Daniel Harlow on Quantum Gravity, Black Hole Information, and the Holographic Principle

Daniel Harlow

Well, hi Sean. Thanks for having me.

Daniel Harlow

I would put it this way: we know a lot about the world already, from gravity and from quantum mechanics, and it's very hard to write down a theory that's consistent with the data we already have. So most ideas are ruled out almost immediately — including many of the ideas that I work on, because it's very hard to write down a theory that's consistent with everything. Sometimes I work in a world with 1+1 space-time dimensions instead of 3+1. Sometimes I have the wrong sign of the cosmological constant.

But the reason I nonetheless feel progress is possible is that there is something universal about gravity that is not so much the case for the other forces of nature. If you look at the standard model, some of the fields feel electromagnetism, some don't; some feel the strong force, some don't, and they feel it in different ways. If you wanted a very simple overarching description of that, it seems hard. People try, but so far we haven't succeeded. The standard model of particle physics is kind of a smorgasbord — I think there are 19 dimensionless parameters that you just fit to data, plus more discrete parameters, like the representations that things transform in.

Gravity doesn't seem to have all that mess. That goes back to Newton and Galileo and the equivalence principle — everybody basically feels gravity the same way. And that leads to a generality of arguments about gravity. For example, I often talk about black holes. The only reason black holes can exist is because everything feels gravity in the same way. If you had a particle that didn't feel gravity, it could escape from a black hole — gravity is what's pulling stuff in, but if you could be neutral under gravity, you'd get out. So there's a kind of inevitability to some of the features of gravity. We try all these unrealistic models — the wrong number of dimensions, the wrong cosmological constant, too much supersymmetry — and somehow the gravitational part of all those theories looks similar even though the other details look different. That gives us confidence that whenever we do find the theory that's actually consistent with everything and also includes gravity, the things we learn from these other models will carry over, and maybe even help us find it.

For me that's the hope. I'm trying to find that theory, and I have to practise a certain amount of humility, despite being a theoretical physicist, because this problem has been around for a hundred years and it hasn't been solved. Who am I to think I'm going to solve it? Probably I'm not. But I feel like every day I learn a little bit more, and each year I know things I didn't know the previous year. They don't feel arbitrary; they don't feel like artifacts of the particular model I was studying — or at least I try to focus on the parts that feel more general.

Daniel Harlow

Right. In some of these models of gravity it does go like that. For quantum gravity with negative lambda and too much supersymmetry, it's dual to some fairly conventional quantum field theory that has a Lagrangian, and you can quantise it using essentially high-school physics if you're good enough at it. But I'm actually somewhat sympathetic to the broader thrust of your question — that maybe that's a special feature of those models which isn't true more generally.

Daniel Harlow

This gets into one of the questions a lot of us are thinking about now. If you do quantum gravity, there are two things you want to think about in terms of making contact with the real world — at least the two most obvious things — which are black holes and cosmology. Quantum gravity should be important inside a black hole, and maybe also outside if you look at it long enough and want to understand the evaporation process. And it should be important near the Big Bang, when the universe was very dense, when we ask where the initial conditions came from.

Something we've learned again and again over the last fifty to a hundred years of trying to do quantum gravity is that black holes are easier than cosmology. For black holes, you can sit outside and drop things in, and naively nothing comes out — but once you include quantum mechanics, something does come out, the Hawking radiation, and you can see what comes out. Those are fairly conventional experiments, the same kind people do at the Large Hadron Collider; they just cost a bit more, so we haven't done them yet. But in cosmology you're always part of the system. It's not like you're on the outside looking in. You're in the system; the system is interacting with you all the time. It's not even clear what it would mean to have something outside the system.

That was an issue going back to the early days of quantum mechanics. If you look at the discussion of Bohr and Einstein, or to pick someone widely respected, Landau — right at the beginning of Landau's quantum mechanics textbook he says quantum mechanics is a theory for a quantum system interacting with a classical apparatus or observer. Which is the same thing Bohr says. And I would say — though you might disagree — that I don't really think quantum mechanics makes sense without that external observer. I don't think it's really science; it's mathematics, but it's not science. For me quantum mechanics is an emergent theory in the limit where you have this external observer and apparatus, arbitrarily big and slow and careful, and that's who gets to test the probabilistic predictions of quantum mechanics to arbitrary accuracy. It's the person outside the system. That relates to all these puzzles about Wigner's friend.

