because in a one-dimensional Hilbert space, that's the only inner product you got. You multiply the numbers. 00:45:10.000 --> 00:45:12.000 With a complex conjugate, of course, right? That's what this is. 00:45:12.000 --> 00:45:15.000 Yeah. Um… 00:45:15.000 --> 00:45:30.000 So, so you see the difference… the difference with Baby Universe Field Theory comes if you consider products of overlaps, right? So if you consider, uh, some quantity that requires two overlaps in the fundamental Hilbert space, right, like something like a product of 2, 1 universe… 00:45:30.000 --> 00:45:33.000 overlaps, okay, and then you put a bar. 00:45:33.000 --> 00:45:40.000 then that's in the… from the path interval point of view, that's a transition amplitude between two universe states. 00:45:40.000 --> 00:45:47.000 It's not the… which, as we just discussed, is not the same as the square of overlaps between one universe states, okay? 00:45:47.000 --> 00:45:51.000 And that's the main thing that is confusing lots of people in this business. 00:45:51.000 --> 00:45:57.000 Um, is you have to decide which interpretation you're using. Are we talking about overlaps with the square brackets? 00:45:57.000 --> 00:46:03.000 Or are we talking about averages of overlaps with the angle… with the double angle brackets? Okay. You have to pick. 00:46:03.000 --> 00:46:14.000 And, you know, you can… I mean, at this stage, you can pick whichever you want, but you have to pick, and you know, you can't change your mind halfway through the calculation which one you're doing. 00:46:14.000 --> 00:46:17.000 So, um… 00:46:17.000 --> 00:46:24.000 So let me just emphasize kind of how bizarre this double angle bracket one-dimensional Hilbert space is, right? So, um… 00:46:24.000 --> 00:46:26.000 So we… I showed you that factorization 00:46:26.000 --> 00:46:30.000 you know, the inner product doesn't factorize in baby universe field theory. 00:46:30.000 --> 00:46:33.000 Um, but it does… it does… 00:46:33.000 --> 00:46:40.000 factorize in a fixed alpha in the sort of… in the sort of average holography sense, where 00:46:40.000 --> 00:46:43.000 Because since, you know, if you have two of these universes, right, 00:46:43.000 --> 00:46:54.000 Um, well, it's just very clear here. The state's factorized, right? So the, you know, if you remove the bar, then this thing is equal to that thing, and so then, since it's true without the bar, it's also true with the bar. 00:46:54.000 --> 00:46:56.000 Um, and so from this point of view, 00:46:56.000 --> 00:47:03.000 Uh, the point of view of average holography, the non-factorization of baby universe field theory, you interpret it as coming from ensemble 00:47:03.000 --> 00:47:13.000 fluctuations of the one universe inner product, right? The fact that in baby universe field theory, these two things were not equal, right? The 2-2 transition amplitude and the product of the… 00:47:13.000 --> 00:47:16.000 One-to-one transition amplitude. 00:47:16.000 --> 00:47:21.000 So, you know, here that's just, you know, we interpret it something like we do in string theory, as I'll say in a minute. 00:47:21.000 --> 00:47:27.000 But here, you really say it arises from fluctuations in the alpha ensemble, right? That these 00:47:27.000 --> 00:47:30.000 this single universe overlap, 00:47:30.000 --> 00:47:34.000 you know, has substantial fluctuations as a function of alpha, okay. 00:47:34.000 --> 00:47:41.000 So they're not quantum fluctuations, they're like ensemble fluctuations in the space of theories that you're thinking about. 00:47:41.000 --> 00:47:54.000 And, you know, and this leads to some pretty crazy things. So, like, here's a fun thing that it leads to, right? Like, say in the fundamental Hilbert space, you know, this one-dimensional fundamental Hilbert space, you want to compute the overlap from J1 to J2. 00:47:54.000 --> 00:47:57.000 Uh, and so… 00:47:57.000 --> 00:48:02.000 And you… and you want it to be a transition amplitude, so you compare it to the norms of the states, 00:48:02.000 --> 00:48:06.000 But actually, if you sort of compute on average, 00:48:06.000 --> 00:48:17.000 how much this differs from just the product of the norms, it doesn't differ at all because of factorization. I can just sh… you know, everything is a product of function of J1 and function of J2, so I can just… 00:48:17.000 --> 00:48:28.000 shuffle things around. Uh, and so what this is showing is between any two states in the one-dimensional Hubbert space, the overlap is one. Of course, that's what you expect, it's a one-dimensional Hilbert space, but this is a way that the path integral is sort of 00:48:28.000 --> 00:48:37.000 Quite clearly telling you that if you want to interpret it as an average over holographic theories, then it's going to be an average over holographic theories with a one-dimensional Hilbert space. 00:48:37.000 --> 00:48:42.000 Okay, and so here's where the one-dimensional Hilbert space is, you know, from the context of the path integral. 00:48:42.000 --> 00:48:48.000 Okay? 00:48:48.000 --> 00:48:56.000 Alright, so that's it for the three ways of interpreting the path integral, right? So let's recap. There's the canonical Hilbert space, that was the angle bracket. 00:48:56.000 --> 00:49:01.000 Okay, there's the baby universe field space, Hilbert space. That's the square bracket. 00:49:01.000 --> 00:49:08.000 And then there's the… the fundamental Hilbert space of average holography, which is one-dimensional and has the double angle bracket. 00:49:08.000 --> 00:49:15.000 And I can give the path integral a quantum interpretation using any of those approaches, but 00:49:15.000 --> 00:49:19.000 Depending which one I want to use, the topologies that I sum over might be a bit different. 00:49:19.000 --> 00:49:22.000 Sound good? 00:49:22.000 --> 00:49:23.000 Yeah. 00:49:23.000 --> 00:49:31.000 Let me try and understand this. So you're saying that there's no physical interpretation to linear combinations of alpha states with different alpha? 00:49:31.000 --> 00:49:32.000 Um, in this approach, yeah, that's right. 00:49:32.000 --> 00:49:35.000 That's right. You just say there are different theories. Exactly. Exactly right. 00:49:35.000 --> 00:49:43.000 I mean, you know, given… you know, the spectral theorem, as as you said, you know. 00:49:43.000 --> 00:49:50.000 automatically, you know, decomposes a Hilbert space. as a sum over the eigenspaces. 00:49:50.000 --> 00:49:53.000 Yes. 00:49:53.000 --> 00:49:54.000 Yeah. 00:49:54.000 --> 00:50:03.000 associated with the spectrum. Now, for some operators, the degeneracy is one, for some it's not. But you have a commutative algebra that you're saying is simultaneously diagonalizable. 00:50:03.000 --> 00:50:04.000 Mm-hmm. 00:50:04.000 --> 00:50:07.000 with the generous degeneracy spaces which are one dimensional. 00:50:07.000 --> 00:50:08.000 Yes. 00:50:08.000 --> 00:50:11.000 I think I missed why it's one dimensional. Actually, for each alpha. There's only one. 00:50:11.000 --> 00:50:22.000 Oh, yeah, I can tell you why that is. Yeah, that's because, um, if I give you the eigenvalues… oh, yeah, sorry, I didn't write the equation, I should have written it, but if I give you the eigenvalues, I can actually also give you the wave function. 00:50:22.000 --> 00:50:28.000 So, think of it this way. You can compute the overlap of alpha with any 00:50:28.000 --> 00:50:32.000 product of the Js, just in terms of the eigenvalues. 