- (Information) Paradox Lost

Tim Maudlin

arXiv:1705.03541 [physics.hist-ph]

Here is the problem. The dynamics of quantum field theories is always reversible. It also preserves probabilities which, taken together (assuming linearity), means the time-evolution is unitary. That quantum field theories are unitary depends on certain assumptions about space-time, notably that space-like hypersurfaces – a generalized version of moments of ‘equal time’ – are complete. Space-like hypersurfaces after the entire evaporation of black holes violate this assumption. They are, as the terminology has it, not complete Cauchy surfaces. Hence, there is no reason for time-evolution to be unitary in a space-time that contains a black hole. What’s the paradox then, Maudlin asks.

First, let me point out that this is hardly news. As Maudlin himself notes, this is an old story, though I admit it’s often not spelled out very clearly in the literature. In particular the Susskind-Thorlacius paper that Maudlin picks on is wrong in more ways than I can possibly get into here. Everyone in the field who has their marbles together knows that time-evolution is unitary on “nice slices”– which are complete Cauchy-hypersurfaces –

*at all finite times*. The non-unitarity comes from eventually cutting these slices. The slices that Maudlin uses aren’t quite as nice because they’re discontinuous, but they essentially tell the same story.

What Maudlin does not spell out however is that knowing where the non-unitarity comes from doesn’t help much to explain why we observe it to be respected. Physicists are using quantum field theory here on planet Earth to describe, for example, what happens in LHC collisions. For all these Earthlings know, there are lots of black holes throughout the universe and their current hypersurface hence isn’t complete. Worse still, in principle black holes can be created and subsequently annihilated in any particle collision as virtual particles. This would mean then, according to Maudlin’s argument, we’d have no reason to even expect a unitary evolution because the mathematical requirements for the necessary proof aren’t fulfilled. But we do.

So that’s what irks physicists: If black holes would violate unitarity all over the place how come we don’t notice? This issue is usually phrased in terms of the scattering-matrix which asks a concrete question: If I could create a black hole in a scattering process how come that we never see any violation of unitarity.

Maybe we do, you might say, or maybe it’s just too small an effect. Yes, people have tried that argument, which is the whole discussion about whether unitarity maybe just is violated etc. That’s the place where Hawking came from all these years ago. Does Maudlin want us to go back to the 1980s?

In his paper, he also points out correctly that – from a strictly logical point of view – there’s nothing to worry about because the information that fell into a black hole can be kept in the black hole forever without any contradictions. I am not sure why he doesn’t mention this isn’t a new insight either – it’s what goes in the literature as a remnant solution. Now, physicists normally assume that inside of remnants there is no singularity because nobody really believes the singularity is physical, whereas Maudlin keeps the singularity, but from the outside perspective that’s entirely irrelevant.

It is also correct, as Maudlin writes, that remnant solutions have been discarded on spurious grounds with the result that research on the black hole information loss problem has grown into a huge bubble of nonsense. The most commonly named objection to remnants – the pair production problem – has no justification because – as Maudlin writes – it presumes that the volume inside the remnant is small for which there is no reason. This too is hardly news. Lee and I pointed this out, for example, in our 2009 paper. You can find more details in a recent review by Chen

*et al*.

The other objection against remnants is that this solution would imply that the Bekenstein-Hawking entropy doesn’t count microstates of the black hole. This idea is very unpopular with string theorists who believe that they have shown the Bekenstein-Hawking entropy counts microstates. (Fyi, I think it’s a circular argument because it assumes a bulk-boundary correspondence ab initio.)

Either way, none of this is really new. Maudlin’s paper is just reiterating all the options that physicists have been chewing on forever: Accept unitarity violation, store information in remnants, or finally get it out.

The real problem with black hole information is that nobody knows what happens with it. As time passes, you inevitably come into a regime where quantum effects of gravity are strong and nobody can calculate what happens then. The main argument we are seeing in the literature is whether quantum gravitational effects become noticeable before the black hole has shrunk to a tiny size.

So what’s new about Maudlin’s paper? The condescending tone by which he attempts public ridicule strikes me as bad news for the – already conflict-laden – relation between physicists and philosophers.

## 1,285 comments:

«Oldest ‹Older 201 – 400 of 1285 Newer› Newest»Tim,

I don't know what you asked Ellis, but anyone who has studied the Hamiltonian formulation of GR will know that the Hamiltonian is a sum of two terms, the first of which is a volume integral over a Cauchy surface and that vanishes on shell, and the second of which is a boundary term. Try the following exercise: go to google and type in something like "hamiltonian general relativity boundary term". You will find a number of lecture notes which discuss this.

In any case, the point is intuitively obvious: to measure the mass of the sun do I need go inside it and count up all its constituents? No, I can stand at infinity and measure the asymptotic falloff of the gravitational field that it produces. I can do this because the Hamiltonian is a boundary term.

You also seem to think that the notion of a boundary of AdS is something mysterious and poorly understood. Wow. I suppose this explains why you think AdS/CFT is so vague and poorly understood.

Regarding your other message, was it not obvious that I was referring to a disconnected Cauchy surface, since that is the entire point of the discussion? There is no connected Cauchy surface that connects to a sufficiently late time on the boundary.

I apologize for my tone here, but try to imagine the reverse situation in which a physicist was making certain claims about well known philosophical issues, and you will understood how it looks from my point of view.

black hole guy,

Don't worry about the tone. I'm sure you will find mine annoying at times, but the issue is the arguments.

I asked Ellis just what I wrote: whether the bulk Hamiltonian can be built out of boundary operators. We seem to be having the miscommunication because you think that the fact that the bulk Hamiltonian has a boundary term implies that it is equivalent to a boundary operator, or maybe is a boundary operator. These are clearly different claims. Indeed, the fact that the bulk Hamiltonian has a surface term implies that it is a surface operator. The fact that the integral vanishes does not mean that there is no operator on the surface.

This confusion seems to be compounded by the following: the ADM mass is a method of assigning a total mass (and hence total energy) to a spacetime. The ADM mass is defined on the boundary of an asymptotically flat space-time to which a boundary has been artificially added. It is therefore not surprising that some boundary operator corresponds to the ADM mass. But that operator is not the Hamiltonian of the bulk. Although Hamiltonians are, in many cases, associated with the total energy of a system it does not follow that every operator that measures the total energy is a Hamiltonian. We are principally interested in the Hamiltonian as the generator of time evolution in the bulk since we want to understand what happens to the information in the bulk under time evolution. You may be able to measure the mass of the Sun at a distance, but you can't specify the time evolution of the interior of the Sun at a distance.

The comments I made about the boundary referred to asymptotically flat space-times, not AdS, as is obvious. The conditions for appending a boundary to an asymptotically flat space-time are quite stringent, and would not be met by a space-time in which a black hole forms in a normal way. Look, for example, at Wald's General Relativity.

I do not at all follow what you are claiming about points in the interior of the event horizon. They all sit on connected Cauchy surfaces in the Penrose diagram I analyze. And every connected Cauchy surface connects to the boundary at space-like infinity. What do you mean by "a sufficiently late time on the boundary"? Look at the diagram again.

I imagine that his will be difficult for you to accept, but the dynamics here is not philosophers vs. physicists. It is people who have devoted themselves to foundations of physics vs. people who have not, if anything. Great physicists can make shocking mistakes on foundational issues. Feynman screws up the explanation of the Twins paradox in his Lectures. Weinberg is confused about the status of rotation in General Relativity in his recent book. Murray Gell-Mann does not understand Bell's theorem. Countless physicists are under the misapprehension that decoherence solves the measurement problem. These are pretty basic points to people who work on foundations, whatever department they sit in.

Tim,

Your message is essentially one error after another. I will not address them all here, so that we can stay focussed.

First, regarding Cauchy surfaces, for some reason you are not staying on topic. The issue at hand is an evaporating black hole in AdS. In the standard Penrose diagram, the only Cauchy surfaces that attach to sufficiently late times on the boundary are disconnected. This is what we are discussing. The asymptotically flat case is not relevant for AdS/CFT.

I am now going to accept the challenge of getting you to understand Hamiltonians in GR. I think you will find this illuminating. Let's start with the basics:

1) Applying standard canonical methods to the action of GR in a space with specified asymptotics yields a Hamiltonian H that is the sum of two types of terms. The first is an integral over a Cauchy surface of N_t C_t + N_i C_i, where the N's are the lapse and shift, and the C's are the constraints of GR. The second term is built out of the asymptotic data of the metric, and so is a sboundary term that we can call M_ADM, though we have in mind here that we could be in AdS in which case the nomenclature is a bit nonstandard.

2) We now canonically quantize (formally, so that we ignore UV issues). The constraint operators C_mu annihilate physical states, C_\mu |\psi_phy> =0. Furthermore, physical operators commute with the C_mu. So the GR Hamiltonian H acting on any physical state is equal to M_ADM acting on that state. And the commutator of H with any physical operator equals the commutator of M_ADM with that state.

3) So H and M_ADM are the same operator when we restrict to physical states. This is the precise sense in which they are the same operator. The fact that they differ on non-physical states is totally irrelevant since such states are not in the Hilbert space.

You should now say whether you agree or disagree with any of the above statements, which are totally standard and can be found in many places.

At this point, I am pretty sure you are confused as to how the boundary operator M_ADM can generate bulk evolution. I will be happy to explain this, but first I want to make sure the above points are crystal clear.

Regarding physicists vs. philosophers, I think your view is quite distorted, and I may expand on this later. One point I do agree with is that decoherence does not solve the conceptual problems of QM.

Tim,

Let me add: rereading Feynman's explanation of the twin paradox, I think you really have to be a nitpicker to say that he "screws up the explanation." The effect is only "paradoxical" if you erroneously assume a symmetry between the two people. As soon as you realize that acceleration breaks the symmetry there is no longer a paradox, and you just have to go and compute the ages of the two travelers. Feynman conveys this essential point well.

What recent book of Weinberg are you talking about, and which point? His most recent book is a text on QM, I believe. Weinberg rarely makes errors.

b h g,

Let me comment first on the Feynman. You say that his explanation is fine and complaining about it is nitpicking. He writes:

But in order for them to come back together and make the comparison Paul must either stop at the end of the trip and make a comparison of clocks, or, more simply, he has to come back, and the one who comes back must be the man who was moving, and he knows this, because he had to turn around. When he turned around, all kinds of things happened in his space-ship—the rockets went off, things jammed up against one wall, and so on—while Peter felt nothing.