In cosmology you don't have that crutch. You don't get an arbitrarily big, cold, slow, careful observer outside the system, interacting with it only if and when it chooses, publishing papers in journals somewhere outside the universe. So it's not really the setting where quantum mechanics makes sense. Until recently it sounded a bit like philosophy — which is not necessarily a bad thing; I was a liberal-arts major and read philosophy too — but you feel better if you have equations to back up your philosophy. And until recently I didn't have equations to back up this idea that the observer being part of the system in cosmology is really different from the observer being outside in the black hole, or the LHC, or any other situation where we test quantum mechanics.

Daniel Harlow

Let me give a little background, going back to black holes for a minute. Fifty years ago Hawking proposed this amazing black hole information paradox. He just took the laws of physics as best we understood them in the early '70s — and more or less as we understand them now: general relativity interacting with quantum field theory, with the gravitational interaction very weak, so you can work in an expansion in weak gravitational coupling. Hawking argued that this seemingly obvious way of combining gravity and quantum mechanics leads to paradoxes when you apply it to black holes.

In particular, a black hole behaves a lot like an ordinary quantum system with a finite number of degrees of freedom — the number being given by the area of the horizon divided by Newton's constant. It has an entropy and an energy. If you throw things in, they equilibrate, the same way things equilibrate if you throw them into an oven. It even radiates thermally, the way an oven would. It obeys the laws of thermodynamics. That all sounds good, except that if you really buy this approximation of quantum field theory with gravity treated weakly, it tells you that's fake — because although the black hole behaves as if it has a finite number of degrees of freedom, it actually has an infinite number. You can fit an arbitrarily large amount of stuff inside. So if you count the ways of preparing the system on some time slice, you get too many.

Hawking quantified that beautifully. If you let the black hole evaporate, its area gets smaller and smaller, so its number of internal degrees of freedom seemingly gets less and less — if the entropy is really counting degrees of freedom, as Boltzmann told us it should. Eventually the black hole is gone and it seems there should be no degrees of freedom left. But then where did the information go about how the black hole was created? If you burn a piece of paper in a well-isolated oven, what you wrote is still there in the oven. But Hawking showed that this approximation of gravity weakly interacting with matter gives the answer that the information is just gone. So somehow, if that information had to go somewhere, the black hole still has degrees of freedom somewhere. In the relativity community, people usually say it left behind a baby universe — the interior pinched off from our world when it shrank to zero size, and that's where the information is. If you talk to your old colleague Bob Wald, he'll tell you that's the resolution of Hawking's paradox. But then why did the black hole behave as if it only had a finite number of states, if it could actually store an arbitrarily large number by sending it off into the baby universe?

This drove many people in the field over the last fifty years, and over the last ten years it's a puzzle we've actually learned quite a bit about — what I view as positive progress. One always has to be careful about claiming progress here, because we don't have a theory of quantum gravity, at least not one consistent with everything. So when I say we learned things, I mean it's more of a mathematical thing. Hawking said there are three things you might want: that the black hole has a finite number of degrees of freedom; that it evolves in a way that preserves information, which in quantum mechanics we call unitarity; and locality, meaning that what I do here in this room can't instantaneously affect what's going on in your office in Baltimore. Hawking said you can't have all three. That's how I like to think about the paradox — three things you might want, and you get to pick two.

Daniel Harlow

He might have said you give up unitarity. The more modern version is Bob's, where you give up the finite entropy — you say there are so-called remnants, and the baby universe is a kind of remnant. But the view many of us have gradually converged on over the last twenty years is that you have to give up option three: locality. The slogan is that spacetime is emergent. Spacetime tells you where things are and when, and when we say it's emergent, we mean that notion is only true in some approximation, in certain situations. It's easy to say that — it could have been said fifty years ago. What's newer is that we've developed the mathematics of emergent spacetime over the last ten or fifteen years.

So we're feeling pretty good about ourselves. We showed at least that it's possible to have one and two, and then three-star — where Hawking was right that you can't have one, two and three, but you can have three-star, which says it's not local, spacetime is emergent, but in order to detect that non-locality you'd have to do something exponentially complicated in the entropy of the black hole. We've never done anything like that. We've tested locality a lot, but not by doing something exponentially complicated in the black hole entropy. So maybe that's okay. And we showed you can have a mathematical model with one, two and three-star, and they're not in contradiction.