00:50:32.000 --> 00:50:36.000 So, if you know the eigenvalues, then you know the eigenstate. 00:50:36.000 --> 00:50:37.000 So there's… so there's no degeneracy. 00:50:37.000 --> 00:50:38.000 I see. Okay. So there's no degeneracy. Okay. 00:50:38.000 --> 00:50:41.000 Yeah. Yeah, yeah, that's important, yeah. 00:50:41.000 --> 00:50:56.000 So… so, you know, I mean, but this is… this is something that's pretty standard, right? I mean, if I can have a… it's just the spectral theorem, as you said. And usually we don't conclude that, um. 00:50:56.000 --> 00:50:57.000 No, no, and you don't… you don't have to, right? I mean, that's one of the less… 00:50:57.000 --> 00:51:04.000 Our Hilbert spaces are one-dimensional. We can… because we can take physical linear combinations of states with different eigenvalues. 00:51:04.000 --> 00:51:05.000 Yeah, so I'm not saying… 00:51:05.000 --> 00:51:07.000 But here, I think you're saying that you can't do that. You can't take linear. 00:51:07.000 --> 00:51:13.000 No, I'm not saying… I'm not saying you can't. I'm saying there's a choice of how you want to interpret this. 00:51:13.000 --> 00:51:14.000 Okay. 00:51:14.000 --> 00:51:21.000 So, so, so there's a… I mean, you could just choose to do canonical quantum gravity, and then you wouldn't sum over these other topologies at all. And, I mean… 00:51:21.000 --> 00:51:26.000 Just purely from a 2D point of view, why not, right? Like, that seems fine, okay? 00:51:26.000 --> 00:51:36.000 Similarly, you can do this baby universe field theory, okay? The only things that you could complain about were the assumption that the path integral is well-defined and that it's positive semi-definite. 00:51:36.000 --> 00:51:52.000 And I will, in a moment, complain about both of them, but for now, let's just say that they're true, okay? And then we can just do this. We use the spectral theorem, and this defines the quantum mechanics, and we can talk about this quantum mechanics. Okay. 00:51:52.000 --> 00:51:53.000 Yeah. 00:51:53.000 --> 00:51:54.000 Okay, so you have a Hilbert space, which is roughly speaking, a direct sum over alpha of one dimensional spaces for each alpha. 00:51:54.000 --> 00:51:57.000 Uh, yeah, but no, but from that point of view, I wouldn't say it's one-dimensional. 00:51:57.000 --> 00:52:03.000 Can I take the linear combination of alpha plus bracket alpha plus bracket beta for different out. 00:52:03.000 --> 00:52:11.000 Yeah, so in Baby universe field theory, you absolutely can, and that's why I said that the Hilbert space is not one-dimensional in baby universe field theory. 00:52:11.000 --> 00:52:12.000 Got it. Mm-hmm. 00:52:12.000 --> 00:52:20.000 Okay? Yeah. The claim is that there's this alternative interpretation of the path integral, where I don't try to interpret it in this third quantized Hilbert space. 00:52:20.000 --> 00:52:32.000 I try to interpret it in terms… as I say rather than alpha being different states, it's different theories. You know, which is what Saad, Schenker, and Stanford did, right? Like, I didn't make this up, right? It's the thing… 00:52:32.000 --> 00:52:36.000 that, um, Saad, Shankard, and Stanford do. 00:52:36.000 --> 00:52:37.000 And, uh, 00:52:37.000 --> 00:52:41.000 For them, for them, these were, like, traces of… exponentials of Hamiltonians on one dimension. 00:52:41.000 --> 00:52:45.000 Yeah, that's because they… that's because their universe had an ADS boundary. 00:52:45.000 --> 00:52:46.000 So then the Hover space is not one-dimensional. But this is Saudshanker Stanford without the ADS boundary. 00:52:46.000 --> 00:52:52.000 Okay. 00:52:52.000 --> 00:53:03.000 Okay? Uh, and then you get that it's an average over one-dimensional things. SYK is also like that, right? Like, the things you average over… the Js in SYK are the alphas. 00:53:03.000 --> 00:53:06.000 Okay? Um… 00:53:06.000 --> 00:53:13.000 Okay, so that's the three ways of interpreting the path integral. Now let's try to apply that to learn some things. Okay. 00:53:13.000 --> 00:53:23.000 So, um… so let me do a few applications. So one example that I like that illustrates the difference between baby universe field theory and average holography is perturbative string theory, right? So, uh… 00:53:23.000 --> 00:53:32.000 you know, okay, we all learn in whatever elementary school that to compute a string scattering amplitude, you sum over topologies, connecting initial and file states. 00:53:32.000 --> 00:53:42.000 Um, so to decide what the quantum interpretation of that calculation is, you want to decide, so this looks like the kind of path integral we were just doing, right? You have some boundaries, you sum over initial… 00:53:42.000 --> 00:53:44.000 And final states. Um… 00:53:44.000 --> 00:53:50.000 So then, um, you sum over everything that you put in between, that's the transition amplitude. 00:53:50.000 --> 00:54:00.000 The question is, now, when you square the amplitude to compute a cross-section, which is what you're supposed to do in string theory, do you include world sheet that connects between S and S star? 00:54:00.000 --> 00:54:03.000 Okay? And if you do, 00:54:03.000 --> 00:54:12.000 then your scattering amplitude is the scattering amplitude in the one-dimensional Hilbert space of a fixed, you know, fixed alpha that's then averaged over alpha. 00:54:12.000 --> 00:54:16.000 And if you don't, then you're doing baby universe field theory. 00:54:16.000 --> 00:54:21.000 And, you know, of course the answer is that you're doing a baby universe field theory, right? You're not doing 00:54:21.000 --> 00:54:23.000 Average holography, you know, when you… 00:54:23.000 --> 00:54:28.000 compute S times S star, you don't connect world sheets between… 00:54:28.000 --> 00:54:37.000 S and S-star, and so you're doing baby universe field theory, and indeed, the Hilbert spaces string scattering states is not one-dimensional, and it's, in fact, 00:54:37.000 --> 00:54:43.000 Probably spanned by alpha states, although I'm not really sure. It would be kind of fun to understand what the alpha states really are. 00:54:43.000 --> 00:54:52.000 in this context, I mean, in some sense, their eigenstates of string… of the string field and string field theory. So the Z-hat is like the string field in closed string field theory. 00:54:52.000 --> 00:54:55.000 in this, uh, version of the story. 00:54:55.000 --> 00:54:58.000 Okay, and so this explains why the string… 00:54:58.000 --> 00:55:06.000 scattering and covert space is not one-dimensional, it's because you're doing baby universe field theory instead of average holography, and there's lots of alpha states. 00:55:06.000 --> 00:55:12.000 Okay? 00:55:12.000 --> 00:55:14.000 So, um, good. 00:55:14.000 --> 00:55:17.000 I think was there… there was a question in the chat, um… 00:55:17.000 --> 00:55:19.000 Let me see, oh, uh… 00:55:19.000 --> 00:55:21.000 Wait, uh, how do I… 00:55:21.000 --> 00:55:29.000 Here. Oh, how does the physics change if you add an ADS boundary? A Lorenzian ADS boundary. Well, then there's a real Hilbert space, because, you know, which… 00:55:29.000 --> 00:55:31.000 from the CFT, there had better be, right? 00:55:31.000 --> 00:55:36.000 Um, so this puzzle of the one state is only there if there are no asymptotic boundaries. 