So the way to make the rule is to say that the man who has felt the accelerations, who has seen things fall against the walls, and so on, is the one who would be the younger; that is the difference between them in an “absolute” sense, and it is certainly correct.

Feynman clearly wants to tie the difference in ages to acceleration. But 1) Even in SR we can set up examples of the twins experiment in which both twins accelerate exactly equal amounts, but still one is older that the other when they get back together. 2) In SR we can set up situations in which the twin who accelerates more is older rather than younger when they meet again. 3) In GR we can set up situations in which neither twin accelerates at all, no rockets go off, nothing jams against the wall for either twin, but they are different ages when they get back together.

Tying the difference in aging to acceleration is just incorrect conceptually. At best it is an accidental correlation in this particular circumstance, but when one looks at the full scope of twins cases appeal to acceleration as explaining anything is just demonstrably false. Feynman is giving his readers the wrong idea here. Acceleration is a measure of how much a world-line *bends*, i.e. deflects off of a geodesic. The differential aging is a matter of the *length* of the world-lines in proper time. These are two completely different geometrical characteristics.

If you think that this is an OK explanation of the paradox, and of why one twin ends up older than the other then this illustrates the problem I have been talking about. This is just conceptually off base, in a way that will eventually confuse someone. Pointing out how it is wrong is not nitpicking.

I'm sure that Feynman would agree immediately if this was pointed out. In other places it is clear he understands the theory perfectly. But that somehow did not keep him from this mistaken account of the single most iconic phenomenon in SR.

The Weinberg passage is in To Explain the World. The passage is:

"The success of Newton's treatment of the motion of planets and comets shows that the inertial frames in the neighborhood of the solar system are those in which the Sun rather than the earth is at rest (or moving at constant velocity). According to general relativity, this is because that is the frame of reference in which the matter of distant galaxies is not revolving around the solar system. In this sense, Newton's theory provided a solid basis for preferring the Copernican theory to that of Tycho. But in general relativity we can use any frame of reference we like, not just inertial frames. If we were to adopt a frame of reference like Tycho's in which the Earth is at rest, then the distant galaxies would seem to be executing circular turns once a year, and in general relativity this enormous motion would create forces akin to gravitation, which would act on the Sun and planets and give them the motions of the Tychonic theory.”

The last sentence is nonsense. I asked him about it and he never responded (having responded to some other points). Do you think that this claim conveys anything true about GR at all?

I hope we can at least agree about when accounts like these are fundamentally accurate and fundamentally wrong. If you find this sort of thing acceptable, then we will certainly not agree about much of the literature on AdS/CFT.

Can we try to hold our explanations to the standard of clarity and accuracy I am requiring?

black hole guy,

OK, let's get into your account. Point 1 is fine, but a little sketchy. If we canonically quantize GR in the usual way we just end up with Wheeler-deWitt. You add "with specified asymptotics" without comment or explanation. This requires adding a boundary to the space-time, a procedure that is non-trivial both for asymptotically flat space-time and for AdS. Your extra boundary term in the Hamiltonian reflects this addition. To understand the meaning and technical limitations of this move may well be essential in the sequel.

Point 2) has some extremely important and questionable claims that go by too fast. We normally associate the Hamiltonian of a system with time translations, i.e. with how the system develops in time. That will be essential to understand if we want to see if, and how, information is preserved through a process like black hole evaporation. The weird thing about Wheeler-deWitt—which can be taken to show that it is a failed attempt to quantize GR—is exactly that the Hamiltonian annihilates the states. We want to know the evolution of the bulk, but the bulk term does not obviously get at that. (I can talk about why this happens, but let's just note the fact for now.) Let's assume that the bulk terms nonetheless do encode the time evolution in the bulk. You assert that "Furthermore, physical operators commute with the C_mu.", intending by that to assert that the bulk term of the Hamiltonian commutes with any "physical operator". Since in regular WdW there is no boundary term, this sort of thing leads to the problem of time, and I don't see that adding a boundary term helps. Any operator that commutes with the Hamiltonian represents a constant of the motion, so every local "physical operator" in the bulk is a constant of the motion. Why in the world should I accept that? What criterion are you using for a "physical operator"? Essentially you are saying that every bulk property that commutes with M_ADM, i.e. which does not affect the ADM mass, is a constant of the motion. So either this approach already is a non-starter for understanding how information is preserved in the bulk or else every local degree of freedom in the bulk has to effect the ADM mass (or Bondi, or whatever boundary operator you have). That's absurd. As I have said, the ETCRs in QFT imply that any local operator in the bulk commutes with every boundary operator at space-like separation. All local operators inside the event horizon of a black hole will be space-like separated from every boundary operator. So you've got the problem of time again. Do you claim that nothing inside the event horizon changes? To be direct: 1) what is the criterion of a "physical operator"? 2) Do you acknowledge that by your criterion every physical operator localized inside the event horizon is a constant of the motion? 3) If so, do you think this could possibly be an acceptable physical account of the physics inside the event horizon?

Point 3) goes seriously wrong. The fact that H and M_ADM act the same on solutions does not imply that they are the same operator: to be the same operator they have to act the identically on all states. Your claim that there are no non-solutions in the Hilbert space is just bizarre.The Hilbert space is defined before we ask for solutions, or even define any operator. Offhand, I see no reason to believe that the set of solutions even forms a Hilbert space. Do you have a proof of this? Even if it does, it is not the Hilbert space we started with. That's the one relevant for defining operators.

These points I am making are not trivial and not nitpicking. If you have clear answers to the questions I have raised already then I am certain to learn quite a lot, since these questions are central to the difficulties for canonical quantization. As I mentioned, if there were ways to answer all these then we would not be looking for a theory of quantum gravity: we would have one.

Tim,

Re Feynman: Of course, everyone here understand the physics involved, and the issue is one of pedagogy. I read the passage differently than you. The ONLY paradox arises if one is under the belief that there is a symmetry between the twins (because only relative speed matters, or something). I see that in your quotation you left out the key sentence directly prior: "By symmetry the only possible result is that both should be the same age when they meet". So to remove the paradox the key is to break the symmetry, and it is certainly correct to point out that acceleration breaks the symmetry. That is the point he wants the reader to take away.

Now, of course I agree that it's not the case that the acceleration per se "causes" the differential aging. But one has to be careful here. Your statement

"The differential aging is a matter of the *length* of the world-lines in proper time. "is also strictly false. Nothing in special relativity tells you that humans age according to the proper times of their worldlines. How they age is a matter of human biology -- how cells behave under acceleration and so on. It could well be that two people on the same worldline could age differently due to their different responses to acceleration. Similarly, two clocks that tick at the same rate when moving inertially will generically tick at different rates when they move together on the same non-inertial worldline -- it depends on the how the clocks are made. All that special relativity says is that if two clocks (or people) tick at the same rate when together at rest, they will also do so when moving together at constant velocity.Regarding Weinberg, I would have to read the surrounding passages for context, and I don't have his book handy. I would like to know whether the emphasis here is a point of history or one of physics.

In all discussions of the Twins paradox the clocks are taken to be ideal and measure the proper time. Biological clocks are treated the same way. Of course given the sort of acceleration typically attributed to one twin, that twin would end up dead, but that really is nitpicking.

As I said, and as you know, we can set up the case in SR with the twins equally accelerated but still different ages (or different times on their clocks) and we can set up cases in GR with no acceleration of either twin and different times on their clocks. So if you thought that acceleration breaks the symmetry you will be unable to explain these cases. It is not hard to make the point accurately.

There is no more relevant context for the Weinberg. He is not trying to explain GR here, but simply puts in this remark when comparing Tycho and Copernicus. No context could make the last sentence even vaguely true. It gives entirely the wrong idea about GR. A wrong idea that has been repeated through many years and ought to have been cleared up. But instead Weinburg repeats it. (I just met another physicist who made the same mistaken claim and got very huffy when I told him it was wrong.) This error is connected to a misunderstanding about the content of the Strong Equivalence Principle and the incorrect claim that in GR it is always possible to interpret the same phenomenon as due to an acceleration or due to a gravitational field. Physicists ought to have figured out that these claims are wrong years and years ago. But attention to conceptual clarity (as opposed to calculational technique) is not part of the physics curriculum.

Tim,

I am glad that you would like to focus more on these basic points. The following remark will sound very cocky, but the difference between us here is that I have a lot of experience calculating and thinking about these issues -- getting my hands dirty -- while I presume you do not. I know very well how the pieces and logic fit together. Our discussion is like a debate between an experienced car mechanic and someone who has read (or perhaps skimmed) a book about cars.

