Okay, yes, all right, great.
Um,
Yeah, so I'm gonna talk about a number of related topics in the context of
quantum cosmology from the point of view of path integral and holography.
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.
Um, and I guess this stuff is…
confusing enough that…
questions and discussion are good.
So, by way of introduction, let me begin by saying that
you know, not everyone maybe agrees, but from my point of view, I would say we now have a fairly robust
framework for thinking about the black hole interior and the information problem.
Don't know the answer to every question, but um…
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
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.
So, if it was…
Four years ago, I would have been…
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.
But today, I'll just…
say that this is true, and um…
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
see if we can use it to learn something about cosmology.
So, unfortunately,
Uh, when we take this slogan here, and we try to…
generalize it or apply it to cosmology, we're led to a rather disturbing conclusion.
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.
Um, and…
you know, then you immediately, of course, ask, you know, what are we to make of this, right? You know, how can you account
For the richness of human experience in a one-dimensional Hilbert space, or
Alternatively, is this an argument that we don't live in a…
in a one-dimensional Hilbert space. You know, sorry, that we don't live in a closed universe.
Daniel, isn't quantum mechanics projective?
Isn't quantum mechanics projective? Yeah, so it means that it's very boring in a one-dimensional Hilbert space, right?
So the projective Hilbert space. If the Hilbert space is one dimensional, projective space is a point.
Yeah, that's correct, yes. So that's why I call it zero degrees of freedom.
Yeah. Yeah, sounds bad, I agree.
So, um…
So last year, um, several of us proposed a way to try to make sense of this problem,
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…
To deal with it, okay, whatever we do.
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.
Now, you know, this is something that actually, for example, Bohr would have been fine with, but uh…
you know, there's a…
in the years since the 1920s, probably there's been a pushback against that point of view, that the observer is just…
Made out of a bunch of quantum particles, just like everything else, and there's decoherence and so on. Now…
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.
Um, but still, you know, trying to…
it's always a bit concerning when you try to…
give yourself special treatment in the laws of physics.
Um, so… so maybe this is all wrong for that reason, I don't know.
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.
Um, the idea was that what you're supposed to do is…
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
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.
And what we argue in this paper is that, um,
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.
Um, and we had some philosophy about how that's the only… that's a…
good enough. You know, if you're an observer, you don't need semi-classical physics better than e to the minus your entropy.
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.
limit of this one, to the extent that that makes sense.
So, um…
So that was the talk I would have given last year.
But it's this year, so I won't give that talk either. Instead, what I'll…
talk about is some potential weak points, or at least questions that I felt were left over by this proposal, and uh…
In this talk, I'll try to…
address those questions that I felt were left over. And so let me say a bit about that, right? So…
So, I'll have some more concrete questions on the next slide, but first let me say the rough idea. So, um…
you know, the way these papers try to phrase the emergence of quantum mechanics in a closed universe is…
In terms of transition amplitudes between arbitrary closed universe states,
Um, and…
In the discussion, um, we both ignored the possibility of disconnected contributions to transition amplitudes between
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.
So we just essentially ignored that
possibility. For example, you could say we impose some sort of
approximate global symmetry that suppressed the…
production of the observer with the universe in it out of nothing.
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
figure out what to say about it. And so that was… that was the sort of…
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
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
How do we understand the gravitational path integral? And then the observer will sort of…
appear towards the end, uh, in… to help us…
resolve one of the issues that comes up.
And so I'll also, at the very end, I'll point out a puzzle that relates these ideas to the string landscape.
Um, which, again, is sort of very, uh, yeah, we'll see when we get to it. It's puzzling.
And so this talk is based on a paper I put out, um, maybe 2 months ago,
Uh, the same week, there were two other papers, one by Ying Zhao, my collaborator on this work, and then another one by…
some superset of the Berkeley people from the other…
Berkeley paper.
Yeah, I wouldn't say these three papers are all totally aligned. There are various points of agreement and disagreement between all of them.
I think you should view all of this as, you know,
up for debate and, uh, not settled. I'll tell you my perspective on all these issues.
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.
Um, so here, here are four questions that I would…
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…
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.
Um, but then we do want to understand how they're related.
Secondly, in the third quantized approach to quantum cosmology, you know, which Duda Coleman Giddings, Strominger,
et cetera. Um…
There's a large closed-universe Hilbert space that's spanned by alpha states, which I'll explain that in a little bit.
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?
In a closed universe.
Also, uh, you know, in perturbative string theory, right, we sum over topologies that connect
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,
compatible with the one-state…
Claim. Okay.
Um, and then finally,
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…
The one-dimensional Hilbert space problem, are they really the same? You know, what's going on? Okay, and so, um…
And what we'll see is that the answers to all these questions depend on how we interpret
the gravitational path integral, that's kind of going to be the…
the common thread going through this talk, um,
And so we'll now talk about that. But so first, any questions about the introduction before we get into it?
Everyone's happy.
Okay. Um…
So let's start with path interval.