Daniel Harlow

One should always be careful about declaring victory on fifty-year-old problems, so I don't want to declare victory. The precise claim is that Hawking said you can't have one, two and three, and we showed you can have one, two and three-star. Now there are two remaining questions. One is: how bad is it to only have three-star and not three?

Daniel Harlow

Yes, unless you do something really complicated, and then they can be large. One could worry that leads to other problems. You have to construct a theory that realises this possibility and convince yourself it doesn't have other fatal problems. We have constructed such theories, but they're not very realistic, so you could worry that making them more realistic would introduce additional problems not present in toy models. I can't rule it out. I bet against it, but I can't rule it out. That's a mathematical question. To really declare victory, there's also the physics question: even if I have a theory that works, that doesn't mean it's right. Eventually we'd want to use it to predict something we can actually test, and we're far from that currently.

Daniel Harlow

That's one way of thinking about it. There are several on-ramps to the idea. My favourite is quantum-mechanical: thinking about how the approximate quantum mechanics of the black hole interior emerges from the fundamental degrees of freedom. I phrase that as a relation between Hilbert spaces, so it's phrased in the language of quantum mechanics, and wormholes don't appear. There's another approach, which I'd say is less fundamental but consistent with it, where you base everything on the gravitational path integral.

Daniel Harlow

Of course. Quantum mechanics was originally formulated in a somewhat opaque way, where you take the things you observe — the locations of particles, how fast they're going — and realise them as really big matrices, which you can multiply and add. Those are Heisenberg's ideas, and I'd say Schrödinger is implicitly doing that too. There's nothing wrong with it; in fact I like that way of thinking because it makes the physics clearer, but it's less intuitive than Feynman's path integral approach. Instead of talking about big matrices, you formulate the calculation — say, the probability that a particle here at time t1 ends up there at time t2 — by considering all the possible trajectories the particle could have taken, and summing over them with some carefully chosen weight. The weight isn't so bad; it's something you might have guessed from the classical physics of the particle, and Feynman showed that guess is correct.

So in ordinary quantum mechanics these approaches are equivalent — there's no big mystery, you can pick either one. When we teach, we usually teach both, because some problems are easier one way and some the other. But in gravity the situation is more mysterious, because the path integral approach is not something you can derive from the big-matrix approach — usually we call that the canonical approach. Somehow the path integral is stronger. It knows things the canonical approach doesn't. For example, it knows the number of degrees of freedom of a black hole. Gibbons and Hawking showed in the late '70s that by summing over all the geometries to get from the initial state to the final state, if you sum in the way that seems most natural, the path integral actually knows how many states the black hole has — even though if you tried to count those states in the canonical formalism, it wouldn't work. That was there in the '70s, but it's been even more appreciated in the last five or ten years, because people keep finding more stuff it knows. One exciting thing, from 2019, was realising the path integral knows the evaporation of the black hole is a unitary process, that information gets out — which is exactly Hawking's paradox.

Now, for me, my approach to the gravitational path integral is that I think of it like the oracle at Delphi. It's something you consult and it tells you the answer, but it's not clear you always understand the answer you're given. I heard a talk by Don Marolf last summer with a nice historical analogy. In the second Persian invasion of Greece, Xerxes was coming to attack Athens, and the Athenians went to the oracle at Delphi for advice. The oracle first gave a very depressing answer — basically, Athens will be destroyed, you're going to lose. They weren't happy, so they asked for something more, and the oracle said the heart of Athens would be protected behind wooden walls. There were two camps on what that meant. One said the Acropolis originally had a wooden hedge around it, so the defenders should go there and hold the line and stop Xerxes. The other — I think it was Themistocles — said the wooden walls are ships, and they should go out and attack Xerxes at sea. The second interpretation was correct. The defenders who stayed in the Acropolis were all killed, and Athens was burned to the ground, but the navy won a great victory over the Persians at the battle of Salamis. Misinterpreting the oracle proved fatal. With the gravitational path integral it's the same — you have to know the lesson you're learning from it. So I always want something more fundamental, from which I can realise the consequences of the path integral, rather than having it be the end of the story.