Point 1)

" You add "with specified asymptotics" without comment or explanation. "OK, to be definite let's say we are in an asymptotically AdS spacetime with standard Fefferman-Graham asymptotics. To be specific, the conformal boundary metric is in the conformal class of the standard metric on S^2 x R. I didn't mention this because these asymptotic boundary are so standard that if you say "asymptotically AdS" anyone familiar with AdS/CFT will understand what you mean. I was also keeping this open so that we could discuss the asymptotically flat case in parallel where appropriate, but I am happy to be more specific. If this is unfamiliar, just type "Fefferman-Graham expansion" into google and you will find lots of hits.Point 2)

"The weird thing about Wheeler-deWitt—which can be taken to show that it is a failed attempt to quantize GR—is exactly that the Hamiltonian annihilates the states."Wrong, and in fact this represents precisely the kind of basic conceptual confusion that you seem to criticize physicists for. This is entirely due to the fact that the GR action is invariant under coordinate reparametrizations. In fact, ANY theory (e.g. Maxwell E&M) can be made reparameterization invariant and then the Hamitonian will similarly annihilate states. The reason for this is simple: if the time coordinate is an arbitrary parameter physical quantities can't depend on it. What this is telling you is you first need to define a physical notion of time and only then can you ask how physical quantities depend on time. So if you think about this correctly you realize that the situation is precisely opposite to what you say: it would have been very weird if H did not annihilate the state so that the wavefunction depended in a specific way on an arbitrary time parameter.The rest of your comments about my point 2 are such an intertwined collection of confusions that it's hard to know where to start. There are a few different issues. One is that I can see that you are not familiar with the canonical quantization of theories with gauge invariance. Most of your questions would be answered if you worked through the simplest example of this sort, which is ordinary electrodynamics. You will learn such things as the following. The theory has constraints, and the physical Hilbert space is defined as the space of (normalizable) wavefunctionals of the gauge potentials which are annihilated by the constraints. The analog of the Hamiltonian here is the electric charge generator (U(1) gauge transformations here being the analog of time translations in GR). The electric charge operator Q can either be expressed as a surface integral of the electric flux at infinity, or as the spatial integral of an electric charge generator. These two different forms of Q are equivalent on the physical Hilbert space. Similarly, in GR once you choose coordinates and solve the constraints M_ADM can be equivalently expressed as an integral over the Cauchy surface. You can work this out totally explicitly in perturbation theory in Newton's constant, and at lowest order you will find that the surface integral M_ADM is equal to the standard form of the Hamiltonian operator one would write in QFT for a theory on a fixed background.

cont

Tim, black hole guy,

May I kindly suggest that you drop the issue of how Feynman once explained the twin paradox and what Weinberg might have meant with that sentence in his recent book.

cont

1) what is the criterion of a "physical operator"?It is the standard one: the operator should be gauge invariant, which in the quantum theory means that it should commute with the constraint operators (since they generate gauge transformations). You surely don't want to draw physical conclusions from non-gauge invariant operators do you?2) Do you acknowledge that by your criterion every physical operator localized inside the event horizon is a constant of the motion?No. You don't understand the distinction between evolution in physical time versus that in arbitrary coordinate time. See above3) If so, do you think this could possibly be an acceptable physical account of the physics inside the event horizon?Well, order by order in 1/G_N (Newton's constant) I can show that this reproduces standard perturbative gravity, so the answer is yes (once you understand what's going on).Point 3) goes seriously wrong. The fact that H and M_ADM act the same on solutions does not imply that they are the same operator: to be the same operator they have to act the identically on all states. Your claim that there are no non-solutions in the Hilbert space is just bizarre.The Hilbert space is defined before we ask for solutions, or even define any operator. Offhand, I see no reason to believe that the set of solutions even forms a Hilbert space. Do you have a proof of this? Even if it does, it is not the Hilbert space we started with. That's the one relevant for defining operators.

Two operators are the same if they have the same matrix elements between all states in the Hilbert space. The Hilbert space is the space annihilated by the constraint operators. H and M_ADM are most definitely the same operator. What you say is "bizarre" it totally standard. Again, your confusion here comes from not having any experience with the canonical quantization of gauge theories. For a pedagogical reference I refer you to Dirac's classic lectures on quantum mechanics given at Yeshiva in 1964.

I carefully wrote my three points so that they would only refer to standard and uncontroversial results in the field, with a huge literature to back them up. There is little hope of moving to the more uncertain terrain of black hole evaporation in the AdS/CFT correspondence if you are not familiar with these bedrock principles.

Sabine,

These examples are relevant to the question of how clearly even great physicists understand the foundations of the theories they have spent their lives working with. It speaks to the question of whether people working in foundations are likely to be in a position to correct errors committed by the many physicists who have never thought about foundations. More particularly, it speaks to the question of whether my own comments here are better characterized as a philosopher making out-of-place claims about physics or as someone working in foundations bringing a specialized knowledge to the situation. If you want to defend Feynman or Weinberg, please do. If you acknowledge that even they can be confused about very basic issues then that is an important observation. I think that physicists do not at all appreciate how bad the situation with respect to foundations is in the field.

Tim,

I am sure there are a lot of topics you would like to discuss with someone that are important for something, but we're at comment 212 and I hope you'll get done with the black hole guy before the end of the millenium. So please stick to the topic which is, to remind you, black hole information loss. Thanks.

Black Hole Guy,

Yes, the Hamiltonian on the surface shows up as a constraint essentially because the choice of a foliation is treated as a choice of gauge, and then all the states on any Cauchy surface fall into the same gauge orbit. But this is a use of the term "gauge" which is not like that in other gauge theories. So, for example, the state on Sigma 1 is, in this sense, gauge equivalent to the state on Sigma2in U SIgma2out (or just on Sigma2out if the information somehow gets out), even though in any normal sense of the term those states are physically quite distinct. So that's all fine formally, but we need to keep in mind that "gauge equivalent" in this context means something quite different than in the usual gauge theories one deals with.

We still have a very sharp disagreement about the conditions for two operators to be the same, which originates from a disagreement about exactly which Hilbert space the operators are defined to operate on. And here I really cannot understand at all what you are claiming. The whole point of defining a Hamiltonian is to have an operator that distinguishes solutions from non-solutions. The Hilbert space over which the Hamiltonian operates is much bigger than the space of solutions, otherwise there would be no point in defining the Hamiltonian in the first place. From the lectures of T. Thiemann, for example, we find that the original kinematical Hilbert space H is a product space of smooth functions and smooth vector fields on the classical configuration manifold. This Hilbert space is certainly not at all the space of solutions! The physical space, the space of states annihilated by the Hamiltonian, does not even carry a natural Hilbert space structure. I give you the references below, but just stop and think about it. If we *started* with the space of solutions (the solution space) then there is no work for the Hamiltonian to do: we have somehow *already* identified the physical states. But on the initial Hilbert space just mentioned the Hamiltonian simply is not the same operator as M_ADM. The Hamiltonian determines the dynamics in the bulk exactly by imposing the constraints on the bulk. The M_ADM operator does not contain the constraints. So there is no sense in which the two operators are the same.

You make the same mistake when you write "Two operators are the same if they have the same matrix elements between all states in the Hilbert space. The Hilbert space is the space annihilated by the constraint operators. H and M_ADM are most definitely the same operator." Again, the solution space not only is not the relevant space for defining the equivalence of operators, it isn't even a Hilbert space at all. The account in Thiemann's "Introduction to Modern Canonical Quantum Gravity" is perfectly clear on this, if you want to check, but just the logic should convince you. If we can't settle this we will get nowhere, but it surely is something we can settle.

M_ADM does not even operate on the bulk. How could it possibly place any constraint at all on the dynamics in the bulk? It can't, and doesn't.

If you look back at our exchange, this point comes up over and over. The relevant passage in Theimann is on pp. 44-5. You can get the lectures online at https://arxiv.org/abs/gr-qc/0110034v1.

Tim,

This discussion has become rather pointless, at least from my end. I am pretty sure that you have never worked through a single example of canonical quantization of a gauge theory along the lines we are discussing, which is why your recent comments are almost completely incoherent. If you are at all serious you will at least go through the simple example of Maxwell theory, which will resolve most of your confusions. You will find: The theory has a Hamiltonian and two constraints. One constraint says that the wavefunction is independent of A_t, and the other says that the wavefunction is annihilated by the Gauss' law operator. The physical Hilbert space is the space of wavefunctionals (normalizable with a suitably defined inner product) that are annihilated by the constraints. Physical operators commute with the constraints. Carrying this out you will recover the standard Hilbert space of free photons, with two helicity states for each momentum. You will also find it useful to construct the electric charge generator in the case where charged matter is present, noting that it can equivalently be expressed as a surface integral of the electric field at infinity, or as a volume integral of the matter charge density. These two objects represent the same operator when acting on the physical Hilbert space. Everything here has a parallel in the gravity case.

You are not going to get anywhere until you understand this basic example. I can give you a reference if you'd like. Otherwise I don't see any point in continuing this discussion, so I will just wish you "good luck".

Also, I am not sure what your point is in referring to the Thiemann lectures is, since the issues he is raising are technical ones that have basically nothing to do with the general conceptual issues at hand.

Black Hole Guy,

Well, you are free to leave the discussion at this point. Everyone can see where it is. You have repeatedly made a false claim about what the kinematical Hilbert space is, from which follows another false claim about the identity of operators. I have backed up everything I said with direct citation from a publicly available source *which is about the quantization of gravity*, and directly contradicts your claims. So you deflect to bringing up Maxwell theory, which has nothing to do with the discussion.

Here is how silly your claim is. Take the normal case of Wheeler-deWitt, where there are no imposed asymptotics and no added boundary. Then the Hamiltonian simply annihilates all the solutions in the kinematical Hilbert space. If you insist that the relevant space of functions for defining operators is the *physical* space, the space of *solutions* (which again need not even be a Hilbert space), then you will argue that the Hamiltonian is identical to the zero operator: the operator that annihilates every state of any kind! Just try to recover some dynamics from that.

The entire discussion has been about whether the physics of the bulk is somehow encoded at the boundary. To make such an argument you have repeatedly insisted that the boundary Hamiltonian *is* the bulk Hamiltonian, and it is now completely clear how you fell into such an error: you tacitly switched the kinematical Hilbert space for the space of solutions. This led you to a false equivalence between quite different operators.

You claim over and over that what you have written is just standard material that everyone knows. The Thiemann lectures are explicitly introductory lectures on the very topic at hand written for graduate students. The points he is making are not at all "technical": he explicitly defined the kinematical Hilbert space and explicitly says that the space of solutions does not have a Hilbert space structure. So you are in direct contradiction to Thiemann. Since what he says makes sense and what you say does not, it's pretty clear how to go here.

You also have taken to claiming that my comments are "almost completely incoherent" without actually pointing out anything incoherent or incorrect. I have just analyzed, in precise detail, where you have gone wrong. If you think my claims are incoherent you can point out exactly where and exactly why. "Go do an example in Maxwell theory" is not a criticism, it is a deflection.

I'm sorry if you find the way this has come out embarrassing. But if you can't actually point out any error in what i have just written, rather than making vague disparaging remarks, then it is clear where things stand.

Tim,

Since you asked, I will point out one glaring example of your complete lack of comprehension. Consider your statement

Here is how silly your claim is. Take the normal case of Wheeler-deWitt, where there are no imposed asymptotics and no added boundary. Then the Hamiltonian simply annihilates all the solutions in the kinematical Hilbert space. If you insist that the relevant space of functions for defining operators is the *physical* space, the space of *solutions* (which again need not even be a Hilbert space), then you will argue that the Hamiltonian is identical to the zero operator: the operator that annihilates every state of any kind! Just try to recover some dynamics from that.Go and read sections 2 and 3 of

T C P, Quantum Gravity, the Cosmological Constant and All That...