Now, I'm going to first make an assumption that some people like and some people hate. I actually more hate it, but… but…
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.
Okay. Now, as far as we know, that's only really true in 1 plus 1 dimensions.
you know, maybe in 2 plus 1, but even that's kind of borderline, because of the sum over topologies.
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…
try to think about how much of what we learn will…
go through in a higher dimensional context where it's not well-defined.
Why do you need it? Why can't you just assume it's an effective theory, like in string theory?
Um, well, I think you have to see what I do with it before you see the answer to that question.
Yeah. But I'm going to do… like, I'm gonna do some…
Mathematical arguments here, which are really going to need it to be well-defined.
Uh, and uh…
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…
Yeah, yeah, probably that's the best thing I can say.
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.
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?
Right.
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.
Um,
that that's difficult, despite a lot of claims in the literature and and.
Well, yeah, as I said, I don't… I don't know if it makes sense.
Yeah, well…
4 4 manifolds. Well, any finitely generated group is pi one of some 4 manifold. So good luck.
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.
In 3D, you have the sum over topology problem, but not the renunalizability problem, and somehow in 2D, you can do both.
Okay.
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
And they all appear in the literature for how to relate the path integral to quantum mechanics.
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
options without being careful about it.
So, the most straightforward thing we can do is canonical quantum gravity.
Okay? So, in canonical quantum gravity,
We want to compute… so first, look at the diagram on the left here. So we have some field…
eigenstate in the past, some field eigenstate in the future.
And we want to compute a transition amplitude between them. Now,
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…
canonical gravity computes. Um, and in particular, it computes it by only summing over the cylinder topology here.
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.
Okay, that's what comes out of canonical quantization.
And so that's our first approach to the path interval.
Now, what is this picture on the right? Well…
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…
to not do it. So this is some notation of Meryl and Maxfield. So basically, this issue of the diff constraint is annoying.
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.
And so what Meryl and Maxfeld proposed is that rather than preparing, you know, computing overlaps of states that are prepared on a finite
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.
So when I write a J, I mean the sources for some fields on an asymptotic Euclidean ADS boundary.
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.
Um, I think everything I say in this language, I can also say in this language, it's just more annoying.
And so, for this talk, we'll just pay the price.
of having the negative lambda, but then in return,
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.
Okay? And that defines…
So you're summing over metrics on this shaded object?
Yeah, that's right. Metrics and field configurations, if there are matter fields and so on, yes.
What's the signature? I think you're doing Euclidean quantum gravity?
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…
Yeah, up to that point.
Right. Right? So I mean, that immediately raises all sorts of contour questions, as you just said.
Yeah, yeah, so indeed, there are contour issues, and I think the right way to say it is that, um,
I think you can…
Yeah, yeah.
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.
Hmm.
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.
Um, yeah. But I won't claim that I have a complete understanding of the contour in this.
Mm-hmm.
So, is there something that guarantees that this Scalar product satisfies the rules of a Scalar product?
Um, yeah, yeah, because it coincides with canonical quantization.
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.
Well, actually, I mean, is there some notion that.
What some components of the boundary are ingo and some are outgoing.
Um, I was gonna talk about that later. Um, that has to do with whether you gauge CRT or not.
Um…
Okay.
Uh-huh.
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.
Okay, well, in Bordism theory, you have to say.
Yeah.
you know, some some components are ingoing, and some components are outgoing, and that's true in both unoriented and oriented boardism.
Yeah, so that's this issue of the bras and ketz that Edward and I were discussing, so, uh…
So, I would say that there's a map where you can move
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.
You do have the ability to move final to initial at the cost of a CRT transformation of the source.
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.
Um, yeah, yeah.
even in unoriented Buddhism theory, you choose a co-orientation, which is an orientation of the normal bundle. Again.
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,
You no longer need the co-orientation.
So, for CRT invariant states, you drop this extra label that, um, Edward talks about in his paper.
Yeah.
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?
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.
So, the second approach, which I think was really most
Clearly articulated by Coleman Giddings, and Strominger, although certainly it had antecedents in the string world sheet and so on.
Um, is what I'll call baby universe field theory.
Um, which is, you know, this third quantized approach.
Um, yeah, and then so recently, Mel…
Okay.
Sorry, sorry, sorry, Daniel, I'm still thinking about what you, your previous slide. You could have taken disjoint unions of cylinders.
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.
When you say one boundary, you mean one connected component of the bound.
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.
Yeah.
Let's see the initial and final slices in your picture, at least, those are supposed to be closed manifolds.
Yes, that's correct. So that's why I drew them that way.
Yes, absolutely.
Yeah, that's what I mean by closed universe.
Compact metapholes without boundary.
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.
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.
So, the idea is that now you allow arbitrary numbers,
of these connected components that Greg was just asking about in the future, in the past.
So here I've drawn an example where there's two in the future and one in the past.
Okay? And then you sum over all the ways of connecting them, including both connected and disconnected contributions.
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.
And then you sum over things where each component of the bulk spacetime is connected to at least one of the boundaries.