Daniel Harlow

Yes. In ordinary statistical mechanics, if you want the entropy of an ideal gas, you have to count the states — you treat it as non-interacting particles and count, which is what we teach undergraduates. But somehow the gravitational path integral knows the entropy of a black hole without doing any state counting. It just gives you the answer, and the answer is correct as far as we can tell — consistent with what we get from string theory and AdS/CFT and holography, and with Hawking's indirect arguments from thermodynamics. But it never counts the states. That's why the approach of trying to understand how the interior degrees of freedom are realised inside the fundamental degrees of freedom — treating the states as actual states — feels more fundamental to me than the oracle approach.

Daniel Harlow

Yes, that's what I'd say. Maybe not everyone in the field would agree with me on that.

Daniel Harlow

Of course I'm right and they're wrong. Obviously.

Daniel Harlow

I'd certainly say we learned something — that you can have one, two and three-star and they can be self-consistent. This complaint could be made about quantum gravity altogether. Let me mention two historical examples, one very well known and one less so. Einstein, when formulating special relativity, spent a lot of time thinking about trains that move close to the speed of light. We still don't have trains that move close to the speed of light — especially not in the United States — but it was a good thing to think about, because it was the situation where the paradoxes were most stark. There was a tension between Newtonian classical mechanics and electricity and magnetism. You could see the paradox in some mathematical way — you try to write down the Lorentz transformation of Newton's laws and it doesn't work — but that's too abstract. Einstein found a way to put it right in front of you, in terms of something you could imagine doing. He knew there weren't going to be trains near the speed of light, but he understood that if there were, you'd have a problem, and not a small one — a big problem that needs an answer. Once you understand the problem well enough, you can find a theory that fixes it, and the hope is that theory then makes other predictions, more terrestrial in nature, that you could actually test. When you're trying to come up with the theory in the first place, an undue focus on what can currently be measured is not constructive. That comes later, once you understand the structure better.

The other example is one of my favourites. Before Maxwell invented electromagnetism, he won an essay prize competition in Cambridge, the Adams Prize, and the problem he studied was the stability of the rings of Saturn. This was a well-known problem, because the rings had been observed for more than two hundred years and were still there. If you try to have some giant disc just hovering around the planet, the dumbest things you can think of for how the rings work are not going to be stable — they'll collapse into the planet or dissipate. I recommend reading the essay, especially the introduction; it's quite fun. He begins by talking about how this is a completely useless problem, because astronomy is for navigation and agriculture, and the rings are so far away they're not helpful for those things. But then, on the other hand, we have to know: does classical mechanics work near Saturn? If it doesn't, that's huge — what does work near Saturn? He says that from this point of view the rings of Saturn are the most remarkable astronomical phenomenon, with the possible exception of the spiral nebula.

Daniel Harlow

Then he writes an eighty-page paper going through all the different options for how the rings could work, rejecting almost all of them on the grounds that they're not stable, with the only option remaining being that the rings are composed of individually orbiting satellites. Now we know that's the answer, so it doesn't sound surprising, but it's not clear that should be stable — it's a many-body system. What he does is show that if you perturb the rings, you get dissipative waves that go around the rings but dissipate, so the structure is stable. And do you know how long it took for that theory to be confirmed?

Daniel Harlow

More than a hundred years, because the particles are pretty small. We had to send a satellite to Saturn and shoot a laser back to Earth to measure the sizes of the particles. But that didn't stop Maxwell from thinking about the problem. And what if he'd found there wasn't an option — maybe we'd have had to modify classical mechanics.

Daniel Harlow

This is before he did all the electromagnetism stuff, or the thermodynamics stuff.

Daniel Harlow

So we're feeling good about ourselves — we learned these new things about black holes, whether in terms of the path integral or the more quantum approach. And as I said, there are two things you do with quantum gravity: black holes and cosmology. We think we learned something about black holes, so onward, brave soldiers, to cosmology. And almost immediately we found ourselves in hot water. Because if you take these ideas that work so nicely for black holes and give us all the answers we like, they give us something in cosmology that, when you first think about it, sounds totally crazy.

The path integral can count the number of degrees of freedom of a black hole. So let's use it to count the number of degrees of freedom for the whole universe. Why not? There are various approaches in various contexts where you can try to do that counting, and the answer is always the same. The answer is zero.