Tom Banks (SLAC & Princeton, Inst. Advanced Study). Jul 1984. 29 pp.

Published in Nucl.Phys. B249 (1985) 332-360

wherein you will find worked out in explicit detail exactly what you are claiming is impossible, namely the ordinary *dynamical* time dependent Schrodinger equation for matter propagating on a solution of Einstein's equation comes out of the equation H|\psi> = 0. I await your reply.

I could do the same with pretty much every one of your claims, but the above should suffice to make the point.

Well this has gotten entertaining. I hope you'll both forgive me for jumping in, but some remarks from a disengaged observer might be useful.

It is indeed clear where things stand, Tim -- you don't know the first thing about quantization in gauge theories, and are unable to recognize that such knowledge is necessary for you to understand canonical quantization in GR. You don't even appreciate that the technical problem is the same! This is despite BHG's clear and patient explanations and attempts to address your confusions. I am amazed that he has kept responding for so long, considering the quality of your responses: you do not respond so much as try to either seek some statement that you can twist to support your claims, or ignore him altogether in an appeal to authority (usually one you have misunderstood but occasionally also one who misunderstands) rather than try to engage with and understand his arguments, a courtesy which he has consistently afforded you. The level of discourse has been almost comically lopsided.

At this point I have to wonder what's really going on here. You seem to think that philosophers working on foundations are in a position to correct the conceptual understanding of physicists. This is logically possible, though I personally would expect much less than from the large community of theoretical physicists dealing with basic conceptual issues on a daily basis (a community whose existence you dismiss with the wave of a hand). You clearly are not in a position to make such a contribution personally, given your manifest ignorance of even the most basic aspects of the theory, but I think your situation is actually much worse.

You do not seem to take a scientific approach to reality. We're not stuck in the 20th century; we've understood the basic structure of our scientific theories very well, we know that they are capable of describing everything that we observe, and we have explored their logical consequences in a rigorous mathematical manner. This is what professional physicists do. You, on the other hand, just cherry-pick passages from papers and quotes from authorities that you (most often erroneously) think support your naive argument. Even if they did support your claims, this is not how science works. There is an objective truth (that many people understand, in this case) which BHG has been explaining to you, and which can only be further revealed by achieving a deeper understanding of the theory. This requires understanding the theory in the first place. Nothing you have said has poked a hole in his explanations; literally all of your concerns reflect a lack of basic knowledge of the underlying and extremely well-understood mathematical structure, not some logical inconsistency. Your failure to appreciate the criticisms of your arguments likewise stems from this lack of knowledge, which BHG has valiantly tried to address.

If you had put forth the effort to understand the BHG your confusions would have been resolved, but with your method of approach I doubt you will actually learn anything. Scientifically, it is hard to take you seriously given your evident lack of understanding. That is on you to change. It is readily apparent to anyone who has spent time honestly grappling with these issues that not only do you lack even the most basic understanding, you are more interested in landing pseudo-sociological blows and declaring victory than in actually learning enough to have an intelligent conversation. This strikes me as arrogant and intellectually lazy at best, and if unchanged will probably prevent you from ever attaining a deep enough understanding to make a contribution of any scientific value.

I apologize for the ad hominem remarks; my intent is only to reorient you towards a more scientific approach, since you seem to have an interest in making a contribution to our field, and are not currently in a position to do so.

Black Hole Guy,

You want my reply?I don't even have to look at Banks to reply, since what you cite is completely irrelevant to my point. One more time:

There exists a zero operator, let's call it Z. It is defined as follows: for every state |psi> in the *kinematic* Hilbert space (the thing both Thiemann and everybody else calls script H, or just the Hilbert space) Z|psi> = 0. That is an operator, yes? I just defined it.

Now on the *solution* space, the thing nobody calls script H (Thiemann calls it script Dphysica), the set of solutions to the basic dynamical equations, the operator Z acts identically to the Hamiltonian H. That is, H|psi> = 0 for every state in Dphysical. So by *your* criterion (which is the *wrong* criterion) the Hamiltonian and the zero operator are the same operator. So by *your* criterion, whatever physics is contained in the Hamiltonian is contained in Z. And by analyzing Z you can recover the basic dynamics of the theory. Which is absurd. And has absolutely nothing to do with whatever is in Banks, since he is not making the ridiculous claim that the criterion for identity of operators is that they act the same on the solution space, rather than on the Hilbert space.

Yet again, and one more time: Dphysical is *not* the relevant space of states here, Indeed, Dphysical is not, in general, even a Hilbert space (see Thiemann again). So as soon as the phrase "an operator is defined by its action on the Hilbert space" pops into your mind, a bunch of alarm bells ought to tell you that the relevant space is *not* Dphysical. Thats presumably why Thiemann chose the letter D rather than the letter H for it: so the reader would not get confused.

The difference between the Hamiltonian and Z could not be more stark. The Hamiltonian has the content of GR written into the constraint operators. It contains the basic dynamics of the theory. Z has nothing written into it: its action is just to annihilate every state, willy-nilly. If you can't see that Z and the Hamiltonian are not *different* operators, I don't know how to respond except to ask you to explain what in the world you mean. But you surely must admit that *by your definition* they are the same operator: they have the same action on Dphysical.

Once you see that the Hamiltonian and Z are different operators, you see why what you cite from Banks is irrelevant. I never denied that H|psi> = 0 contains lots of physics. Why shouldn't it? The equations of GR, as well as the dynamics for the matter fields, are packed into H. What I deny is that Z|psi> = 0 contains any physics at all. I am absolutely, 100% certain, without even looking, that Banks is not engaged in the hopeless project of recovering the "ordinary dynamical time-dependent SchrÃ¶dinger equation" from the equation Z|psi> = 0. But if, as you appear to want to insist, Z is identical to the Hamiltonian then that ought to be possible.

I just can't lay this out any more clearly. If I am making an error, it appears in some sentence above. Stop telling me to study Maxwell theory or work through some other paper: just point out the error. I have pointed out yours.

I await your reply.

Dark Star,

So that was a long rant with no content. You would not have to apologize for making ad hominem remarks if, well, you just refrained from making them. Your rhetoric about "a scientific approach to reality" and my "evident lack of understanding" are just empty words until you actually back them up. So I issue exactly the same challenge to you as to Black Hole Guy. If there is a single error of any kind in the response I just gave to him, point it out. Point out the exact sentence and the exact mistake. I have done him that courtesy, and pointed out that defining the equality of operators by the sameness of their action on the space of solutions is an error, and is further exactly the error I have pointed out repeatedly in this discussion. I have explained in painful detail, both conceptually and by example, why it is an error.

Black Hole Guy is evidently confused. It is a confusion I have not come across before, and is so plain and obvious that I am astonished anyone could make it. Apparently, you are also incapable of recognizing it, which astonishes me even further. If I have made a mistake, point it out.

Socrates said in the Apology that the appropriate punishment for someone who is ignorant is to be taught. To use your phrase, it is logically possible that there is some error in what I just wrote to Black Hole Guy. If it contains an error, then the scientific thing to do is to point it out so I can correct it. I will forthrightly admit that I am wrong if either you or Black Hole Guy can actually find a mistake.

But instead of that you just blather on about nonsense, larded with insults. I have not insulted either you or Black Hole Guy. I have called arguments silly when they are silly, and I have shown why they are silly. You, on the other hand, simply assert without offering a shred of evidence that I lack some basic knowledge of something.

To be honest, your rhetoric smacks of a sort of quasi-religious self-regard. Physicists have some deep knowledge not vouchsafed to anyone else. All you have to do is label someone else a non-physicist, and you can simply disregard any argument being made. You have lots of fancy names for my supposed errors but oddly no actual actual examples. I engage in "appeal to authority" you say. OK, just exactly how do I do that? The only person I have even mentioned is Thiemann, and I hadn't even read Thiemann until five days ago. Black Hole Guy kept saying to go on the web to get basic knowledge, and I went on the web and Thiemann was literally the first thing I hit. Not surprisingly (to me) Thiemann happened to clearly make the very same point I was making to Black Hole Guy, so I cited him. But that is not an "appeal to authority" in any case, since I explained the error: I did not just state is was an error and appeal to someone else. Since you charge that I often appeal to authority (some of whom are right and some wrong) you could actually back up that claim by mentioning the authorities I have appealed to and sort them into the reliable and the unreliable. The whole discussion is recorded, nothing is hidden. I literally have no idea what you are talking about.

I am interested in landing "pseudo-sociological blows". Well that's a fancy thing to do. How about an actual example? Again, I have no idea what you are talking about.

Reread your own post. I dare you to point out in your post any actual, contentful criticism of anything I have written. You just repeat that I am making all sorts of errors and twisting things and so on without one single concrete example. And you could have jumped in any time to point an error out.

You would like nothing more, it seems, than to show that I have no understanding of the physics. Well, you have all the ammunition you need recorded here. Please cite chapter and verse the errors I have made or the bad arguments or the deflections. That would be useful. If you can't, there is an obvious conclusion to be drawn.

BHG,

Sorry: a typo. I wrote "If you can't see that Z and the Hamiltonian are not *different* operators," Of course I meant "if you can't see that Z and the Hamiltonian are *different* operators".

Tim,

The content of my post is that your responses to BHG are woefully inadequate, both in the sense that your criticisms of his mathematical logic (which underpins our understanding of physical theory) are not themselves grounded in mathematical logic, and in the sense that you ignore his arguments to cite "expert" opinions which you have misunderstood (Theimann, Feynman, Weinberg, Jacobson, and that's off the top of my head-- should I go on?)