So that's the usual, sort of, Feynman diagram type.
combinatorics, um…
And so then the idea is you, um…
you use this path integral, which we're assuming is well-defined, including the sum over topology,
Um, to define, I guess what I should call a sesquilinear form.
on the linear span of, uh, states, you know, of these objects that are labeled by sources for some definite number.
of boundary, uh, boundary components.
And so this, uh, yeah, this vacuum bubble thing here, we refer to as
following Meryl Finn Maxfield. Okay, so that's the definition.
Um, so, um…
Now, here's where Tom's question comes, which is, how do you know that this…
Sesquilinear form is actually an inner product, and in fact, in general, it's not, because, um,
It can have zero modes, so there's two comments to make about that, so what…
So, so that issue was ignored in the old literature. What Marilyn Maxfield, the whole paper, is about that issue.
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.
to define a Hilbert, an actual Hilbert space, uh, that lives on the… whose elements are these equivalence classes?
So that's the construction of the baby universe field theory Hilbert space.
So, Daniel, if you're doing anything real as opposed to…
Well, even for Maroff and Maxfield, I… I forget…
Do they integrate over the moduli?
Yeah, well, I mean, in principle, yes. I mean, in the models where they actually did calculations, no?
Yeah, so…
Okay.
They just, uh, had a topological model. I think… but I think… I think we could probably…
In some sense, put Sodschenk or Stanford in this category, and they summed over the moduli.
Yeah, so if you're… if you're summing over the moduli, this sum is not Burrell summable.
So, there's… there's another mathematical question.
of, you know, what this scalar product means.
Yeah, yeah, so, I mean, that's right, so our…
after you… I mean…
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.
Right.
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.
Okay.
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.
Um…
Okay, so in Baby Universe field theory, um, there's a particular state that is…
nice that we like to think about, which is the Harle-Hawking state, which is the state where there's no past boundaries.
So here's… here is the sum…
Contributions to the two-boundary Hartle-Hawking estate.
Okay, and then there's more, right? You can put a handle, etc., you know.
So, the norm of this state is the sum over vacuum bubbles, this allott.
Which I'll remind you was the exponential of the sum over connected, uh, spacetime, uh, closed connected space-times.
really compact connected space-times.
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.
So here's the key step of Merrillfin Maxfeld.
is we introduce an operator Z-hat,
that acts on this pre-Hilbert space by creating an extra universe with a
source configuration J.
Okay? And, um…
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.
So, if you think about it, you can…
quickly convince yourself that these C hat operators all commute with each other.
Essentially, that's because we treat the closed universe as bosons, so when you create a J, it doesn't matter which slot
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
you know, state with the product of J's afterwards.
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?
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.
I strongly advocate that everyone who works in this business adopts this notation, because otherwise people get horribly confused.
Okay, uh…
Back to where we were, oh yeah?
Oh. Yeah. Well, as you know, I wrote a paper with Benergy.
because I found the mayor of Maxfield paper. to my mind, extremely confused.
Mm-hmm.
where we. in that language we coupled a topological an arbitrary topological field theory to.
Yeah.
topological gravity by topological gravity. You just. Wait, wait topologies by some.
Yeah.
some factor to the Euler character. And you know we defined a harder locking state and a dual harder locking state.
You don't need to assume any kind of inner product, and the inner product is not Aleph.
Is it what you don't like this question?
the the the I mean, that really surprised. Yeah, that equation right there. That really, really, really, really surprised me.
But I think all our formulas are well defined.
Well, I think… I think this follows.
And that. Not really true.
I mean, I think this follows from the definitions, right? I defined the…
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.
Yeah.
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.
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…
sorting out the difference between the two proposals.
Well, we could talk about it. But. You know, I think.
I mean, I think the way I define this here, this is true by definition, right? Because I just defined it to be this,
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?
Well, look, if you write, if you write precise formulas.
Using the formalism of topological field theory and summing over the topologies, then you can write precise formulas for the Hartle-Hawking state.
But I think it… the only thing I assumed here is that… so when I said I assumed the path integral is well-defined,
And just not.
Uh, the weak version of that is that I meant that the terms in this sum
are in these two sums are well-defined, right? And then, of course, the strong version is that the sums converge.
Um,
Well, the sums do converge for general. General, semi-simple general semi. Well.
Yeah, but then since I defined this…
For generic topological field theories. Where the string coverage is.
But I defined this using this picture, right? So it's just a fact about this picture.
Anyway. Look, what do you mean? Daniel, what do you mean by the Jays?
Uh, yeah, so for me, the Jays are sources at these Euclidean ADS boundaries.
How are you?
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.
Well, you need sources on the boundary?
Well, you need a little more than that. Any Frobenius algebra associated with a circle.
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.
I'm labeling a Hilbert space that's spanned by, uh…
lists of boundary conditions.
for the bulk fields, modulo null states, and I had to assume to make a Hilbert space, that it was positive, that this
Sesquilinear form was positive semi-definite.
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…
Okay.
discuss whether the assumptions are true.