Daniel Harlow

Right — and it's not infinity, so zero. There are various versions of the argument. The most flippant one, and the least precise, but the easiest to say: I mentioned holography. Inspired by the quantum mechanics of black holes, holography is the idea that if you're doing quantum gravity in a region of spacetime, the fundamental degrees of freedom really live at the boundary of that region. This is inspired by the fact that the black hole entropy goes like the surface area of the black hole, and all these toy models — often in the context of AdS/CFT — are in some sense holographic. So let's take the principle and apply it to the whole universe. Holography says the fundamental degrees of freedom live at the spatial boundary. When I say cosmology, I really mean you're living in a closed universe — one with no spatial boundary. For example, space could be a big three-sphere.

Daniel Harlow

That's right, but it's not so much about the spatial curvature, because we don't know the large-scale structure of the universe. If there's eternal inflation going on, then what we measure is an accident of the bubble we're in. Even in de Sitter space you can choose to slice it with closed slices, or flat slices, or open slices. Conceptually the closed ones are easiest to think about, because you don't have the IR divergences you'd worry about in the other two cases. So for now, when I talk about quantum cosmology, I mean I'm in a closed universe, and then there's no spatial boundary. If holography says that's where the fundamental degrees of freedom live, then no spatial boundary means no fundamental degrees of freedom. Somebody probably made that argument at a bar twenty years ago, and everyone laughed. If that were all we had, it wouldn't be enough to convince me. But what we learned over the last five years is that all these ways we now have of doing calculations about black holes back up the argument. They quantitatively tell you, yes, indeed, that's what's going on.

Daniel Harlow

In some sense. But if you take the laws of physics as we currently understand them and study them in a closed universe, there are definitely infinitely many degrees of freedom.

Daniel Harlow

That's exactly right. That's why this is so shocking. We could very well be living in a closed universe — it's consistent with all the data we have — and how does the richness of human experience fit into a theory with zero degrees of freedom? That's the real question. There are various responses. The first is: there must be a mistake somewhere. Absolutely could be. But then you immediately have to explain why that doesn't kill all the stuff we thought we learned about black holes, so I don't necessarily like that option. I try hard to find the mistake, because I should, and I haven't yet. Another possibility is to say this is just an argument that we don't live in a closed universe. But then the only response I can have is that it's pretty wacky for us to learn something about the global structure of the universe without even looking out the window — never mind building a telescope — just waking up in the morning and saying, guess I don't live in a closed universe. You might say the arrow-of-time problem has some aspect of that too, and that's something we're worrying about.

More conservatively, if you go by the usual philosophy that we shouldn't be able to tell what's going on arbitrarily far away just by looking around the room where we woke up, and you want to say the argument isn't wrong, then the only option remaining is that somehow the richness of human experience is consistent with the universe having zero degrees of freedom. And I'm currently on that team. When I'm counting degrees of freedom, I'm using the usual quantum count, which mathematically is the dimension of the Hilbert space.

Daniel Harlow

Well, I'd say it's one, because for qubits I usually define it as the log base two.

Daniel Harlow

That's a little bit subtle. We have to decide whether we're counting discrete or continuous degrees of freedom. Roughly speaking I'd say they both have one, but there are different kinds — continuous and discrete. When I say that, it's because I think a continuous particle really has a cutoff or something.

Daniel Harlow

That's correct. Thermodynamically we'd say that's one degree of freedom — when you do the thermodynamics of a gas, continuous variables count as one degree of freedom in the equipartition theorem. But at the practical level, when we say something that sounds technical like the dimensionality of Hilbert space, we can think of it as how many completely distinguishable quantum states there are in the system. The best way to say this, Sean, is that it's the number of distinct states of the universe. And I'm saying the number is one. There's only one possible state of the universe. It could not be in any other state.

Daniel Harlow

That's why you have to think more about what you really mean by state. The key question is what happens when you do a measurement. In the quantum mechanics I learned, when you do a measurement the state changes — collapse of the wave function, projected onto the result, now in a different state. You don't have to say it that way; I'm happy to say you got entangled with the apparatus. But that's still a different state. So being a many-worlder won't get you out of that one.

Daniel Harlow

Yes. Although, as you said, because the energy is zero in a closed universe, the time evolution is not so interesting.