Emphatically, my intent was not to criticize the content of your posts, since that has been done amply, and ably, by BHG. I am instead criticizing the honesty of your scientific engagement with BHG. He can defend his own, extremely well-founded position. But just to humor you, let's talk about Thiemann. The passages you quoted refer to the quantization of quantum fields on a fixed spacetime manifold, i.e. quantum field theory in curved space, rather than quantum gravity. It is beyond uncontroversial that the basic principles of gauge theory* imply that H|psi> = 0 on any state in the physical quantum gravity hilbert space, which is not just the space of solutions to Einstein's equations, and includes states that are not solutions so long as they are diffeomorphism-invariant. You will find, if you read Thiemann's notes with a modicum of care, that this is part of his treatment: see the discussion around I.1.1.35 and its followups. BHG has already explained to you how this is compatible with dynamical bulk evolution, but since you are unwilling to try to understand quantization in gauge theories, which furnish a more basic example of the phenomenon of constrained quantization without the additional conceptual complexity of a constraint involving an unphysical coordinate time, your prospects of understanding this in the more subtle setting of GR are essentially nonexistent.

I assert your lack of knowledge without citation because it is manifest in the thread of conversation, and has been pointed out by both me and BHG, over and over again. Your inability to understand or even appreciate the content that is being presented to you is depressing.

I find your accusation of quasi-religious self-regard quite entertaining. Of us two, there is one of us who recognizes the humility necessary for a rigorous study of physics, under which one's beliefs are subjugate to the mathematical structure of the theory, while the other thinks himself capable of upending the field without even a basic understanding of its rigorous underpinnings. Regarding your comments about the sanctimony of physicists, BHG and I have both engaged seriously with you, despite the fact that you are not a physicist; we have both deemed your arguments invalid on their merits. I see serious scientific engagement with non-physicists as an integral part of our social responsibility and have often found such conversations fruitful for my own understanding, but seldom when the other party refuses to engage or deflects as you have done here.

By pseudo-sociological blows, I mean statements along the lines of "Feynman [or other Serious Physicist X] must have been confused about the fundamentals of [GR or Theory Y], because of my (extremely distorted mis-)reading of some of his lecture notes, therefore most physicists must be confused about fundamental issues". You haven't pointed out any actual scientific issues, man. Learn canonical quantization in gauge theories, then in the canonical formulation of GR, and after that you might be able to have a semi-intelligent conversation. (To cite one example of your confusion I found particularly egregious, WdW is not something you choose to do, or not, it is an inevitable consequence of the symmetries. Your statement is literally like saying we can choose whether or not to impose Gauss's law in electrodynamics.) Right now you are barely even spewing nonsense. I suggest you choose to engage seriously while we are still indulging you.

This thread seems to be coming to an unfortunate contentious end. Before it does, I want to thank Bee for patiently hosting this discussion; and darkstar, black hole guy and Tim Maudlin for all that they have written.

Apart from rekindling my dormant interest in physics, this thread has served as a reminder of the need for conceptual clarity and of extreme precision in expression when talking about difficult problems in physics. In my opinion, in the current pickle that particle physics finds itself in, both of these are needed, if physicists are not to go around in eternal circles without forward movement.

I'm also glad that it proved possible to fruitfully examine the logical structure of arguments (as in why is something likely to be true (or false)) without getting lost in the details. Thus, I'm also glad that a philosopher is bringing his touchstone to physics.

I hope the conversation can proceed, but if not, so be it!

Tim,

First, I would like to lower the temperature of this discussion. Second, I admit that in my last message I misdiagnosed what was apparently bothering you, and hope that you find this more on target.

Let's again consider the case of gravity in a closed universe with some matter. We start with some space of wavefunctionals \Psi, which are functionals of the spatial 3-geometry and matter configurations on those 3-geometries. We then encounter the constraint equations H_i \Psi =0 and H \Psi =0. H_i and H are generators of coordinate transformations (there is a slight technical subtlety in this statement, but I really do not think it is relevant here). Wavefunctionals \Psi that are not annihilated by H_i and H have no physical meaning: they are not gauge invariant, meaning in this case that they assign different amplitudes to geometries which differ merely in their choice of coordinates. So what we would like to do is to solve these equations to reduce the space of \Psi's down to those that only depend on coordinate invariant data. This is the "physical Hilbert space" (I can hear you claiming that this is not a Hilbert space, but this is a bit of a red herring and will be addressed below). What is sometimes done is to first solve H_i \Psi =0 and label this the "kinematical Hilbert space". That's fine, but keep in mind that most of these states have no physical meaning since they are coordinate dependent. The remaining equation H \Psi = 0 has deeper significance than being "just" a dynamical equation. Non-solutions of this equation simply have no physical interpretation since the value of \Psi changes if you do nothing but change your coordinates. So you shouldn't think of H\Psi= 0 as being like Newton's equation of something like that, but instead as an equation that restricts the space of wavefunctionals to those with physical meaning. H\Psi = 0 is a difficult equation to solve in general, so the discussion can quickly get pretty formal. But the thing to stress is that the only physically relevant \Psi are those that are annihilated by all the constraints, and likewise the only physically meaningful operators are those that commute with all the constraints, so that they keep you in this subspace. If we could solve all the constraints we would have no need to refer to the larger space of unphysical wavefunctionals, and furthermore when considering operators we would only be interested in how they act on the physical wavefunctionals that obey the constraints -- everything else is coordinate dependent garbage.

cont

cont

There are situations in which we can complete the program and go on to solve H\Psi = 0, and these are extremely illuminating. One case is the semi-classical limit as discussed in the Banks paper. One sees explicitly how this equation becomes the time dependent Schrodinger equation for the matter field, with a time coordinate built out of the metric. The Hilbert space structure in this limit is then evident.

But an especially relevant example for present purposes is the case of spherically symmetric configurations ("mini-superspace") of gravity coupled to a scalar field in asymptotically flat or AdS spacetimes. This is the BCMN model, nicely discussed in an appendix in Unruh's famous 1976 paper "Notes on black hole evaporation". We have the constraints as before, and the Hamiltonian is a sum of constraint terms and the M_ADM surface integral at infinity. Due to the restriction to spherical symmetry, one can very explicitly solve the constraints to express the metric variables in terms of the scalar field variables. So we're left with a space of wavefunctionals \Psi that depend on the scalar field configuration and that contains only physical coordinate independent information. Since the constraints annihilate these wavefunctions, the Hamiltonian operator reduces to M_ADM on this space, so it really is a true statement that H = M_ADM on the physical Hilbert space. Now, I think you have been asking how it is that if M_ADM is a surface integral at infinity it can generate bulk evolution? Well, it was a surface integral when expressed in terms of the gravitational field, but now that we have solved the constraints it explicitly becomes a bulk object. It is very illuminating to expand out M_ADM is power of Newton's constant. The first term becomes nothing but the ordinary scalar field Hamiltonian in flat space, and then one gets a series of corrections representing the gravitational self-interaction. This makes it totally clear how M_ADM, when acting on the physical Hilbert space, generates bulk evolution even though it is a boundary term when expressed in terms of metric. So conceptually, the asymptotically AdS or flat space case is conceptually a lot clearer than that of a closed universe, basically because the asymptotic structure specifies a physical time coordinate, and then the equation id/dt \Psi = M_ADM \Psi can be readily interpreted. So if I were to suggest one example for you to go through to clarify things, it would be this one.

The above was for spherically symmetric spacetimes, but I think pretty much everyone would agree that the same story will hold after relaxing this condition, although it is technically much harder. At the very least, order by order in perturbation theory there should be no obstruction to carrying this out: solving the constraints, defining a scalar product so that the Hilbert space structure is well defined and so on. One then manifestly has the physical Hilbert space, and the Hamiltonian equals M_ADM on this space. There is then no more need to refer to the larger space of unphysical wavefunctions.

My suggestion to consider Maxwell theory was not an attempt to deflect but rather to be helpful, since it really is a good analogy in a more intuitive setting for many of these issues. This is how most people learn to think about the subject.

Tim,

Re: the Unruh paper, I actually don't mean the appendix, but rather section IV

Dark Star,

The first rule of holes is: stop digging. You keep getting in deeper and deeper.

I asked you to point our any errors in my reply to Black Hole guy. You refuse to do that, even though you continue to say that my posts are full or errors. I'd say that it would be rhetorically more effective to actually point one out. But since you refuse, how about this: BHG and I have a straightforward disagreement. He claims that two operators are identical if they have the same action on the space of solutions, the physical space. I claim that they must have the same action on the kinematical Hilbert space. (This is certainly a necessary condition. Leave aside whether it is sufficient.) So who do you think is correct here, BHG or me. Commit to one side or the other.

Now let me demonstrate what it is to respond to a post by citing what it actually says. You claimed that I am constantly appealing to authority. I asked for examples, and you replied with four names: Feynman, Weinberg, Thiemann and Jacobson. What this list demonstrates is that you don't even know what appeal to authority is. The only time I mentioned Feynman and Weinberg was to assert that they made fundamental conceptual or physical mistakes. That is not only not appeal to authority, it is the exact opposite. Thiemann I have already explained. I have from time to time mentioned what, e.g., Ellis said. But that sort of appeal to authority is not any kind of logical error. On the relevant points I have also provided independent arguments. Black Hole Guy is constantly making assertions about what every physicists knows. Pointing out accomplished physicists who disagree is obviously relevant. And I have to try to break through your dismissive attitude somehow.

It is you and BHG who are constantly making appeals to authority, but not to any actual named individual. You appeal to the amorphous authority of "what every physicist knows" (and what no philosopher knows). To fight against that is to fight against shadows. Your attitude is that all I should do it sit quietly at your feet, even if what you say strikes me as nonsense. But the only way to actually learn anything is to say when something makes no sense to you. As Socrates says in the Apology, the proper punishment for someone who is ignorant is to be taught. Teaching is more than mere assertion: it requires giving a clear account of the matter. We have actually managed, after all this, to get to a perfectly precise question. On it, a tremendous amount of what Black Hole Guy has said rides. On my side I have both a sharp argument (given above) and reference to what seems to be a standard presentation. On his side there is...his bare assertion. No argument. No citations. But if I don't just accept what he says you think I am not paying attention.

As for Feynman and Weinberg, let's do the same thing as with Black Hole Guy. I assert that Feynman's account of the asymmetry in the twins case is conceptually mistaken, and I assert that Weinberg's claim about GR and the Tychonic system is completely confused and wrong. Sabine does not want us to discuss these cases. But at least commit yourself: who is right, Feynman or me? Who is right, Weinberg or me? Even if we don't discuss the cases, one day it will hit you that I am right about them all. And maybe you will learn something from that.