Yeah.
Okay, I insist that the formulas that I wrote with Banerjee are mathematically precisely well-defined, and are the closest.
thing I could make use I could find that could make some kind of sense of what Marolf and Maxfield were saying.
Well, I think in their topological model, this all works, right? Like, it's, you know, it's a pretty boring model, but…
There was so much even in their topological model, there was a great deal of.
There was a lot of confused statements. I'm going to be blunt.
I don't disagree, I mean…
There were a lot of really deeply confused segments.
I mean, Greg, there's a reason I'm reviewing this myself.
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.
Yeah, so, okay, so let me go on. So now you define an operator Z hat,
that acts on this Hilbert space?
are well related to the pre-Hilbert space, but then it survives the quotient, because it's by definition orthogonal to null states.
Um, uh, well, no, what you have to show is it maps null states to null states.
Um,
So this operator, uh, what it does is it creates an additional universe with this boundary condition J.
Um, and so these operators commute with each other because, um, because closed universes are bosons.
And so now here's where Greg's other question about CRT comes in, okay?
is… I assume that for every future boundary J,
there's a past boundary, J star, which gives the same path integral.
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.
And so that's the… that's the CRT operation.
So, if that's true, which I assume it is, then you further show that the dagger of a Z
is also a Z, which means that the Zs are normal operators.
They commute with their own daggers.
So they can be diagonalized, and in fact, since they're all mutually commuting normal operators, they can be simultaneously diagonalized.
Um, leading to a set of eigenstates, alpha, these are the alpha states,
that are eigenstates of all the z-hat operators at once, with eigenvalues Z alpha of j.
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.
Um, and, you know, you don't get to use the spectral theorem if everything has giant error bars on it.
Well, you have a community of C star algebra.
Well, you didn't show a C star, actually. You have a commutative algebra with a star.
Uh, yeah, yeah. Um, it would…
That's not it. That's not quite a sea star algebra.
Yeah, there's something about the norm or something, you mean?
Yeah.
Mm-hmm.
you know.
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.
is that the norm of AA star is the norm of A squared.
Uh, yeah, yeah. Yeah, but I mean, in higher dimensions, like…
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…
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.
if the path integral is not well-defined.
And, you know, and this is not like, you know,
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.
Um, for this conclusion to follow up.
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.
Well, I've already done some of the complaining now, but I'll do more of it later in the talk.
Okay. So, why is, why are these alpha states so interesting? Why are certain…
people jumping up and down, excited about them, you know, going back to Coleman and continuing
To the present.
It's because by inserting a complete set of these alpha states into one of these
Um, overlaps.
We can give it an average interpretation, right? So we take these…
this J state, and we view it as a product of a bunch of Z hats acting on the Hartel-Hawking state.
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.
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,
Over alpha, and then a product.
of these functions, these eigenvalues, that just depend on the boundaries separately.
Each boundary gets its own factor.
And then this probability distribution, P alpha, that you average over comes from the overlap between the Harle-Hawkins state and the alpha states.
When you say they're complete, could they be overcomplete or are they orthonormal, the alpha states?
Um, I was assuming orthonormal, because I'm really using the spectral theorem.
Yeah.
Uh, unclear, right?
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.
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.
Okay. And then there would be a density of states.
That's correct, then it could be an integral, that's right, yeah.
Yeah, I don't want to commit on that one.
Actually, I'll comment a bit more on that later.
Okay.
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.
Um, but I really want to emphasize that this is not an average over theories. This is a conventional,
quantum theory. It has a Hilbert space that's spanned by the alpha states. You can have superpositions of states with different alpha.
And, uh, you have operations that take you from one alpha to the other, acting on this…
Hilbert Space, um…
And in particular, if you multiply two transition amplitudes in baby universe field theory,
Um, you don't include…
Topologies that connect them.
So, like, say here I have two one-universe overlaps, J1 prime to J1 and J2 prime to J2.
Um, and I, uh, I multiply them.
Um, alright, this should have been Aleph-2 minus 2 here, but anyway, it doesn't. Sorry about that.
Um, so this is, uh, you know, this is… this one is a product of things like… sum over things like this.
And this one is a sum over things like that.
But then you don't include, you know, geometries that connect to J1 prime here to J2 over there.
or J2 prime here to J1.
Over there. Because, you know, these overlaps are just numbers, and you're multiplying them, right? So this is a factorized
quantity.
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
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.
Okay.
People fight with me about this, but I don't know why, because it's just obvious. This is the definition of this quantity.
So, Daniel?
Yeah.
In the original Coleman paper, and the ones that followed it, and also in the Saudshanker-Stanford paper,
Yeah.
Um, it's… it's kind of important that the average that you're doing is an average over…
Um, local couplings in a conventional
field theoretic treatment of a single universe.
And I don't see anything like that in this description here.
Um…
Well, yeah, so in com… because common was… in common's thing, you're not in the closed universe.
In common Sting, you're living in an asymptotic region, and you're summing over microscopic wormholes?
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.