00:55:36.000 --> 00:55:38.000 Okay. 00:55:38.000 --> 00:55:42.000 Okay, so another… another application that I like is the evaporating black hole. 00:55:42.000 --> 00:55:47.000 Um, so this, I have to be pretty terse, sorry, I mean, we're getting pretty far behind here. 00:55:47.000 --> 00:55:50.000 So, in Baby Universe Field Theory, the… 00:55:50.000 --> 00:55:59.000 there's these little papers of Coleman, and then also Polchinski and Strominger, you know, Coleman had this black hole, this red herring paper, and Polchinski and Strominger. 00:55:59.000 --> 00:56:02.000 Had some story relating the information problem to alpha states. 00:56:02.000 --> 00:56:10.000 Uh, I won't try to defend this, because there's not time, but the… to explain it, but the idea is that the state, if you prepare a… 00:56:10.000 --> 00:56:16.000 N identical black holes in a pure state collapse, and then you evolve them to fully evaporate. 00:56:16.000 --> 00:56:25.000 The idea is that what's left is some radiation cloud that's entangled with the alpha states for the closed universe that the black hole left behind. 00:56:25.000 --> 00:56:30.000 Um, that looks like this, okay? And so then the Rennie entropy is something like this. 00:56:30.000 --> 00:56:41.000 Um, and the main point of Coleman, Polchinski, and Strominger was that if you take the number of black holes to be large, then the Rennie entropy of the cloud doesn't just become N times the… 00:56:41.000 --> 00:56:47.000 Entropy of one cloud, it becomes something that's independent of N. The second Reny of the alpha ensemble. 00:56:47.000 --> 00:56:52.000 Okay, and so from this point of view, the first black hole you evaporate gives a mixed state. 00:56:52.000 --> 00:56:58.000 But as you evaporate more of them, then you pin… gradually pin down the value of the alpha parameters. If you measure the… 00:56:58.000 --> 00:57:07.000 Radiation Cloud, and then once you've pinned them down enough, now you're in an alpha state, and then from then on, the… any more black holes you make, the evaporation is pure. Okay, so that was the… 00:57:07.000 --> 00:57:10.000 story of Coleman, Polchinski and Strominger. 00:57:10.000 --> 00:57:21.000 Um, uh, that's what you get from baby universe field theory. That's not what you get from holography, right? In holography, you're in some fixed alpha from the beginning, you may not know which one, but you're in one of them. 00:57:21.000 --> 00:57:24.000 And so the very first black hole you make is in a pure quantum state. 00:57:24.000 --> 00:57:30.000 And so, actually, that's what I think should be true, that's what's true in ADS-CFT, and so for me, that's a… 00:57:30.000 --> 00:57:33.000 A reason to be suspicious of this baby universe field theory. 00:57:33.000 --> 00:57:40.000 Okay. And so, in fact, yeah, now that I presented… we've spent time presenting this, now let me complain about it. Okay. 00:57:40.000 --> 00:57:50.000 So, the first complaint, which we've already discussed, is that outside of 1 plus 1 dimensions, uh, you probably can't do this algorithm, right? The path integral is not normalizable, 00:57:50.000 --> 00:57:54.000 You don't know how to do the sum over topology, it's probably not convergent. 00:57:54.000 --> 00:58:01.000 Uh, and so, you know, this application of the spectral theorem is probably sort of vacuous, you know, it's just not well-defined enough to… 00:58:01.000 --> 00:58:07.000 argue that the alpha states really exist, and from my attitude, that's probably for the best, because, um… 00:58:07.000 --> 00:58:13.000 you know, we probably shouldn't be able to non-perturbatively construct string theory just starting from supergravity. That seems like… 00:58:13.000 --> 00:58:16.000 Something that would be too good to be true. Okay. 00:58:16.000 --> 00:58:26.000 Um, secondly, you know, in baby universe field theory, the Hilbert space is not holographic, right? I mean, because it has fundamental degrees of freedom, this alpha parameters, 00:58:26.000 --> 00:58:30.000 Um, which don't live at the asymptotic boundaries. 00:58:30.000 --> 00:58:32.000 they somehow live… 00:58:32.000 --> 00:58:34.000 I don't know where they live, just… 00:58:34.000 --> 00:58:41.000 you know, God knows them, I don't know, but we don't. Like, I don't know, like, they're just somehow outside the universe, wherever these alpha parameters are living. 00:58:41.000 --> 00:58:44.000 Um, and uh… 00:58:44.000 --> 00:58:54.000 you know, I don't think that's consistent with holography, which says that the fundamental degrees of freedom should live at the boundaries. You know, and as we just discussed, it also leads to black hole radiation that's not pure. 00:58:54.000 --> 00:58:57.000 Um, and so for me, that's not acceptable. 00:58:57.000 --> 00:59:07.000 And then finally, in string theory, right, you know, every parameter that we know about is the expectation value of some field. There aren't parameters that are just there. 00:59:07.000 --> 00:59:14.000 Um, and, you know, in particular, like, in N equals 4, this includes the gauge coupling, the number of colors, right? Those are boundary conditions for fields. 00:59:14.000 --> 00:59:28.000 Um, and so, you know, in the discussion we just had, these should be thought of as part of J instead of part of alpha, right? You know, so sometimes people talk about averaging over CFTs, and they think it's like averaging over alpha. You can't believe how many times I heard it, but it's not true. 00:59:28.000 --> 00:59:40.000 Averaging over CFTs is averaging over parameters in the CFT Lagrangian, and those are boundary conditions for fields, okay? And so that's averaging over J, it's not averaging over alpha. There's no alpha. 00:59:40.000 --> 00:59:47.000 an ADS-CFT. And that's reflected in the fact that the partition function factorizes in ADS-CFD. It doesn't have this sum over P-alpha. 00:59:47.000 --> 00:59:50.000 Uh, giving a non-factorizing answer. 00:59:50.000 --> 00:59:55.000 You know, and so, like I said, I mean, you can have… you can have a bulk geometry that interpolates between 00:59:55.000 --> 01:00:03.000 N equals 1,000, and N equals 1,001 with the Euclidean D3 brain, so it can't be an alpha, which would just be the same everywhere for all time. 01:00:03.000 --> 01:00:12.000 Okay, so some… so yeah, this is what I just said. So some people wanted to average over this and said it was like alpha, but I disagree with them. 01:00:12.000 --> 01:00:15.000 So, my view, in light of these objections, um, 01:00:15.000 --> 01:00:21.000 remains that we should look for some definite holographic theory with no alpha parameters, such as string theory. 01:00:21.000 --> 01:00:28.000 Um, from which we can hope to have this baby universe field theory, or maybe conical cone gravity, emerge in some appropriate limit. 01:00:28.000 --> 01:00:35.000 As an effective field theory with a limited regime of validity. Okay, that's what I think we should actually ask for. 01:00:35.000 --> 01:00:38.000 Uh, you know, I think there has been… 01:00:38.000 --> 01:00:44.000 some taking the low-dimensional models too seriously in the literature, and so I'm trying to push back on it. 01:00:44.000 --> 01:00:47.000 here, and say what I think we can actually hope to learn. 