Con't

More details from your post. You say that I don't accept BHG's "mathematical logic". Mathematical logic is a particular subject that we have not discussed at all.

You say that the "physical quantum gravity hilbert space" of states such that H|psi> = 0 "includes states that are not solutions so long as they are diffeomorphism-invariant". But every state is supposed to be diffeomorphism-invariant: that is one understanding of what it is to have a background-free theory. So are you claiming that H annihilates every state in the kinematic Hilbert space, even if they are not solutions to the EFEs? Then we don't have a theory of gravity at all.

"I assert your lack of knowledge without citation because it is manifest in the thread of conversation". That's a neat trick. If there are so many errors, pick the most egregious and cite it and explain it. Anything else is just an excuse.

As for WdW: it is a very natural sort of approach to quantizing GR. Put it in Hamiltonian form and turn the canonical quantization crank. Maybe it is even right.. But it has rather severe conceptual difficulties. If it didn't, then it is hard to see why physicists have been saying that it is hard to quantize gravity: about the first thing you try works! Again, I have an open mind here, but note that if it is correct then you are arguing that physicists have spent 50 years chasing something they already had. That is kind of ironic.

Tim,

I am truly puzzled about your comment:

"He claims that two operators are identical if they have the same action on the space of solutions, the physical space. I claim that they must have the same action on the kinematical Hilbert space."Do you agree that the only states that can ever occur in the physical theory are those that are annihilated by all the constraints? If so, then note that if two operators have the same matrix elements between all states in this physical space then they are identical insofar as any conceivable experiment can detect. Their difference has no physical significance. This is just to say that the larger unphysical kinematical Hilbert space is just scaffolding one uses in the process of constructing the eventual theory, and that one eventually discards in favor of the smaller physical Hilbert space. Said differently, one can quantize gauge theories or gravity in, say, light cone gauge, and then the entire Hilbert space is physical from the very beginning. This larger kinematical Hilbert space never makes an appearance. Do you really wish to say that there can be any physically relevant distinction between two operators that agree on all states in the physical Hilbert space? That is certainly a novel claim.

BHG,

Do you really wish to say that there can be any physically relevant distinction between two operators that agree on all states in the physical Hilbert space?I wouldn't want to do so, but how do you distinguish between the two operators,

H |physical state> = 0

and Tim's zero operator,

Z |physical state> = 0, in fact, Z |any psi> = 0

since they both agree on all states in the physical Hilbert space?

Tim is saying that the distinction is that H and Z have different kernels in a larger-than-just-physical-space, and that is the only way to distinguish them.

Now, I think you have been asking how it is that if M_ADM is a surface integral at infinity it can generate bulk evolution? Well, it was a surface integral when expressed in terms of the gravitational field, but now that we have solved the constraints it explicitly becomes a bulk object.IMO, there is a sleight of hand here. Trying to put my finger on it :)

Black Hole Guy,

I am happy to bring the temperature down, and I will look at the Unruh paper, which may take me some time to work through. Let me return the favor of your clear post with what will be a rather long discussion of constraints and "gauge invariance" from a very conceptual point of view, because I think that the terms "gauge theory" and "gauge equivalent" and "gauge degree of freedom" can be highly misleading here. First a brief prologue, which is not meant to annoy you and will sound insufferably smug, but which I really think is necessary.

Above you made the analogy between someone who had just read a manual and someone who has been actually fixing cars for years. It's a fine analogy. I can't fix cars at all, if by that you mean actually solve equations. It's not in my skill set, and if there are forms of insight into a theory that come from being able to solve the equations, I don't have them. If you do, and can convey them, that would be a great help.

But you have to add this to the analogy. The manual has been somewhat shoddily written. There are places where the terminology is not observed, where instructions are not precise, and so on. Just by studying the manual you can identify some of these problems and try to clear them up. Some of these imprecisions are benign and don't make much difference, some are quite serious and make virtually impossible to understand how the engine actually works. On a day-to-day basis, these don't bother the mechanic, who has enough experience to know what to do in most cases. But for novel problems not covered in the manual, for attempts to expand the manual, these confusions can be quite deadly. And the guy who has been only studying the manual can be of service there. Even if he can't fix an engine to save his life.

My own experience in graduate-level physics was this. For the sorts of basic conceptual questions I am interested in, something helpful might have been said on the first day of class. But the rest of the class, about how to use Green's functions to solve equations, for example, just had no relevance at all. If you are interested in Newtonian mechanics at a conceptual level and are given F = mA, you immediately ask questions like "What is F?" "What is A?""How much am I committing myself to physically by accepting this equation?". These are not merely mathematical questions (What mathematical object is used to express F?) but questions that lie between physics and mathematics. Newton thought that making physical sense of A required commitment to Absolute Space. He turned out to be wrong about that. This sort of inquiry is just different from learning how to solve F = mA to actually predict the trajectories of objects. Maybe someone who has been actually calculating trajectories for a living sees this other activity as pointless. But when the manual no longer even covers the problems, and needs to be expanded, there might be some really good advice coming from the guy who has been studying it.

For what it's worth, I recently was reading an article published in an excellent physics journal and picked up a fundamental error on a quick skim. A very important error given the aims of the paper. I was able to do this because of having thought about certain conceptual issues a lot, which focuses your attention to particular parts of the paper (not the mathematical derivations). I read the Weinberg passage above to several of my co-workers in foundations and they all were immediately agog at how he could have written it. The issue is not mathematical: it is conceptual. In order to keep the conceptual issues clear, it is often useful to introduce new terminology, which I will do below. All I am asking is that you read this with some care, and at least give me the benefit of the doubt that I know what I am talking about.

Con't

Now let's take a view of our situation from 40,000 feet. We want a quantum theory of gravity that is somehow based on GR. We are going to have several conceptual/technical problems to solve just to have something coherent. Let's assume that at the very least we are going to want to be able to talk about quantum gravitational states, and that using the basic insight of GR we mean superpositions of states with different "gravitational fields", i.e. different matter-fields-cum-space-time geometries that solve the Einstein Field Equations. Notice here that I am not starting by postulating a flat background space-time and then a perturbation (h) from it. It seems quite plausible that no perturbation from flatness will give us black holes, much less evaporating black holes. I am doing this in a “background free” manner in that sense.

In non-relativistic QM the wave-function is a function on the configuration space of the classical analog. Here, the classical “configuration space” is trickier to even specify. Points in the configuration space should certainly correspond to specification of a geometry and matter fields on the geometry. But now we run into two problems: a problem that has to do with co-ordinates and a problem that has to do with diffeomorphisms. (You may think of these as really the same problem, which is fine, but let me exposit it this way.)

For the moment (but wicked complications to come!) let’s just consider the state on a Cauchy surface in some solution to the EFEs. So this is a purely spatial object. In order to specify it in the classical setting we need to specify 1) the intrinsic Riemannian geometry of the surface 2) the distribution of the matter fields on the surface and 3) the embedding of the surface in the ambient space-time, i.e. the exterior curvature. How am I even going to begin to mathematically represent these?

One possibility is to lay down a co-ordinate system on the thing and then give co-ordinate dependent mathematical expressions for 1), 2), and 3). If I do that, then clearly what I get by laying down a different co-ordinate system is mathematically different but physically the same. We can say that these two different mathematical objects are “gauge equivalent”, implying that they do not represent physically different states at all. The differences in the mathematics are “merely gauge”. So what do we do?

There are three choices here. One is to work in a mathematical setting in which the two different co-ordinate-dependent descriptions are still there in the mathematical structure, they are distinct mathematical points in the configuration space, but we add a constraint on the wavefunctional: it must assign the same value to these two different points because they represent the very same physical situation. Any wavefunctional that assigned different values to these two points would be “unphysical”, or not make any sense. Notice that we haven’t actually done any physics yet! As far as I have said, some of these states of three-metric-cum-matter-field-cum-exterior-curvature may be physically impossible given the EFEs. So let’s be really careful here, and say that such a wavefunctional, that assigns different values to co-ordinate-dependent descriptions that correspond to the same physical situation are “ungeometrical” rather than “unphysical”. If I am following your thoughts accurately, you take one of the functions of the constraints in the Hamiltonian to be to eliminate these ungeometrical wavefunctionals, since we could not make any sense of them.

con't

Now I get the sense that you think that is all the constraints in H are doing: they are ruling out ungeometrical wavefunctionals. You write

“Wavefunctionals \Psi that are not annihilated by H_i and H have no physical meaning: they are not gauge invariant, meaning in this case that they assign different amplitudes to geometries which differ merely in their choice of coordinates. “

This sounds like you are saying that all the constraints do is eliminate ungeometrical wavefunctionals in the sense I just defined. If that is what you mean, I certainly want to deny it. There are wavefunctionals that are geometrical (they never assign different values to states that differ only in their choice of co-ordinates) but are not annihilated by the Hamiltonian. For example, take a 3-surface that is intrinsically flat and whose exterior curvature is zero, but that has lots of matter fields distributed over it in whatever way you like. Different co-ordinate systems will yield different co-ordinate-dependent descriptions of such a surface-cum-matter-fields. For a wavefunctional to be geometrical, it must assign the same value to all of these. But such a surface is unphysical in GR: it can’t occur. More generally, there are complete space-times with 4-metrics and matter fields that are not physically allowed by the EFE’s. But a wavefunctional that assigns them a high probability can be perfectly geometrical. Such a wavefunctional had better not live in the space of physical wavefunctionals, but it does live in the space of geometrical wavefunctionals. The sense in which you might say that these wavefunctionals “have no physical meaning” is that they do not correspond to physical possibilities, not because they assign different amplitudes to geometries that differ merely in their choice of co-ordinates.

So on your picture, as I understand it, the situation is this: the thing that stands in for the configuration space in non-relativistic QM, the thing that the quantum state is defined over, is a set of co-ordinatized descriptions of a space (or space-time) with matter fields. Some of these descriptions are related in that they arise from different co-ordinatizations of the same space (or space-time) plus fields. You want to rule out as meaningful any wavefunctional that assigns different amplitudes to such points as being physically uninterpretable, and so want a constraint on the wavefunctionals. Let’s call the result of imposing such a constraint the “geometrical space” of wavefunctionals. I do not want to call it the “physical space” because we have not yet done any physics! All our problems so far have arisen from having co-ordinate-dependent mathematical descriptions of objects that admit of different co-ordinate systems. That’s all geometry. In particular, if that were all the Hamiltonian in our theory were doing, ruling out the ungeometrical wavefunctionals, then it could clearly not be a theory of gravity! Somewhere the EFEs had better play a role, and so far they have not showed up at all.