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…
Yeah, it's a bit different than the context that Coleman was thinking about.
Well, I certainly understand that Coleman made an extra assumption.
about the wormholes being microscopically small, that's not being made here.
Yeah.
But, um, it… it also seems to me that you're…
Um, what you're getting here is… is…
quite different than has… than occurred in either of those two examples.
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?
Okay.
So that one's gonna be the third one.
Okay, good.
So just hold off on that for a sec.
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
be different between this and the previous one.
Okay? Um…
So this average here, we said that all… we said that we compute an overlap
You know, each overlap has an average interpretation, but we're not averaging over overlaps, okay, right? Like, if we square
this two overlaps is just the product, okay? There's no average over the products, right?
So, in this third approach,
We decide to switch, instead of viewing the Hilbert space spanned by alpha states as fundamental,
We say that we view alpha as
a parameter that labels…
fundamental theories.
And, um, so we…
we define some states, so now the double angle bracket are these states that live in the holographic dual at fixed alpha.
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…
numbers. So each state with some number of j's, it just gets mapped to this number.
Okay? Um…
And so then, in this language, this…
would have made it a very large dimensional space, but I think, saying its functions of J's.
Um, well…
Yeah, it's a bunch… it's a bunch of states. I mean, we…
I mean, you know, any. Any, any Hilbert space could be written as a sum over one-dimensional Hilbert spaces, right?
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?
So, I diff… I… I take…
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.
Well, I mean, if I if I consider the set of complex valued functions on a manifold.
I could say that, you know, for each point I just number.
No, no, but then you allow a superposition, then you allow super… right, I mean,
The thing is, Jay is not, like, uh…
Yeah, these are not wave functions, right? These are the states themselves. I didn't compute the overlap with anything.
Right?
Our Jays were like boundary values of fields. Aren't they points in some infinite dimensional manifold?
Um, yeah, yeah, but I'm just, uh…
See, if I put a psi here, overlapped with J, then what you're saying is correct?
But I'm not doing that. I'm saying the state itself is a number.
Not the wave function. The wave function is always the number, but the state itself is a number.
Okay?
Um, and then if you look at this, so this averaging equation that I wrote here,
In this language, what we say is that this path-integral overlap with the square brackets
is equal to an average over an overlap in a one-dimensional Hilbert space with the double angle brackets.
Okay, this is the key equation relating average holography, and uh…
And the Baby Universe Field Theory.
Okay? Um, oh, sorry, one time my phone is ringing, let me see what that is.
Um, so, um…
So this is a key equation, right? We say that instead of interpreting this…
thing as a sesquilinear form on a Hilbert space of large dimension.
We say it's an average over inner product in a one-dimensional Hilbert space.
Okay, that's what this equation says. That's this average tallography.
The average is the average over alpha.
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…
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.
With a complex conjugate, of course, right? That's what this is.
Yeah. Um…
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…
overlaps, okay, and then you put a bar.
then that's in the… from the path interval point of view, that's a transition amplitude between two universe states.
It's not the… which, as we just discussed, is not the same as the square of overlaps between one universe states, okay?
And that's the main thing that is confusing lots of people in this business.
Um, is you have to decide which interpretation you're using. Are we talking about overlaps with the square brackets?
Or are we talking about averages of overlaps with the angle… with the double angle brackets? Okay. You have to pick.
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.
So, um…
So let me just emphasize kind of how bizarre this double angle bracket one-dimensional Hilbert space is, right? So, um…
So we… I showed you that factorization
you know, the inner product doesn't factorize in baby universe field theory.
Um, but it does… it does…
factorize in a fixed alpha in the sort of… in the sort of average holography sense, where
Because since, you know, if you have two of these universes, right,
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.
Um, and so from this point of view,
Uh, the point of view of average holography, the non-factorization of baby universe field theory, you interpret it as coming from ensemble
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…
One-to-one transition amplitude.
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.
But here, you really say it arises from fluctuations in the alpha ensemble, right? That these
this single universe overlap,
you know, has substantial fluctuations as a function of alpha, okay.
So they're not quantum fluctuations, they're like ensemble fluctuations in the space of theories that you're thinking about.
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.
Uh, and so…
And you… and you want it to be a transition amplitude, so you compare it to the norms of the states,
But actually, if you sort of compute on average,
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…
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
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.
Okay, and so here's where the one-dimensional Hilbert space is, you know, from the context of the path integral.
Okay?
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.
Okay, there's the baby universe field space, Hilbert space. That's the square bracket.
And then there's the… the fundamental Hilbert space of average holography, which is one-dimensional and has the double angle bracket.
And I can give the path integral a quantum interpretation using any of those approaches, but
Depending which one I want to use, the topologies that I sum over might be a bit different.
Sound good?
Yeah.
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?
Um, in this approach, yeah, that's right.
That's right. You just say there are different theories. Exactly. Exactly right.
I mean, you know, given… you know, the spectral theorem, as as you said, you know.
automatically, you know, decomposes a Hilbert space. as a sum over the eigenspaces.