01:00:47.000 --> 01:00:53.000 Um, so in the remainder of the talk, what I was planning to do was argue that introducing an observer 01:00:53.000 --> 01:00:59.000 allows for Baby Universe Field Theory to emerge from a fixed holographic theory with one alpha, 01:00:59.000 --> 01:01:04.000 Um, but only up to effects which are of order e to the minus S observer. Uh, and then I was going to make some… 01:01:04.000 --> 01:01:17.000 comments about, you know, holography, the path integral, and the landscape of string theory. Now, that said, we're at time, and I have, like, 10 more slides, so I don't know how you guys want to handle this. 01:01:17.000 --> 01:01:18.000 Um, the way… 01:01:18.000 --> 01:01:23.000 Well, I asked a lot of questions, so I'll just be quiet. I'd like to hear the rest. 01:01:23.000 --> 01:01:28.000 Yeah, the way we usually handle it is anybody who wants to leave can leave. 01:01:28.000 --> 01:01:29.000 Okay. 01:01:29.000 --> 01:01:34.000 And anybody who wants to listen can listen, and the speaker can decide when to quit. 01:01:34.000 --> 01:01:41.000 Yeah, I mean, the rest of this is mostly calculation, you know, it's trying to put a little meat on it. So I just told you some general story, but uh… 01:01:41.000 --> 01:01:49.000 Yeah, so, okay, well, let me see how far I get in 10 minutes, okay? And if it's, like, egregious, then we can… I'll try to wrap up. 01:01:49.000 --> 01:01:50.000 Okay. 01:01:50.000 --> 01:01:51.000 Okay? Yeah. 01:01:51.000 --> 01:01:54.000 Um, okay, um… 01:01:54.000 --> 01:02:03.000 So, okay, so I want to… so let's… let's now do some actual calculations. So to bring this down to earth, um, I don't want to have a model where I can actually compute stuff. 01:02:03.000 --> 01:02:09.000 So I… this… I'll basically use the Meroff-Maxfeld model, where the action is the Euler character, 01:02:09.000 --> 01:02:11.000 Uh, times minus S naught. 01:02:11.000 --> 01:02:15.000 But then I'm going to do a generalization due to Ying and Misha and this Wang. 01:02:15.000 --> 01:02:21.000 Um, where you add a world… some worldline matter that carries a species index i. 01:02:21.000 --> 01:02:26.000 So somehow, if you have pure topology, it's too boring to be interesting, so you need to have some… 01:02:26.000 --> 01:02:31.000 stuff that you can try to do physics with, and so I'll use Worldline Matterist stuff. 01:02:31.000 --> 01:02:45.000 And so the dynamics for the matter index is just that it… the flavor and the species index has to agree at the start at end of a world line. So, you know, the propagator is delta IJ, very boring. 01:02:45.000 --> 01:02:48.000 So, in this paper with Ying and Misha, um, 01:02:48.000 --> 01:02:53.000 We studied the states of this model with one matter particle in a closed universe, okay? 01:02:53.000 --> 01:03:00.000 But, um, for my purposes today, so I had this four questions which I… which I started with, this is too much, this is too simple, because 01:03:00.000 --> 01:03:07.000 The one particle part of the Hartle-Hawkins state is zero in this model, because there's nowhere for the propagator to go. 01:03:07.000 --> 01:03:16.000 And uh… in particular, it makes it hard to ask this question of how is the Harle-Hawkins state related to the one state, which is one of the questions that I wanted to answer. 01:03:16.000 --> 01:03:22.000 Um, so to fix this, I need to… the easiest thing to do is instead consider two particle states. 01:03:22.000 --> 01:03:32.000 Right, so, like, here's a… in this model, say I have two of these flavor indices, okay, then they… the propagator can just connect them like that with the delta IJ, right? And so that's here, okay? 01:03:32.000 --> 01:03:37.000 And then there's some sum over handles that leads to some factor like this. 01:03:37.000 --> 01:03:39.000 Okay, involving this S0. 01:03:39.000 --> 01:03:42.000 So the… 01:03:42.000 --> 01:03:43.000 Yeah. 01:03:43.000 --> 01:03:45.000 Sorry, could you go back? Could you go back one slide? I know you're. 01:03:45.000 --> 01:03:47.000 So what are these I's and J's? 01:03:47.000 --> 01:03:53.000 Um, yeah, so it… I had… I have matter that's worldline matter. 01:03:53.000 --> 01:03:59.000 And then each wall line carries an index i. So just think of it as I have a particle which has a species label. 01:03:59.000 --> 01:04:00.000 Okay? 01:04:00.000 --> 01:04:08.000 But does the world line now give a one dimensional submanifold inside this space time? 01:04:08.000 --> 01:04:16.000 Um, yeah, but since it's topological, I'm not going to try to sum over that. 01:04:16.000 --> 01:04:19.000 Yeah, so, yeah, so in principle, 01:04:19.000 --> 01:04:29.000 Yeah, yeah, so you can choose… you can choose to worry about that. My attitude, though, is that… so then you need to have ad handles for that… yeah, or actually, I already hear it matters, you can think about the winding. 01:04:29.000 --> 01:04:33.000 But my philosophy is that… is that if you want to do that, then I should add a mass term. 01:04:33.000 --> 01:04:41.000 And then the higher windings will be suppressed exponentially in the mass times the length. And so my… so that's my excuse for not, uh, including the windings. 01:04:41.000 --> 01:04:48.000 Yeah. Um, in principle, we should if we really want to get good answers, and that'll put corrections on this that are exponential in the mass of the thing. 01:04:48.000 --> 01:04:51.000 Okay, um… 01:04:51.000 --> 01:04:56.000 So, uh, good. So the Hartle-Hawking state, the overlap with one, two-particle state, 01:04:56.000 --> 01:05:02.000 is just this delta IJ, because the worldline propagator just hooks the i and the j together here. 01:05:02.000 --> 01:05:08.000 Um, if we do an overlap between two, two particle states, IJ and IJ prime, 01:05:08.000 --> 01:05:13.000 Then there's a disconnected contribution, which is the square of the Harle-Hawking state. 01:05:13.000 --> 01:05:19.000 And then there's also a connected contribution from the cylinder. So I feel like this is canonical gravity here, 01:05:19.000 --> 01:05:30.000 This is something that's there in baby universe field theory, but not there in canonical gravity. Um, you know, and the thing that's kind of scandalous is that this term is bigger, right? Like, if you compare these two, this has an e to the 2s naught in it, 01:05:30.000 --> 01:05:37.000 And this has a 1, okay? So this contribution actually dominates, and this is the thing that we ignored. 01:05:37.000 --> 01:05:43.000 Uh, in the paper a year ago, and the primary thing that I wanted to think more about in this context is, what am I supposed to say? 01:05:43.000 --> 01:05:55.000 about this disconnected contribution to the inner product, because it… you know, at least naively, it leads to a huge deviation with canonical quantum gravity, you know, between canonical quantum gravity and, uh, and baby universe field theory. 01:05:55.000 --> 01:06:03.000 You know, in Chronicle quantum gravity, I mean, I didn't really try to defend it because I thought it was maybe obviously good, but I mean, that's the thing that goes over to quantum field theory in a fixed background, right? Like, when you… 01:06:03.000 --> 01:06:15.000 When you compute the CMB using, uh, inflation, you do canonical quantum gravity. You don't include contributions like this, and if you did, they would give you answers that are totally inconsistent with the experiment, alright? So, uh… 01:06:15.000 --> 01:06:20.000 So that's the thing for us to worry about. That's the thing that I was worried about. 