Daniel Harlow

That one has a more conventional answer: what do you really mean by time? You don't mean some God-given time ticked by a divine clock. You mean the time ticked by your watch, and that has evolution even in a closed universe. So that one usually isn't so mysterious. But saying there's only one state, that no other state is possible —

Daniel Harlow

There's no question to be answered about which state you're in, because there's just one.

Daniel Harlow

It sounds bad. As we go on, what I say becomes more speculative and idiosyncratic — now we're getting into the regime where there are lots of people with lots of opinions, and I'm just giving you mine.

Daniel Harlow

I'd say the reason we got this answer of one state is that we're trying to apply quantum mechanics to the whole universe, and that's only correct if there's an observer outside the universe looking in. That's who quantum mechanics is for. The one state is telling you there isn't an observer outside looking in for a closed universe. You just have whatever is in the universe, and you have to make a theory out of that. That can be you, me, all of us, the apparatus — everything is part of the system. We need to develop a theory that lets us do physics that way, and that theory won't quite be quantum mechanics. It'll approximately be quantum mechanics in some limit we can discuss, but it won't be quantum mechanics on the nose.

Daniel Harlow

There are models that have been proposed, where we can do calculations. One aspect I like a lot, though it's controversial, is the question of how good the emergent description of the cosmological world needs to be. What are the errors in that description? A naive guess — if you talk to a randomly chosen theoretical physicist — is that the mistake should be something like e to the minus the entropy of the de Sitter space we currently find ourselves in. It should be exponentially small in Newton's constant, in cosmological units. The entropy of the de Sitter space we live in is about 10 to the 120, so we're talking about e to the minus 10 to the 120 effects — the usual scale we'd guess for non-perturbative quantum cosmology effects. But the models we have, where we try to build in a rule for accounting for the observer as part of the system, don't work to that level of accuracy.

And they can't, for a very interesting reason. Imagine we have this theory with one state, and we take the limit of G Newton going to zero — Newton's constant, the strength of gravity. We gradually turn off gravity. Make it half as much as it is currently: the Hilbert space dimension is still one. A quarter: still one. Divide it by ten trillion: still one. And then somehow, when it gets exactly to zero, you go from one to something big. There's some crazy discontinuous thing that happens when G goes to zero. To me that's the really fun thing about this problem — it persists all the way to turning off gravity. My interpretation is that this problem is really getting at foundational questions in quantum mechanics, problems that exist even when G Newton is zero. The way that works out in our models is that the entropy of the observer becomes important.

I'll do a sociological experiment with you today, Sean. I've observed a bimodal response from physicists to the following statement, and I think I can guess which camp you're in, but let's try. I claim that science as a concept is approximate, with a lower bound on the error given by e to the minus the entropy of the observer.

Daniel Harlow

To motivate this a little: you can't even write down a number to better precision than that, because you just don't have enough bits. And you can't believe anything to better accuracy than that, because it could be a figment of some crazy fluctuation in your brain — your memories can just fluctuate into totally different configurations at that scale. The standard response is: fine, but just make the observer bigger. When you're not doing quantum cosmology, that's fine — the observer is outside the system, you make them bigger, slower, more careful, and there's a limit where all these effects can be removed. But in quantum cosmology there isn't. So my claim is there's a fundamental limit. When you said the best possible science, you were taking the observer to infinity, and my comment is that you're not allowed to do that in quantum cosmology. So we've got to deal with the possibility of your memories fluctuating in weird places, and the ink in the journal you're reading rearranging to say something else.

Daniel Harlow

I am including us. It's crucial to include us as part of the system — for me, the whole thing is that we are part of the system. It's just a question of the rules. The rules I'm thinking about do treat us as part of the system. The question is whether they treat us the same way they treat everything else in the universe, or whether we get some special treatment because we're the ones doing the science. My current understanding is that we should be treated differently, because we're the ones doing the science. And there's a quantitative way of doing that: I keep the observer as part of the system, but in doing calculations using the one state, I always subject the observer to a decohering channel that averages over their microscopic degrees of freedom to make them classical. Not because I think the observer actually is classical, but because I don't think the concept of an observer makes sense if you don't do that.