OK, so putting a constraint on allowable wavefunctionals is one way to deal with these problems. But there are other ways as well. One obvious thing to do is not to put a constraint on the space of wavefunctionals, but to change the space over which the wavefunctionals are defined in the first place. If we can define a relation between the various co-ordinate dependent descriptions that holds exactly when they arise from re-coordinatization of the same space-plus-fields, then we can pass from our original configuration space to a reduced configuration space by quotienting out. Each point in the new configuration space now corresponds to a collection of points in the old one, and we no longer have to constrain the wave-functionals: every one will be geometrically interpretable automatically.

This same choice point—constrain the wavefunction or change the underlying state—already comes up in non-relativistic QM. If there are identical particles, how does that change the physics? One approach is to keep the configuration space appropriate for non-identical particles and impose a restriction on the wavefunctions: they must be symmetric or anti-symmetric under exchange, for example. A completely different approach is to change the configuration space: for N identical particles in a D-dimensional space, the relevant configuration space is NRD rather than RDN, where a point in NRD corresponds to a set of N points in RD. This is a prettier solution than to impose constraints on the wavefunctional, in my opinion. We can call the relevant space the geometrical Hilbert space rather than the physical one because, as already mentioned, we have not done any physics.

The third approach to solving this problem is to leave out the co-ordinates altogether. Describe the geometry and fields on our surface with completely co-ordinate free methods, so the issue of coordinate systems cannot arise. This gets tricky, though, because of diffeomorphism invariance. If all of the possible physical states are defined on the same manifold, then descriptions that assign different fields to different points in the manifold can nonetheless describe the same state if the two states are related by a diffeomorphism. This problem comes up independently of any co-ordinates. Once again, we have two choices. Either constrain the wavefunctional to assign the same amplitude to states that differ by a diffeomorphism, or cut down the configuration space by quotienting out over diffeomorphisms. In some sense these different approaches my turn out to be equivalent, but at least at first glance they appear to be different.

No matter which of the approaches we take, at the end we should have a geometrical space of wavefunctionals: each of the wavefunctionals ascribes an amplitude to a configuration in a coherent way. So far, no mention of physics at all. This is all just geometry.

(to be con’t, but it will be a while)

Tim,

"This sounds like you are saying that all the constraints do is eliminate ungeometrical wavefunctionals in the sense I just defined. If that is what you mean, I certainly want to deny it."No, I do not think, and did not say, that that is *all* the constraint equations do. But that is a big part of what they, do and I believe you are missing some of the story here. There are 4 constraints: H_i (i=1,2,3) and H_t. The H_i are easy to understand: they generate purely spatial coordinate transformations on the spatial 3-geometries. The final constraint H_t is more subtle, as it does double duty in terms of imposing invariance under coordinate transformations that involve time reparametrizations and in imposing the dynamical Einstein equations. Full 4-dimensional coordinate invariance has to come from somewhere, and it's certainly not coming from H_i so it must come from H_t. The details of how this works is fairly intricate (Thiemann comments on it briefly, and for more you can look at papers by Teitelboim from the 70s). But let's put it this way. Suppose at the classical level you start with some initial data that does not respect H_t = 0, and then you evolve it forward in time via Hamiltonian evolution. There is no guarantee that the resulting 1-parameter family of 3-geometries and extrinsic curvatures labelled by t will have any interpretation in terms of a 4-dimensional geometry, and indeed it won't generically. In fact, the basic structure of the equation H_t = 0 can be understood as resulting from the requirement that the initial data it corresponds to does evolve into a bona fide 4-geometry. This is all reflected at the quantum level (but it's not a simple story, and I don't clam a complete understanding myself). So I think you are wrong in saying that quantum states that fail to obey H_t \Psi = 0 can be "unphysical" but still "geometrical", as I don't think there will be any consistent interpretation of such states in terms of 4-dimensional spacetime geometry.

Irrespective of this, for sure the only states that physically occur obey H_i \Psi = H_t \Psi = 0, so I am still baffled by why you object to the criteria that two operators should be regarded as equivalent if they agree on this space.

Irrespective of this, for sure the only states that physically occur obey H_i \Psi = H_t \Psi = 0, so I am still baffled by why you object to the criteria that two operators should be regarded as equivalent if they agree on this space.The H_i and H_t agree on this space, annihilating all \Psi, so they should be regarded as equivalent.

If I had something other than the Einstein action, the H_i would be the same, but H_t would be different? That is, the H_i represent simply the equivalence of different coordinates on a folio of Cauchy surfaces, while H_t represents not only reparametrizations of the coordinate normal to the Cauchy surfaces, but also acceptable dynamical changes?

Tim,

In that case you must be very close to the core ;)

As I said, BHG has ably addressed your confusions. I agree with all of his physics statements, except for what I believe to have been a typo in an earlier post when he referred to gauge-variant operators in the CFT. It is not useful for me to reiterate his remarks, nor my own.

You are correct that I conflated appeals to authority with questions about the conceptual understanding of certain authorities. The former is certainly irrelevant, the latter relevant only to your claim that even great physicists are confused about the basics of GR. You have not identified such a mistake, though, so it is hard to take this seriously.

The correct way to break through dismissal by any scientist is to respond with scientific arguments, not appeals to authority. Any scientist ought to evaluate such arguments on their merits, and I believe the BHG and I have done so here.

The appeal is not to "what every physicists knows (and no philosopher knows)" but to the actual mathematical structure of the theory. That is what ultimately determines the correctness of any of the arguments we are making, and so it is the relevant thing to appeal to. There are philosophers who know it much better than you, and some physicists who do not properly understand it at all (though not the ones who make actual progress in the field). If you had pointed out something overlooked in the mathematical structure, you would have the attention of the entire community, but instead you seem confused about most of the basics.

"On his side there is...his bare assertion. No argument. No citations. But if I don't just accept what he says you think I am not paying attention."

This is laughable. He is presenting you the arguments, and you are either failing to engage with them to the point of understanding or launching off on meaningless diatribes.

"You say that the "physical quantum gravity hilbert space" of states such that H|psi> = 0 "includes states that are not solutions so long as they are diffeomorphism-invariant". But every state is supposed to be diffeomorphism-invariant: that is one understanding of what it is to have a background-free theory. So are you claiming that H annihilates every state in the kinematic Hilbert space, even if they are not solutions to the EFEs? Then we don't have a theory of gravity at all."

This is severely confused. Diffeomorphism-invariance has nothing to do with background-independence; it's just the statement that it doesn't matter what coordinates you put on your manifold. Next, there are spacetime manifolds that do not satisfy the EFE, but they are part of the quantum theory because the path integral sums over all geometries. They generate quantum corrections to the classical action. This is entirely analogous to the effects of electric field configurations that satisfy Gauss's law but not Maxwell's equations; such fields are certainly physical, and play a crucial role in QED. For example, they lead to the astoundingly accurate prediction of the anomalous magnetic moment of the muon. I agree that the situation is more confusing in GR (even though technically identical) because the constraint involves the generator of time-translations, but BHG has explained how H=0 is both compatible with dynamics and required by diff-invariance, hence must be imposed on what you call the kinematic Hilbert space.

"I assert your lack of knowledge without citation because it is manifest in the thread of conversation". That's a neat trick. If there are so many errors, pick the most egregious and cite it and explain it. Anything else is just an excuse."

I did, Tim! You don't believe the Wheeler-de Witt equation! That's like not believing Gauss's law!

Also, you are confusing the ability to quantize a theory with a full understanding of the dynamics of the quantum theory. Yeesh!

Dark Star,

In case you hadn't noticed, BHG and I are back to a discussion of the structure of the theory. He acknowledges that he misunderstood the point I was making, and so was not being responsive to it. We still haven't resolved that question (as Arun has noted) but I need to get some more distinctions on the table before returning to it. I infer from your post (although you do not straightforwardly say it) that you agree with BHG that two operators that agree in their action on the solution space are the same operator, at least for all purposes relevant to physics. I also infer that you side with both Feynman and Weinberg against me. Let's at least record those decisions. I have explained Feynman's error above. I have cited the Weinberg claim that is completely wrong. Why don't you at least think about what Weinberg said and see if it makes any sense. If you understand GR, you will see easily that it doesn't. And conversely, if you can't see that it means you don't understand GR.

I don't know what you mean by "You don't believe the Wheeler-deWitt equation." The equation is what it is, and it is derived by a plausible approach to quantizing GR. But not every plausible approach is correct, and there is certainly no general agreement that this equation is the right way to quantize gravity. There are all sorts of presuppositions built into the approach—e.g. that we are still dealing with a space-time manifold—that can be questioned. Many people think that the right way involves discretizing space-time. Many people think it require string theory. Neither of these is implemented in WdW. If you can't see the difference between the status of WdW and the status of Gauss's law then most of this discussion will go over your head.

Black Hole Guy,

OK, picking up the tale (I’m pretty sure you agree with all this so far) we have talked about characterizing a single Cauchy surface in a way that does not privilege or depend on a co-ordinate system. If we keep all of the equivalent co-ordinate-dependent descriptions in our space of configuration states (rather than quotienting them out before quantizing) then we must characterize which set of states lie in the same gauge orbit, and then demand that the wavefunctional assign the same amplitude to all of them. If I am following, that is how you are thinking of the effect of the Hi constraints in the Hamiltonian: a wavefunctional that does not assign the same amplitude to all the configuration states that lie in the same gauge orbit will not be annihilated by the Hamiltonian and hence not make it into the space of solutions. Is that correct?