01:06:20.000 --> 01:06:33.000 I'll comment more on that in a bit. Um, so in this model, if i is not equal to j, or I prime is not equal to J prime, then you don't have this contribution, right? Because the propagator can't connect, and then it's dominated by this. But if they're equal, then you have to include it. 01:06:33.000 --> 01:06:36.000 Okay. Um… 01:06:36.000 --> 01:06:40.000 Now, um, in order to see how this, uh, 01:06:40.000 --> 01:06:47.000 you know, factorization of the inner product. So, you know, somehow the inner product basically factorizes onto the Hartle-Hawking state. 01:06:47.000 --> 01:06:53.000 You know, in some sense, it's some semi-classical avatar of the one-state problem, okay? 01:06:53.000 --> 01:07:05.000 So, in order to see that in more detail, though, we need to… so this is the bulk model that I just told you, but we can also have some microscopic model that's dual to this. So this is a bit like the matrix model side of Sodzchenker Stanford. 01:07:05.000 --> 01:07:15.000 Okay, so here's the model that I'll use. I won't try to explain it in complete detail, but… so to give you a microscopic model, I have to tell you what's this state IJ with the double angle. 01:07:15.000 --> 01:07:19.000 Right? So I need a number… so remember, these states are numbers, and I'm telling you the number here. 01:07:19.000 --> 01:07:29.000 You, uh, you take the states of the two particles, you feed them into some random orthogonal matrix O, these are two copies of the same 01:07:29.000 --> 01:07:38.000 random orthogonal matrix, and then you project onto some entangled state chi, where you… and then take the large D limit, maintaining that these, uh… 01:07:38.000 --> 01:07:44.000 this quantity where max is the unnormalized maximally entangled state, is finite in the limit of large D. 01:07:44.000 --> 01:07:58.000 Okay, so that's a lot to drop on you, and I don't necessarily want to make the whole talk be about this model, but you should just think of this as some version of the Sods-Strinker-Stanford matrix model that gives you a formula for these numbers IJ. 01:07:58.000 --> 01:08:06.000 Um, where then we can… we can, you know, look at overlaps in this fundamental Hilbert space and average over O, so O is, like, the alpha parameter. 01:08:06.000 --> 01:08:10.000 and try to connect to this bulk model that we just discussed. 01:08:10.000 --> 01:08:18.000 And indeed, if you do this calculation, averaging over O, you get exactly the same answer we had in the bulk, right? So you get this delta IJ for the Hertel-Hawking, 01:08:18.000 --> 01:08:29.000 And then for this thing, you have this sum over the three ways of connecting, which was, uh, these three things here. Um, with this one being bigger, and this one being bigger is this factor of K here. 01:08:29.000 --> 01:08:40.000 And so this K parameter that has to do with the entanglement of chi is the e to the 2 to the S naught in this microscopic realization of the bulk model that I just told you. 01:08:40.000 --> 01:08:48.000 Okay? Um, and so this agreement continues with more boundaries, right? And so you can think of this O being like the alpha parameters, 01:08:48.000 --> 01:08:56.000 And then this ZO of IJ, right, in the language, this was Z alpha of J in the language we were using before, that's just this, uh… 01:08:56.000 --> 01:09:05.000 encoded version, that's this encoded version of the… of the state that's this, uh… where did I write it? Here, this, uh, this… right, this thing is the same as this angled… 01:09:05.000 --> 01:09:13.000 IJ. I'm trying to suppress the code language here, but the code tells you how to relate the Baby Universe Field Theory Hilbert space to the fundamental Hilbert space. 01:09:13.000 --> 01:09:20.000 Okay. Um, now I want to emphasize that the one state here 01:09:20.000 --> 01:09:21.000 It's just a complex number. 01:09:21.000 --> 01:09:23.000 But the double bracket Ij is just a complex number. Even though even though you're writing it as a kit. 01:09:23.000 --> 01:09:27.000 That's correct, yes. That's correct. 01:09:27.000 --> 01:09:28.000 That's right. Yeah, that's right. 01:09:28.000 --> 01:09:30.000 So, it's a ket in a one-dimensional Hilbert space lay the complex numbers. 01:09:30.000 --> 01:09:33.000 That's right, yes. 01:09:33.000 --> 01:09:40.000 So, um, so this one state, like, if you look at… and that's kind of clear in this picture, right? So, basically, this is saying the encoding map 01:09:40.000 --> 01:09:45.000 From the Baby Universe Field Terry Hilbert space, the two-particle sector of it. 01:09:45.000 --> 01:09:50.000 to the fundamental Hilbert space, is you take whatever state, and then you just hit it with this big projector. 01:09:50.000 --> 01:09:54.000 Okay, so the holographic encoding map is a Rank 1 projection. 01:09:54.000 --> 01:09:58.000 Okay, and so this is the one state right here. This bra here is the one state. 01:09:58.000 --> 01:10:01.000 Okay, so if I write it in equations, it looks like this. 01:10:01.000 --> 01:10:07.000 And in particular, I want to emphasize that this is not the Harle-Hawking state, right? The Harle-Hawking state 01:10:07.000 --> 01:10:18.000 is the one we computed here. It's this delta IJ. Where did I write it? Here, it's this, okay? So here, this delta ij, you see there's no O here, there's no UV dependence here, this is the Hartell-Hawkins state. 01:10:18.000 --> 01:10:25.000 The one state is this, which is strongly UV-sensitive, right? It depends on O or alpha, whichever you want to call it. 01:10:25.000 --> 01:10:32.000 Um, you know, whereas in the fundamental description, the Hartle-Hawkins state is just the number 1, right, because it doesn't insert any boundaries. 01:10:32.000 --> 01:10:41.000 And the effective description, it's this delta IJ state. Okay, but either way, it's not the one state, and so that's one of the things you learn from this model. 01:10:41.000 --> 01:10:50.000 Now, okay, so what I just showed is, um, is that… where did I write it? Yeah, I showed… I showed that the baby universe field theory answer comes from averaging 01:10:50.000 --> 01:10:56.000 the answer in the one-dimensional Hubbard space, right? That's this thing we said before. Baby universe field theory equals an average 01:10:56.000 --> 01:11:01.000 over… over what happens in a fixed holographic theory. Okay. 01:11:01.000 --> 01:11:08.000 But remember, I don't like… I don't like this average, right? I don't like alpha parameters. I don't think there's an average over alpha parameters. 01:11:08.000 --> 01:11:13.000 And so, you know, the real question is, to what extent is the one-dimensional overlap 01:11:13.000 --> 01:11:21.000 equal to Baby Universe Field Theory without doing the average, right? Because you don't have the average. You know, you just have the CFT. 01:11:21.000 --> 01:11:26.000 Right, so you just have one factorizing theory, so to what extent 01:11:26.000 --> 01:11:28.000 Is this true? Okay. 01:11:28.000 --> 01:11:35.000 Um, and so this is… so you can quantify this, you know, how close is a fixed alpha to the average by looking at the variance 01:11:35.000 --> 01:11:45.000 in the ensemble, right? In order for this to be true, you need the variance in the overlap between these states to be small compared to the product of their norms. That's what it means for the… 01:11:45.000 --> 01:11:56.000 overlap to be close to the average. But of course, that's not true. It can't be, because this is a one-dimensional Hilbert space, and this is a many-dimensional Hilbert space, and so how can these overproduct… 01:11:56.000 --> 01:12:00.000 inner products agree with each other, and that's the whole one-state problem, right? It's that… 01:12:00.000 --> 01:12:07.000 is that, you know, we had this semi-classical, nice, big Hilbert space spanned by alpha states, but really there's only one alpha state, and so what are we… 01:12:07.000 --> 01:12:15.000 What are we supposed to do with that? Okay, and that's… that's where the observer rule really comes in. So I've, so far, suppressed the observer rule in the last 01:12:15.000 --> 01:12:22.000 Couple minutes, I'll, uh, I'll use it to explain how to do this. Okay, so we're getting there, so I think we're making good progress here. 01:12:22.000 --> 01:12:28.000 Um, okay, so what does the observer rule do? So it takes this here, 01:12:28.000 --> 01:12:38.000 And it modifies the inner product to include an extra… to include entanglement with some external system. 