I think this is related to the Wigner's-friend stuff, where from a purely Everettian point of view nothing ever happens — the measurement result never actually happens. The system just gets entangled and it could be unentangled, and there's no sense in which the definite outcome was selected. And since you described yourself as having an Everettian heart, I can't resist one snide comment. I think a true dyed-in-the-wool Everettian does not think quantum mechanics happens in Hilbert space — I think they think it happens in a vector space.

Daniel Harlow

Yes, but it has an inner product, and the question is whether we need the inner product or not. Why? You're just evolving unitarily — and we don't even need the word unitarily; you're evolving linearly, evolving invertibly. Why not just evolve invertibly in a vector space? Why bother talking about an inner product?

Daniel Harlow

Why? Who cares? Does the inner product mean anything?

Daniel Harlow

I only know one thing that we use it for, which is the Born rule.

Daniel Harlow

Everettians don't believe in the Born rule.

Daniel Harlow

No, but you can't.

Daniel Harlow

Other people have too, but I think they're always sneaking it in the back door, by using the inner product. As soon as you use the inner product, you've introduced an additional axiom.

Daniel Harlow

I think it's not hard to argue that if quantum mechanics is going to have a probabilistic interpretation, then the Born rule is the one it has to be.

Daniel Harlow

But that's not the question. The question is why it would have a probabilistic interpretation at all. If you just have the Schrödinger equation and nothing else, then I'd say you have invertible evolution in a vector space, and there's no additional structure on that vector space that's physical. I think that's wrong — I think quantum mechanics has an additional axiom that's not the Schrödinger equation, which tells you there's a physical inner product, and the reason it's physical is the Born rule. I don't think that can be escaped. And it immediately begs the question: the Born rule defines these real-valued probabilities with infinite precision. Who gets to use those? I'd say it's the external observer looking at the system from outside, because the person inside the system, with a finite number of degrees of freedom, does not deserve those real-valued probabilities. They can't make use of them — or only approximate use.

Daniel Harlow

But in quantum cosmology it's necessary. When you're not doing quantum cosmology, you have this crutch of an arbitrarily good observer outside, and you have to free your mind from that to do quantum cosmology. I think quantum mechanics crucially relies on that crutch, so what we're doing in quantum cosmology is not going to be the standard quantum mechanics we learned in the textbooks.

Daniel Harlow

I do want to emphasise something I didn't finish saying earlier. In these models — where you have the one state, and then use this decoherence rule where you subject the observer to a decohering channel on the pointer basis — what happens mathematically is that effective field theory emerges, but only up to errors suppressed by e to the minus S observer. The question is whether that's good enough. Are we willing to accept the emergence of semiclassical physics only up to errors of order e to the minus S observer? It sounds totally nuts, a very speculative and radical thing to claim, but it's the least bad of the options I currently see. And in some sense I think it's poetic — why should science be better than e to the minus S observer? Normally we don't have to worry about it, but in quantum cosmology we do, and that's exactly what the one-state rule is giving us: effective field theory up to e to the minus S observer.

Daniel Harlow

Yes, that's right.

Daniel Harlow

In the models it's well defined. As a general framework, maybe less so — but there are different standards of evidence. Is QCD well defined?

Daniel Harlow

I'd say the rules are well defined here in the sense that I can do precise calculations and get answers. What's approximate is the semiclassical physics that emerges from it. But the actual calculations are numbers.

Daniel Harlow

Not fundamentally. But I still get the same result of the experiment, up to e to the minus S observer.

Daniel Harlow

Yes, but it's observer-dependent. For me I always believed the collapse of the wave function is subjective — that's the obvious resolution of Wigner's friend. My rule is that the wave function collapses for you when you learn the result of the experiment. It's theoretically possible that even after you learn the result, I from the outside could recohere you and the apparatus and everything. But still, for you, the result happened when you learned it, because you became irreversibly decohered with it, in the sense that any operation which recohered you would also disrupt this convenient fiction that you have free will and are forming memories that make sense. This idea that there's a shared reality, where we do experiments and write papers and agree — that's all based on mutual decoherence we share together. When you have a world where stuff can be undone, these ideas are not going to be infinitely precise. So I feel the traditional formalism of quantum mechanics is too precise to account for a world where nothing really precisely happens.

Daniel Harlow

I certainly believe clocks are ambiguous.