Yes.
Yeah.
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.
Mm-hmm.
with the generous degeneracy spaces which are one dimensional.
Yes.
I think I missed why it's one dimensional. Actually, for each alpha. There's only one.
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.
So, think of it this way. You can compute the overlap of alpha with any
product of the Js, just in terms of the eigenvalues.
So, if you know the eigenvalues, then you know the eigenstate.
So there's… so there's no degeneracy.
I see. Okay. So there's no degeneracy. Okay.
Yeah. Yeah, yeah, that's important, yeah.
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.
No, no, and you don't… you don't have to, right? I mean, that's one of the less…
Our Hilbert spaces are one-dimensional. We can… because we can take physical linear combinations of states with different eigenvalues.
Yeah, so I'm not saying…
But here, I think you're saying that you can't do that. You can't take linear.
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.
Okay.
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…
Just purely from a 2D point of view, why not, right? Like, that seems fine, okay?
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.
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.
Yeah.
Okay, so you have a Hilbert space, which is roughly speaking, a direct sum over alpha of one dimensional spaces for each alpha.
Uh, yeah, but no, but from that point of view, I wouldn't say it's one-dimensional.
Can I take the linear combination of alpha plus bracket alpha plus bracket beta for different out.
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.
Got it. Mm-hmm.
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.
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…
that, um, Saad, Shankard, and Stanford do.
And, uh,
For them, for them, these were, like, traces of… exponentials of Hamiltonians on one dimension.
Yeah, that's because they… that's because their universe had an ADS boundary.
So then the Hover space is not one-dimensional. But this is Saudshanker Stanford without the ADS boundary.
Okay.
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.
Okay? Um…
Okay, so that's the three ways of interpreting the path integral. Now let's try to apply that to learn some things. Okay.
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…
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.
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…
And final states. Um…
So then, um, you sum over everything that you put in between, that's the transition amplitude.
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?
Okay? And if you do,
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.
And if you don't, then you're doing baby universe field theory.
And, you know, of course the answer is that you're doing a baby universe field theory, right? You're not doing
Average holography, you know, when you…
compute S times S star, you don't connect world sheets between…
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,
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.
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.
in this, uh, version of the story.
Okay, and so this explains why the string…
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.
Okay?
So, um, good.
I think was there… there was a question in the chat, um…
Let me see, oh, uh…
Wait, uh, how do I…
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…
from the CFT, there had better be, right?
Um, so this puzzle of the one state is only there if there are no asymptotic boundaries.
Okay.
Okay, so another… another application that I like is the evaporating black hole.
Um, so this, I have to be pretty terse, sorry, I mean, we're getting pretty far behind here.
So, in Baby Universe Field Theory, the…
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.
Had some story relating the information problem to alpha states.
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…
N identical black holes in a pure state collapse, and then you evolve them to fully evaporate.
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.
Um, that looks like this, okay? And so then the Rennie entropy is something like this.
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…
Entropy of one cloud, it becomes something that's independent of N. The second Reny of the alpha ensemble.
Okay, and so from this point of view, the first black hole you evaporate gives a mixed state.
But as you evaporate more of them, then you pin… gradually pin down the value of the alpha parameters. If you measure the…
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…
story of Coleman, Polchinski and Strominger.
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.
And so the very first black hole you make is in a pure quantum state.
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…
A reason to be suspicious of this baby universe field theory.
Okay. And so, in fact, yeah, now that I presented… we've spent time presenting this, now let me complain about it. Okay.
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,
You don't know how to do the sum over topology, it's probably not convergent.
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…
argue that the alpha states really exist, and from my attitude, that's probably for the best, because, um…
you know, we probably shouldn't be able to non-perturbatively construct string theory just starting from supergravity. That seems like…
Something that would be too good to be true. Okay.
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,
Um, which don't live at the asymptotic boundaries.
they somehow live…
I don't know where they live, just…
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.
Um, and uh…
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.
Um, and so for me, that's not acceptable.
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.
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.
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.
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.
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.
Uh, giving a non-factorizing answer.
You know, and so, like I said, I mean, you can have… you can have a bulk geometry that interpolates between
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.
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.
So, my view, in light of these objections, um,
remains that we should look for some definite holographic theory with no alpha parameters, such as string theory.
Um, from which we can hope to have this baby universe field theory, or maybe conical cone gravity, emerge in some appropriate limit.
As an effective field theory with a limited regime of validity. Okay, that's what I think we should actually ask for.
Uh, you know, I think there has been…
some taking the low-dimensional models too seriously in the literature, and so I'm trying to push back on it.
here, and say what I think we can actually hope to learn.
Um, so in the remainder of the talk, what I was planning to do was argue that introducing an observer
allows for Baby Universe Field Theory to emerge from a fixed holographic theory with one alpha,
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…
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.
Um, the way…
Well, I asked a lot of questions, so I'll just be quiet. I'd like to hear the rest.
Yeah, the way we usually handle it is anybody who wants to leave can leave.
Okay.