01:12:38.000 --> 01:12:54.000 Which is there to make the observer classical. So this line going in is the observer in the closed universe who's looking at our two-particle state that we were discussing before to make the observer classical, I entangle them with their reference system in their pointer basis, so that if I trace out, 01:12:54.000 --> 01:12:58.000 The reference system, I'm essentially averaging over the microstate of the observer. 01:12:58.000 --> 01:13:06.000 Okay, that's the… I could say a lot more about why that is equal to the quantum channel and so on, but that was the talk I would have given a year ago, so for now, I'll just do it. 01:13:06.000 --> 01:13:16.000 Um, and so if we now evaluate this modified version of the inner product, okay, so first here, I'm just doing the average, right? So then I get… 01:13:16.000 --> 01:13:22.000 Um, so it's basically the same result that I had before. Um, you know, I have this… 01:13:22.000 --> 01:13:27.000 three-party thing from the cylinder, and then I have this delta IJ and IJ prime from here. 01:13:27.000 --> 01:13:30.000 But now I have to add an amplitude lambda, 01:13:30.000 --> 01:13:38.000 Which is the amplitude to create an observer out of nothing, right? Because you see in this diagram, the disconnected one, the observer goes in here, and the observer goes in here, 01:13:38.000 --> 01:13:46.000 And there somehow needs to be an amplitude here to create the observer and absorb the observer, and that's this lambda factor here. That's the amplitude to make an observer. 01:13:46.000 --> 01:13:59.000 Now, you might think that that's small, okay, but remember, in this diagram, it's multiplied by e to the 2s naught, and that's probably a lot bigger than the amplitude to make an observer as small, and so this term will still dominate, even 01:13:59.000 --> 01:14:03.000 When you pay the price of nucleating an observer out of the Hartel-Hawkins state. 01:14:03.000 --> 01:14:06.000 Okay. Um… 01:14:06.000 --> 01:14:08.000 So, um… 01:14:08.000 --> 01:14:18.000 Now, I'll take the observer Hilbert space dimension to be KO, that's the e to this S observer that I was talking about before, and to absorb… to avoid just making ridiculous numbers observers, 01:14:18.000 --> 01:14:26.000 I'll say that the amplitude to do it from nothing is small compared to 1 over KO, otherwise you would just sort of have unsuppressed 01:14:26.000 --> 01:14:28.000 Observer production all the time. 01:14:28.000 --> 01:14:38.000 Um, and so, um, yeah, so here's this thing I just said, that although the disconnected amplitude is suppressed by lambda squared, it's enhanced by e to the 2S naught, which means that 01:14:38.000 --> 01:14:42.000 For reasonable-sized observers, the transition amplitude is actually dominated 01:14:42.000 --> 01:14:55.000 by the disconnected contribution, at least if i equals J and I prime equals J prime. So in other ways… in other words, the best way to make a transition between two states of the universe is to deflectuate the entire universe, including the observer, 01:14:55.000 --> 01:15:00.000 Uh, and then re-fluctuate a new universe with the system in the final state. 01:15:00.000 --> 01:15:02.000 Okay? Alright. 01:15:02.000 --> 01:15:16.000 So, um, so this is something like the Boltzmann brain issue, as we'll talk about a bit later. So, I should also say that the other observer role, the Berkeley one, would throw away this disconnected contribution. I think that's wrong. We'll discuss that in a minute. 01:15:16.000 --> 01:15:18.000 Okay. Um… 01:15:18.000 --> 01:15:27.000 So, um, but anyway, Baby Universe Field Theory has both of these contributions, so we don't need to get rid of this one. In fact, if we're trying to match with debut universe field theory, we should include this contribution, we shouldn't throw it away. 01:15:27.000 --> 01:15:30.000 Okay. Um… 01:15:30.000 --> 01:15:42.000 Okay, and so now, good, so that was still the average. Now, what we really want to do is we want to learn what happens at fixed O if we don't average over the alpha parameters. And so for that, we look at the variance of this inner product. So I drew it small so that you don't… 01:15:42.000 --> 01:15:49.000 have to actually try to check, but there are 15 topologies that contribute to the square of the inner products. 01:15:49.000 --> 01:15:55.000 Okay? Um, if you do the calculation, which I will spare you, the result is that using the observer rule, 01:15:55.000 --> 01:16:00.000 Um, all of the non-factorizing contributions to this are suppressed by the observer entropy. 01:16:00.000 --> 01:16:02.000 Which means that, um, 01:16:02.000 --> 01:16:07.000 The ensemble fluctuations, you know, that tell you the difference between being at fixed alpha 01:16:07.000 --> 01:16:21.000 And being in the average are suppressed by 1 over K observer, okay? And so… so here we see that what the observer rule is really doing. It's not that it's getting rid of the disconnected contribution, it's getting rid of the ensemble fluctuations. 01:16:21.000 --> 01:16:24.000 Um, and so when you use the observer rule, 01:16:24.000 --> 01:16:30.000 Then holography at fixed O, or fixed alpha, which is the thing that I think should actually be a good model for higher dimensions, 01:16:30.000 --> 01:16:33.000 Um, actually agrees with the baby universe. 01:16:33.000 --> 01:16:44.000 field theory answer. Okay, and so in some sense, that's the main result of this talk, okay, is that so… so the observer role suppresses the difference between average holography, or really unaverage holography, 01:16:44.000 --> 01:16:49.000 And baby universe field theory. Okay. 01:16:49.000 --> 01:16:52.000 Uh, let's see, I'll see somebody said something, uh… 01:16:52.000 --> 01:16:57.000 Can you at least explain the last two out of the 15? 01:16:57.000 --> 01:17:00.000 2 out of the 15 terms? What, you mean, uh, these two terms? 01:17:00.000 --> 01:17:03.000 Yes, I want to get a feeling of, uh… 01:17:03.000 --> 01:17:10.000 of the schematic and how you generate these terms. 01:17:10.000 --> 01:17:21.000 Um, I'm just summing over all the topologies, right? So, so you… I mean, I just used the rules, right? So there's this Euler character weighting, that's what determines this Snaught factors. 01:17:21.000 --> 01:17:27.000 And then when you make an observer, there's the… there's a lambda factor whenever you fluctuate an observer out of nothing. 01:17:27.000 --> 01:17:34.000 And then these deltas are the ways the world lines connect between the different geometries, okay? 01:17:34.000 --> 01:17:40.000 Yeah, those are the rules, yeah. Yeah, yeah. In the paper, I write the whole formula with no dot dot dots, but it's too awful to put it on this slide. 