Daniel Harlow

Roughly speaking I agree with that, and that's why I think it's important that it's the observer who's treated specially. In our paper we had three axioms for observers. The first is that the observer, semiclassically, should have some finite number of degrees of freedom, and they can do physics up to e to the minus that number. The second is that they should be classical, in the sense that they have a stable pointer basis, stable under interactions with the environment — I don't think it makes sense to have an observer that's not approximately classical. And the third is that observers are never pure. I don't mean that in the Catholic sense — I mean they're always entangled with the rest of the universe. You can't have an observer sitting in a pure quantum state by themselves; that's not how you get observers. They're produced out of the world they live in, entangled with it, and because they're classical, the pointer basis will be the basis where their density matrix is diagonal.

Daniel Harlow

No. Not by the rules we were applying. The calculation showing you get effective field theory up to e to the minus S observer uses all three of these assumptions.

Daniel Harlow

Roughly speaking, you take the quantum mechanics you like, but whatever state you have, you hit it with a decohering channel in the pointer basis of whichever observer is doing the physics, before you compute anything else.

Daniel Harlow

That's why it's radical. If I said that to Bohr, he'd be totally happy — that's what I was saying all along, now you finally gave me the equations for it. Or if you read the beginning of Landau, that's what they're saying; they just didn't write the equations. That's why I like this aspect that even when I take G Newton to zero I still get something non-trivial. On the other hand, in practice, doing this doesn't change anything about what we actually do, because we are indeed very classical, already decohered, so decohering us more doesn't do anything in daily life.

Daniel Harlow

That's part of the laws of nature. That's the change. In the traditional approach to decoherence, this happens because of your interaction with the environment. The problem in quantum cosmology is that the environment is also part of the closed universe, so environmentally-induced decoherence isn't enough. But I can just say it's a mathematical fact that if the laws of physics do the decoherence, that is enough to resolve the one-state problem and give you emergent semiclassical physics. I don't know whether it's correct, but it's a mathematical fact that it works, and it seems better motivated to me than anything else anyone has proposed to make sense of this one-state business.

Daniel Harlow

I think it's all at this scale of e to the minus S observer.

Daniel Harlow

One fun thing, since we're in this part of the conversation. Remember in the '90s there was all this talk about black hole complementarity, where you have the observer inside the black hole and the observer outside, and they don't have to agree on things. One of the problems was that there weren't equations for it, and later the idea was attacked by AMPS and others. But this is actually a rule that gives you different rules for different observers, and you can study that case.

One way of seeing how the one state comes from the evaporating black hole: semiclassically it does leave behind a baby universe, but that baby universe is a closed universe, so it only has one state. The holographic encoding of that one state is just a rank-one projection. The information goes into the black hole, hits that rank-one projection and bounces off, goes through the entanglement in the Hawking modes out to the Hawking radiation, and that's how the state is pure. The fact that the Hawking radiation is pure is crucially tied to the fact that the baby universe has a one-dimensional Hilbert space. That's why I'm hesitant to give it up — if I make it two-dimensional, the Hawking radiation is a little bit mixed; three-dimensional, more mixed. It's intimately connected to unitary black hole evaporation in the one state.

Now do the classical complementarity thing. I have an observer who falls into the black hole and one who stays outside, and I use my observer rule for both, to answer the question: is the Hawking radiation pure? When you measure the entropy, really I mean the second Rényi entropy — you do the swap of the two states. The outside observer measures the unitary S-matrix, and finds the swap expectation value is one, because the state is pure. The inside observer gets e to the minus S observer, which is consistent with e to the minus S black hole, because they can't do physics to better than e to the minus S observer. They get different answers because the laws of physics are different for the two observers. The inside observer gets an answer consistent with having enough entanglement across the horizon to account for their experiences, because there's a smaller observer entropy; the person outside sees it as pure. So now we have a mathematics of black hole complementarity that works, in basically the same way this thing resolves the semiclassical physics of the closed universe.

Daniel Harlow

It connects to various things we like — which again doesn't mean it's correct. Like you said, it's crazy. I don't know if it's too crazy or just crazy enough.

Daniel Harlow

I hope so. I don't know. It'll take a while. Right now this is all pretty much floating around — we have to see to what extent it connects into more concrete theories, string theory and whatever, and how we can turn it into real predictions. It'll be a while before we know, and maybe something better will come along.

Daniel Harlow

Thanks, Sean. Nice to talk, as always.