And anybody who wants to listen can listen, and the speaker can decide when to quit.
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…
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.
Okay.
Okay? Yeah.
Um, okay, um…
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.
So I… this… I'll basically use the Meroff-Maxfeld model, where the action is the Euler character,
Uh, times minus S naught.
But then I'm going to do a generalization due to Ying and Misha and this Wang.
Um, where you add a world… some worldline matter that carries a species index i.
So somehow, if you have pure topology, it's too boring to be interesting, so you need to have some…
stuff that you can try to do physics with, and so I'll use Worldline Matterist stuff.
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.
So, in this paper with Ying and Misha, um,
We studied the states of this model with one matter particle in a closed universe, okay?
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
The one particle part of the Hartle-Hawkins state is zero in this model, because there's nowhere for the propagator to go.
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.
Um, so to fix this, I need to… the easiest thing to do is instead consider two particle states.
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?
And then there's some sum over handles that leads to some factor like this.
Okay, involving this S0.
So the…
Yeah.
Sorry, could you go back? Could you go back one slide? I know you're.
So what are these I's and J's?
Um, yeah, so it… I had… I have matter that's worldline matter.
And then each wall line carries an index i. So just think of it as I have a particle which has a species label.
Okay?
But does the world line now give a one dimensional submanifold inside this space time?
Um, yeah, but since it's topological, I'm not going to try to sum over that.
Yeah, so, yeah, so in principle,
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.
But my philosophy is that… is that if you want to do that, then I should add a mass term.
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.
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.
Okay, um…
So, uh, good. So the Hartle-Hawking state, the overlap with one, two-particle state,
is just this delta IJ, because the worldline propagator just hooks the i and the j together here.
Um, if we do an overlap between two, two particle states, IJ and IJ prime,
Then there's a disconnected contribution, which is the square of the Harle-Hawking state.
And then there's also a connected contribution from the cylinder. So I feel like this is canonical gravity here,
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,
And this has a 1, okay? So this contribution actually dominates, and this is the thing that we ignored.
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?
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.
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…
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…
So that's the thing for us to worry about. That's the thing that I was worried about.
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.
Okay. Um…
Now, um, in order to see how this, uh,
you know, factorization of the inner product. So, you know, somehow the inner product basically factorizes onto the Hartle-Hawking state.
You know, in some sense, it's some semi-classical avatar of the one-state problem, okay?
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.
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.
Right? So I need a number… so remember, these states are numbers, and I'm telling you the number here.
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
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…
this quantity where max is the unnormalized maximally entangled state, is finite in the limit of large D.
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.
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.
and try to connect to this bulk model that we just discussed.
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,
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.
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.
Okay? Um, and so this agreement continues with more boundaries, right? And so you can think of this O being like the alpha parameters,
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…
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…
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.
Okay. Um, now I want to emphasize that the one state here
It's just a complex number.
But the double bracket Ij is just a complex number. Even though even though you're writing it as a kit.
That's correct, yes. That's correct.
That's right. Yeah, that's right.
So, it's a ket in a one-dimensional Hilbert space lay the complex numbers.
That's right, yes.
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
From the Baby Universe Field Terry Hilbert space, the two-particle sector of it.
to the fundamental Hilbert space, is you take whatever state, and then you just hit it with this big projector.
Okay, so the holographic encoding map is a Rank 1 projection.
Okay, and so this is the one state right here. This bra here is the one state.
Okay, so if I write it in equations, it looks like this.
And in particular, I want to emphasize that this is not the Harle-Hawking state, right? The Harle-Hawking state
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.
The one state is this, which is strongly UV-sensitive, right? It depends on O or alpha, whichever you want to call it.
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.
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.
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
the answer in the one-dimensional Hubbard space, right? That's this thing we said before. Baby universe field theory equals an average
over… over what happens in a fixed holographic theory. Okay.
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.
And so, you know, the real question is, to what extent is the one-dimensional overlap
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.
Right, so you just have one factorizing theory, so to what extent
Is this true? Okay.
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
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…
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…
inner products agree with each other, and that's the whole one-state problem, right? It's that…
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…
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
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.
Um, okay, so what does the observer rule do? So it takes this here,
And it modifies the inner product to include an extra… to include entanglement with some external system.
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,
The reference system, I'm essentially averaging over the microstate of the observer.
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.
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…
Um, so it's basically the same result that I had before. Um, you know, I have this…
three-party thing from the cylinder, and then I have this delta IJ and IJ prime from here.
But now I have to add an amplitude lambda,
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,
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.
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
When you pay the price of nucleating an observer out of the Hartel-Hawkins state.
Okay. Um…
So, um…
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,
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
Observer production all the time.
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
For reasonable-sized observers, the transition amplitude is actually dominated
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,
Uh, and then re-fluctuate a new universe with the system in the final state.
Okay? Alright.
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.
Okay. Um…
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.
Okay. Um…
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…
have to actually try to check, but there are 15 topologies that contribute to the square of the inner products.