01:17:40.000 --> 01:17:41.000 Yeah. 01:17:41.000 --> 01:17:43.000 Okay, okay, thank you. 01:17:43.000 --> 01:17:45.000 Yeah, um… 01:17:45.000 --> 01:18:05.000 Okay, um, so, okay, so I… just two more slides and we're done. So, or three more, I think, sorry. So, okay, so we've seen that the observer rule allows baby universe field theory to remove… to emerge from holography without the need for alpha parameters. Okay, that's the main slogan, put that in a box. You know, I don't like alpha parameters, here's a way to get this baby universe field theory without having alpha parameters. 01:18:05.000 --> 01:18:11.000 So for me, that's the most interesting, or at least one of the most interesting consequences of this discussion. 01:18:11.000 --> 01:18:18.000 Okay? Um, so on the other hand, this baby universe field theory, so that's good, we like it, okay, great. 01:18:18.000 --> 01:18:30.000 On the other hand, you know, it does have this disturbing feature that the transition amplitude can be dominated by the disconnected factorization onto the Harle-Hawking state, and so that leads to a tension with canonical quantum gravity, which didn't have that disconnected 01:18:30.000 --> 01:18:38.000 contribution, and indeed, that's a version of the Voltzmann brain problem, right? Everything just happens by fluctuation in a theory where the disconnected contribution is large. 01:18:38.000 --> 01:18:41.000 So, um… 01:18:41.000 --> 01:18:43.000 So, of course, we could just hope that 01:18:43.000 --> 01:18:51.000 The initial or final state we find ourselves in just happens to be almost orthogonal to the Harle-Hawking state, like I not equal to J in this model. 01:18:51.000 --> 01:18:59.000 Okay. Um, but you know, in cosmology, we'd like to have some principled theory of the initial state, and in fact, the Harle-Hawking one, at least naively, seems like a pretty nice one. 01:18:59.000 --> 01:19:05.000 Now, it doesn't work phenomenologically, you know, in some sense, essentially because of this problem. 01:19:05.000 --> 01:19:14.000 But, you know, we should have some principled theory of the initial state, maybe rather than just declaring that we want it to be almost orthogonal to the Harle-Hawking state, so we can suppress the disconnected contribution. 01:19:14.000 --> 01:19:27.000 And so finally, about this, I wanted to say this observer rule, where I decohhered the observer. You can think of it as just a particular averaging over the boundary sources J, and this is actually quite similar. 01:19:27.000 --> 01:19:33.000 to what happens with the spectral form factor at long time in this work of Schenker and Stanford and collaborators. 01:19:33.000 --> 01:19:46.000 Where you go to late times in the spectral form factor, and you get these crazy oscillations, and the answer factorizes, but then if you average over the time window, then you get a non-factorizing answer that's consistent with the wormhole and the bulk. 01:19:46.000 --> 01:19:52.000 So this story I just told you is kind of the same as that, except turned on the side so that the wormhole is going like this, okay? 01:19:52.000 --> 01:19:56.000 And so I think it's sort of… 01:19:56.000 --> 01:20:08.000 not… although the observer rule maybe is a bit scary, the way I presented it, uh… mat, you know, it's kind of the same as this kind of more kosher thing of averaging the spectral form factor over a time window. Those are both… 01:20:08.000 --> 01:20:12.000 Averaging over J, and averaging over J is good, unlike averaging over alpha. 01:20:12.000 --> 01:20:23.000 Okay. Um, so now the last thing, I was going to say this thing about the landscape, so this should be pretty quick, especially because I'm a bit still confused about it, and so then I'll be done after these two slides. 01:20:23.000 --> 01:20:26.000 Um, which is good, because I have office hours in 7 minutes. 01:20:26.000 --> 01:20:35.000 So, um… so let's consider a landscape. So I'm going to try to argue that this business of the disconnected contribution and so on can lead to some tension between 01:20:35.000 --> 01:20:39.000 Baby universe field theory and ADS-CFT. So this is something for us to all 01:20:39.000 --> 01:20:44.000 be puzzled about and try to understand, okay? I'm not sure I understand the resolution myself. 01:20:44.000 --> 01:20:57.000 So, I want to consider a landscape, um, with one ADS vacuum and one metastable desitter vacuum, okay? Tom, if you don't like the word vacuum, I just mean a potential that looks like that, okay? You can use whatever word you want for it. 01:20:57.000 --> 01:21:03.000 And I'm interested in the sector of this theory, where there's one spatial ADS boundary. 01:21:03.000 --> 01:21:10.000 So, what I learned in grad school is that if you take a theory like this, and you look at the sector of the theory with one ADS, 01:21:10.000 --> 01:21:21.000 Boundary, then the fundamental Hilbert space is the CFT Hilbert space, right? In ADS-CFD, the Hilbert space of quantum gravity with one ADS boundary is the CFT Hilbert space. 01:21:21.000 --> 01:21:34.000 And I want to… in particular, I want to consider a state in that CFT Hubbard space, which definitely has an observer sitting in the middle of ADS. So, for this discussion, I'll assume that observers can live in both ADS and DeSitter. Okay. 01:21:34.000 --> 01:21:39.000 That may be a weak point in the argument, but I'm going to assume that. 01:21:39.000 --> 01:21:57.000 So, you know, if you came to me, you know, a few years ago, and you gave me a state in ADS-CFT where you, you know, you inserted some low-dimension primary to put an observer living in the ADS, and you asked me, what's the gravity dual of this CFT state, I would have said, well, it's got an observer sitting in the middle of ADS, of course, that's what the joule is. 01:21:57.000 --> 01:22:01.000 Um, I'm going to now try to convince you that's not the duel, okay? 01:22:01.000 --> 01:22:03.000 Um, so… 01:22:03.000 --> 01:22:11.000 The idea is to use this… the path integral in this context of, say, Bib universe field theory to compute the expectation value of the following observable. 01:22:11.000 --> 01:22:19.000 So Q, integrated over spacetime. So if there's no observer at the place where Q is currently located, you get 0. 01:22:19.000 --> 01:22:24.000 If there's an observer who's in ADS, you get zero, and if the observer is into sitter, you get 1. 01:22:24.000 --> 01:22:29.000 Okay, so this is an observer that tells you… an observable that tells you whether you're in Decid or ADS. 01:22:29.000 --> 01:22:40.000 Okay? And now to condition on having an observer, I should divide the expectation value of this by the expectation value of an observable, which is 1 if there's an observer and 0 otherwise. 01:22:40.000 --> 01:22:44.000 Okay, because if there's no observer, then we don't see anything. 01:22:44.000 --> 01:22:53.000 Um, so here's what the calculation looks like. So, when I compute the expectation value of having the observer be into center, 01:22:53.000 --> 01:22:59.000 Well, I have the ADS here, but then I need to have a desitter here also disconnected with the observer sitting on this dashed line. 01:22:59.000 --> 01:23:06.000 Because if I didn't have this bubble here, then Q would be 0, because the observer wouldn't be in decision. 01:23:06.000 --> 01:23:19.000 Okay? In the denominator, I have… so this blue is the ADS boundary, this blue line, right? This dot is the ADS boundary operator that creates the observer who's sitting in the center, so these dashed lines are the observers. 01:23:19.000 --> 01:23:23.000 So there's an observer, definitely, in the middle of ADS and all these contributions. 01:23:23.000 --> 01:23:31.000 Down here, if the observer is in ADS, I don't need the DeSitter bubble, but then they could also be in DeSitter, so I have the bubble. 01:23:31.000 --> 01:23:34.000 And now we just compute the ratio. What is it? 01:23:34.000 --> 01:23:41.000 Well, the ADS part just factorizes out and cancels from everything, so what I get is the desitter sphere, 01:23:41.000 --> 01:23:50.000 divided by 1 plus the DeSitter sphere, but the sphere with the observer present, so it's e to the s to sitter minus the observer mass over t to sitter. 01:23:50.000 --> 01:23:58.000 And the important thing is that this is large compared to one indicator, okay? So, in other words, that means that the observer is almost certainly in desider space. 01:23:58.000 --> 01:24:04.000 You know, we started just, you know, with n equals 4 or something, or maybe something that's a little better, so it allows observers. 01:24:04.000 --> 01:24:10.000 And we made a state where there's definitely an observer in ADS, and then we use the path integral to tell us 01:24:10.000 --> 01:24:20.000 If you're an observer in this theory, in this state, what do you see? And the answer is you almost certainly live in De Sitter. You live in some disconnected DeSitter bubble that has nothing to do with the ADS boundary. 01:24:20.000 --> 01:24:24.000 Um, and so, um, so somehow the… 01:24:24.000 --> 01:24:29.000 That means the Boltzmann brain problem hasaded ADS-CFD. You know, we take the dual… 01:24:29.000 --> 01:24:36.000 You know, because, for example, in n equals 4, right, the dual is 2B, we think 2B probably does have metastable to sit or vacua, so it does have contributions like this. 01:24:36.000 --> 01:24:41.000 Uh, and this is somehow saying that that's actually… yeah, we're gonna live there, we're not gonna live in ADS. 01:24:41.000 --> 01:24:57.000 Um, so, uh, you know, maybe we have to pick between ADS-CFT and the Harle Hawking state, I don't know, I'm not sure what to make of this, but I find it quite interesting, and I would like to understand it more. Okay, so that's it. Thank you for listening, sorry for going over time. 01:24:57.000 --> 01:25:08.000 Okay, let's thank Daniel. 01:25:08.000 --> 01:25:16.000 So I can answer maybe one more question before I have to do office hours, if anyone wants to… 01:25:16.000 --> 01:25:19.000 Sorry, the last part was a bit rushed. 01:25:19.000 --> 01:25:20.000 Yeah. 01:25:20.000 --> 01:25:25.000 What if you had many desidera vacua? Would you then? 01:25:25.000 --> 01:25:27.000 Select the one with the largest cosmological constants or something. 01:25:27.000 --> 01:25:29.000 Yeah, I think you probably do, yeah. 01:25:29.000 --> 01:25:33.000 Yeah. Sorry, the smallest, the smallest, because you want the biggest consideration. 01:25:33.000 --> 01:25:34.000 You want the… you want the biggest decider entropy. 01:25:34.000 --> 01:25:36.000 the smallest. Yeah, there we go. Okay. 01:25:36.000 --> 01:25:39.000 Yeah, the biggest consider entropy, yeah. 01:25:39.000 --> 01:25:44.000 So so so is this a solution to the cosmological constant problem when it's small? 01:25:44.000 --> 01:25:49.000 Well, I think that's kind of what Coleman was trying to do, right? 01:25:49.000 --> 01:25:50.000 Yeah, yeah, I think this is kind of what Komen was saying, yeah. 01:25:50.000 --> 01:25:53.000 Yeah, that's a good point. That's right. 01:25:53.000 --> 01:25:54.000 Right. 01:25:54.000 --> 01:26:00.000 Yeah, we're just… we're now revisiting, you know, now that I argued that maybe Coleman is right, once you use the observer rule, then maybe now, uh… 01:26:00.000 --> 01:26:11.000 Yeah, but the thing… the thing is, I don't… I'm not sure, like, uh, I had a very toy landscape here, so if it has also flat and ADS and many different ones, I don't… I'm not sure how much of this is still correct. 01:26:11.000 --> 01:26:15.000 Daniel, I don't think you're taking into account the fact that 01:26:15.000 --> 01:26:22.000 All of these configurations have to have asymptotically anti-decidor boundary conditions. 01:26:22.000 --> 01:26:24.000 Um, they do. Every one of them has this blue line, right? 01:26:24.000 --> 01:26:28.000 Yeah, yeah, so then this thing that's sitting 01:26:28.000 --> 01:26:32.000 Partly in decider space is going to look like a black hole. 01:26:32.000 --> 01:26:34.000 from the ADS point of view. 01:26:34.000 --> 01:26:44.000 No, because it's disconnected. It's disconnected. This is the full ADS spacetime here. It's like a vacuum bubble, but then the operator found itself in it, so… 01:26:44.000 --> 01:26:45.000 So, so then you have to keep it. 01:26:45.000 --> 01:26:48.000 Oh, it's a… it's a disconnected topology. 01:26:48.000 --> 01:26:56.000 Yeah, it's a disconnected topology. This was a product of two topologies, and so is this one. That's why it factorizes… that's why this factorizes out. 01:26:56.000 --> 01:26:58.000 Oh, okay. 01:26:58.000 --> 01:26:59.000 So then… 01:26:59.000 --> 01:27:07.000 Yeah, it's just, like, bring in a desider sphere. I have the ADS that you thought we had, but then I just bring in a disconnected decider sphere, and then the operator is allowed to be there. 01:27:07.000 --> 01:27:14.000 Okay. So you… right, okay, so you're just assuming that those… 01:27:14.000 --> 01:27:15.000 Yeah. 01:27:15.000 --> 01:27:16.000 contribute in ADS-CFT? 01:27:16.000 --> 01:27:24.000 Yeah, yeah, because I thought that's what I'm supposed to do. I sum over all the bulk things that are consistent with boundary conditions, and that includes these. 01:27:24.000 --> 01:27:27.000 Yeah. 01:27:27.000 --> 01:27:28.000 That may be wrong, indeed. 01:27:28.000 --> 01:27:35.000 Yeah, and that may be wrong. I mean, as I said, I have no idea whether this argument is correct, but yeah, I mean, several steps here could be wrong, but I feel like it would be instructive to understand 01:27:35.000 --> 01:27:37.000 which ones are, if any of them are. 01:27:37.000 --> 01:27:39.000 Okay. 01:27:39.000 --> 01:27:40.000 Yeah. 01:27:40.000 --> 01:27:45.000 Yeah, just to just to muddy the waters. If you if you have fermions. 01:27:45.000 --> 01:27:51.000 Then in these kinds of nice symmetric vacua, of course you're going to have fermion zero modes. 01:27:51.000 --> 01:27:54.000 Right, right, right. 01:27:54.000 --> 01:27:55.000 Yeah, although it may cancel between the numerator and the denominator, I'm not sure. 01:27:55.000 --> 01:27:58.000 Pacific. Oh, your path integrals will be 0. 01:27:58.000 --> 01:28:06.000 Yeah. 01:28:06.000 --> 01:28:10.000 All right, well, I should probably go have office hours, but it's great to see you all, at least virtually. 01:28:10.000 --> 01:28:12.000 hopefully I'll see you soon, somewhere. 01:28:12.000 --> 01:28:15.000 All right, thanks again for a great talk, Daniel. 01:28:15.000 --> 01:28:26.000 Yeah, but actually, I'll mention, since it's new, you guys, I'm actually going to UPenn on Monday to give the same talk, so if you really can't… they didn't have… their slot does not conflict with my teaching, so…