Okay? Um, if you do the calculation, which I will spare you, the result is that using the observer rule,
Um, all of the non-factorizing contributions to this are suppressed by the observer entropy.
Which means that, um,
The ensemble fluctuations, you know, that tell you the difference between being at fixed alpha
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.
Um, and so when you use the observer rule,
Then holography at fixed O, or fixed alpha, which is the thing that I think should actually be a good model for higher dimensions,
Um, actually agrees with the baby universe.
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,
And baby universe field theory. Okay.
Uh, let's see, I'll see somebody said something, uh…
Can you at least explain the last two out of the 15?
2 out of the 15 terms? What, you mean, uh, these two terms?
Yes, I want to get a feeling of, uh…
of the schematic and how you generate these terms.
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.
And then when you make an observer, there's the… there's a lambda factor whenever you fluctuate an observer out of nothing.
And then these deltas are the ways the world lines connect between the different geometries, okay?
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.
Yeah.
Okay, okay, thank you.
Yeah, um…
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.
So for me, that's the most interesting, or at least one of the most interesting consequences of this discussion.
Okay? Um, so on the other hand, this baby universe field theory, so that's good, we like it, okay, great.
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
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.
So, um…
So, of course, we could just hope that
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.
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.
Now, it doesn't work phenomenologically, you know, in some sense, essentially because of this problem.
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.
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.
to what happens with the spectral form factor at long time in this work of Schenker and Stanford and collaborators.
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.
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?
And so I think it's sort of…
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…
Averaging over J, and averaging over J is good, unlike averaging over alpha.
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.
Um, which is good, because I have office hours in 7 minutes.
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
Baby universe field theory and ADS-CFT. So this is something for us to all
be puzzled about and try to understand, okay? I'm not sure I understand the resolution myself.
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.
And I'm interested in the sector of this theory, where there's one spatial ADS boundary.
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,
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.
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.
That may be a weak point in the argument, but I'm going to assume that.
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.
Um, I'm going to now try to convince you that's not the duel, okay?
Um, so…
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.
So Q, integrated over spacetime. So if there's no observer at the place where Q is currently located, you get 0.
If there's an observer who's in ADS, you get zero, and if the observer is into sitter, you get 1.
Okay, so this is an observer that tells you… an observable that tells you whether you're in Decid or ADS.
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.
Okay, because if there's no observer, then we don't see anything.
Um, so here's what the calculation looks like. So, when I compute the expectation value of having the observer be into center,
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.
Because if I didn't have this bubble here, then Q would be 0, because the observer wouldn't be in decision.
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.
So there's an observer, definitely, in the middle of ADS and all these contributions.
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.
And now we just compute the ratio. What is it?
Well, the ADS part just factorizes out and cancels from everything, so what I get is the desitter sphere,
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.
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.
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.
And we made a state where there's definitely an observer in ADS, and then we use the path integral to tell us
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.
Um, and so, um, so somehow the…
That means the Boltzmann brain problem hasaded ADS-CFD. You know, we take the dual…
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.
Uh, and this is somehow saying that that's actually… yeah, we're gonna live there, we're not gonna live in ADS.
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.
Okay, let's thank Daniel.
So I can answer maybe one more question before I have to do office hours, if anyone wants to…
Sorry, the last part was a bit rushed.
Yeah.
What if you had many desidera vacua? Would you then?
Select the one with the largest cosmological constants or something.
Yeah, I think you probably do, yeah.
Yeah. Sorry, the smallest, the smallest, because you want the biggest consideration.
You want the… you want the biggest decider entropy.
the smallest. Yeah, there we go. Okay.
Yeah, the biggest consider entropy, yeah.
So so so is this a solution to the cosmological constant problem when it's small?
Well, I think that's kind of what Coleman was trying to do, right?
Yeah, yeah, I think this is kind of what Komen was saying, yeah.
Yeah, that's a good point. That's right.
Right.
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…
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.
Daniel, I don't think you're taking into account the fact that
All of these configurations have to have asymptotically anti-decidor boundary conditions.
Um, they do. Every one of them has this blue line, right?
Yeah, yeah, so then this thing that's sitting
Partly in decider space is going to look like a black hole.
from the ADS point of view.
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…
So, so then you have to keep it.
Oh, it's a… it's a disconnected topology.
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.
Oh, okay.
So then…
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.
Okay. So you… right, okay, so you're just assuming that those…
Yeah.
contribute in ADS-CFT?
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.
Yeah.
That may be wrong, indeed.
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
which ones are, if any of them are.
Okay.
Yeah.
Yeah, just to just to muddy the waters. If you if you have fermions.
Then in these kinds of nice symmetric vacua, of course you're going to have fermion zero modes.
Right, right, right.
Yeah, although it may cancel between the numerator and the denominator, I'm not sure.
Pacific. Oh, your path integrals will be 0.
Yeah.
All right, well, I should probably go have office hours, but it's great to see you all, at least virtually.
hopefully I'll see you soon, somewhere.
All right, thanks again for a great talk, Daniel.
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…