Now all of this treatment of our Cauchy surface falls squarely in the usual understanding of “gauge theory”, where the choice of a gauge is a choice between various mathematically distinct descriptions of one and the same physical situation. This is strictly analogous to the situation in classical E & M when using the vector and scalar potentials as a means to describe the electro-magnetic field. If one thinks that what is physically real are the E and B fields, then there are gauge degrees of freedom in the mathematical description. One can kill them off by fixing a gauge, but in the case of choice of co-ordinates on our Cauchy surface there is no analogous way to characterize a unique co-ordinate system with some nice mathematical property. So our choices are to either quotient out and start with a reduced configuration space before quantizing, or else have the wavefunctional range over a space that is physically redundant (different points in the space correspond to the same physical situation) and impose restrictions on the allowable wavefunctionals.

I hope we agree up until now. Because from here on things get much more difficult, for reasons you have mentioned.

Having dealt with the Hi’s, we are left with Ht. The constraint Ht, unlike the Hi’s, has to accomplish two completely different things. On the one hand, Ht has to somehow counteract the existence of multiple co-ordinatizations of the same space-time, where in some sense this multiplicity of mathematical representations is “merely gauge” (WARNING! The use of the term “merely gauge” is already misleading here.) And on the other hand, Ht has to implement the EFEs: it contains the guts of the theory of gravity (via the Einstein-Hilbert action). That is to say, Ht contains real physics in addition to playing a role in compensating for the existence of multiple coordinatizations of the space-time.

You have asserted this as well, and admit that it is not entirely clear to you how all this works. I am in the same situation: it seems unavoidable that Ht somehow has these dual roles, but not clear if one can somehow cleanly separate them mathematically (this feature of Ht does one job and that feature the other). But even if we can’t separate them mathematically, we can separate them conceptually.

Con't

In the terms I have introduced above, one effect of Ht is to restrict all solutions to lie in the geometrical space, i.e. the space that corresponds to co-ordinatizations of 4-dimensional Lorentzian space-times. Once we co-ordinatize such a 4-dimensional space-time we effectively foliate it, with the surfaces of constant t-coordinate being the leaves of the foliation. Each leaf can be characterized by a Reimannian 3-metric and an extrinsic curvature, and the leaves “fit together” to form a 4-dimensional Lorentzian manifold. This, of course, will not generically occur if we just randomly select the Reimannian 3-metric and extrinsic curvature of each slice! If I am following correctly, a sequence of such slices that don’t cohere into a 4-D space-time is what you called “co-ordinate dependent garbage” that needs to be avoided.

Now here we hit one point of disagreement, I think, or at least one that requires some discussion. What I have called the geometrical space is the space of co-ordinate dependent descriptions that can arise from the co-ordinatization of a Lorentzian 4-manifold. In fact, I am requiring even more: it should be a co-ordinatization of the 4-manifold such that the surfaces of constant t-coordinate are all Cauchy surfaces, i.e. it should reflect a foliation into Cauchy surfaces. (The t-co-ordinate certainly has to foliate into space-like surfaces, in order for the surfaces to have a Riemannian 3-metric. I take it that it is also required in solutions that these space-like surfaces be Cauchy. Of course this means restricting to globally hyperbolic spaces-times.) But even requiring all this does not restrict us to the solution space. There are Lorentzian 4-manifolds with matter fields that do not obey the EFEs. If use co-ordinates that slice them into Cauchy surfaces we get something that should lie in the geometrical space but not the solution space. In fact, the geometrical space will be much, much huger than the solution space, in the sense that solutions to the EFEs are set of measure zero in the geometrical space. This returns us to one of our remaining points of dispute. You think that it is sufficient for two operators to be physically equivalent that the have the same action on the wavefunctionals in the solution space. I insist that at the least (as a necessary condition) they must have the same action on all wavefunctionals on the geometrical space. My Z operator counts as identical to the Hamiltonian by you criterion but not by mine since the Hamiltonian does not annihilate all the wavefunctionals in the geometrical space and Z does. I really can’t see how you can continue to hold your view here. Neither does Arun.

Con't.

black hole guy,

I referred to one of your comments above in a recent post because I thought it was a good comparison. Please note that my response is not addressed at you and the content of mentioned blogpost has nothing to do with your discussion here in particular (quantum gravity being one of the few problems in the foundations of physics that imo is an actual problem). That is just to prevent any possible misunderstanding.

Now to the really important part. Consider two coordinatizations of the same Lorentzian 4-manifold that agree completely up to some Cauchy surface and then diverge: after that final surface of agreement, their t-co-ordinates slice up the space-time differently. Clearly, in some sense this is not a real physical difference at the end of the day: these are just two ways of describing the same 4-dimensional object. In some sense, this is just a “difference of gauge”. So one way to deal with this is to say that the slicings in each of these foliations belong to the same “gauge orbit” or differ only by a “gauge transformation”. And (this is the key move here!) it can be tempting to say that all of the individual slices in each foliation are “gauge equivalent” to all of the individual slices in the other. But this is a completely different sense of “gauge equivalent” than occurred in our discussion of the Cauchy slices. In that context we were talking about putting different co-ordinates down on the same 3-surface. The resulting different co-ordinate dependent descriptions clearly have the same physical content. And in our new case, the totality of the slices in each co-ordinatization (i.e. the whole 4-manifold) is the same. But it is not true that the individual slices in the two co-ordinatizations are physically the same! For example, suppose in the full space-time there are two stars that go supernova at space-like separation. In some co-ordinatizations, there will be a single Cauchy slice on which both of the stars are going supernova. In other co-ordinates, no slice contains both supernovas. So when one says that the slice that contains both supernovas is “gauge equivalent” or “lies in the same gauge orbit” as a slice that contains only one supernova (or indeed as a slice that contains no supernovas), the sense of “gauge equivalent” is quite different from the sense in which the same Cauchy slice described in two different co-ordinate systems are gauge equivalent.

There is a deep connection here to the problem that set off this whole discussion: whether information is lost in black hole evaporation. The first point I make in my paper is that the issue is not so much information in the Shannon sense but determinism: in a deterministic theory the state on every Cauchy slice together with the dynamical laws implies the state on every other Cauchy slice and hence the state of the whole 4-dimensional space-time. In that sense, all the Cauchy slices contain the same information, and one can say in that sense that they are all “gauge equivalent”. But despite this sort of equivalence, the states on different Cauchy slices are clearly different physically from one another. This distinction lies at the center of the so-called “problem of time” in WdW.

It is because Ht does this double duty—both accounting for differences in the mathematics that arise from using different Cauchy foliations and also implementing the dynamics of the EFE—that these two different meanings of “gauge equivalent” get confused here. And it is also for this reason that the existence of what I have called the “geometrical space” is easy to miss.

Ideally, we could mathematically separate these two roles of Ht into 2 operators, call them Htgeometric and Htdynamical. (I have no idea if this can be implemented formally: I am making a conceptual point here). Then from some initial space of wavefunctionals that includes even those that do not cohere geometrically we could first impose the condition that the state be annihilated by the Hi’s and Htgeometrical, which would cut the space of wavefunctionals down to the geometrical space, and then, as a separate move, demand that Htdynamical annihilate a wavefunctional in order that it be included in the solution space. Or something like this.

Con't

Now to the really important part. Consider two coordinatizations of the same Lorentzian 4-manifold that agree completely up to some Cauchy surface and then diverge: after that final surface of agreement, their t-co-ordinates slice up the space-time differently. Clearly, in some sense this is not a real physical difference at the end of the day: these are just two ways of describing the same 4-dimensional object. In some sense, this is just a “difference of gauge”. So one way to deal with this is to say that the slicings in each of these foliations belong to the same “gauge orbit” or differ only by a “gauge transformation”. And (this is the key move here!) it can be tempting to say that all of the individual slices in each foliation are “gauge equivalent” to all of the individual slices in the other. But this is a completely different sense of “gauge equivalent” than occurred in our discussion of the Cauchy slices. In that context we were talking about putting different co-ordinates down on the same 3-surface. The resulting different co-ordinate dependent descriptions clearly have the same physical content. And in our new case, the totality of the slices in each co-ordinatization (i.e. the whole 4-manifold) is the same. But it is not true that the individual slices in the two co-ordinatizations are physically the same! For example, suppose in the full space-time there are two stars that go supernova at space-like separation. In some co-ordinatizations, there will be a single Cauchy slice on which both of the stars are going supernova. In other co-ordinates, no slice contains both supernovas. So when one says that the slice that contains both supernovas is “gauge equivalent” or “lies in the same gauge orbit” as a slice that contains only one supernova (or indeed as a slice that contains no supernovas), the sense of “gauge equivalent” is quite different from the sense in which the same Cauchy slice described in two different co-ordinate systems are gauge equivalent.

There is a deep connection here to the problem that set off this whole discussion: whether information is lost in black hole evaporation. The first point I make in my paper is that the issue is not so much information in the Shannon sense but determinism: in a deterministic theory the state on every Cauchy slice together with the dynamical laws implies the state on every other Cauchy slice and hence the state of the whole 4-dimensional space-time. In that sense, all the Cauchy slices contain the same information, and one can say in that sense that they are all “gauge equivalent”. But despite this sort of equivalence, the states on different Cauchy slices are clearly different physically from one another. This distinction lies at the center of the so-called “problem of time” in WdW.

It is because Ht does this double duty—both accounting for differences in the mathematics that arise from using different Cauchy foliations and also implementing the dynamics of the EFE—that these two different meanings of “gauge equivalent” get confused here. And it is also for this reason that the existence of what I have called the “geometrical space” is easy to miss.

Ideally, we could mathematically separate these two roles of Ht into 2 operators, call them Htgeometric and Htdynamical. (I have no idea if this can be implemented formally: I am making a conceptual point here). Then from some initial space of wavefunctionals that includes even those that do not cohere geometrically we could first impose the condition that the state be annihilated by the Hi’s and Htgeometrical, which would cut the space of wavefunctionals down to the geometrical space, and then, as a separate move, demand that Htdynamical annihilate a wavefunctional in order that it be included in the solution space. Or something like this.

Con't

Now one can read that while the WdW equation is essentially impossible to solve in general (and in some cases makes no sense), it can be solved in certain cases of cosmology. One reason is that the cosmologists impose a condition of perfect spherical symmetry on their problems, which kills off a couple of degrees of freedom. And with the remaining 2 degrees of freedom there is even more help: in these special cases there is typically a unique York slicing that has constant curvature on the Cauchy slices. In other words, in this setting (but not in general) we can essentially fix a gauge to cut out the degrees of freedom that go along with alternative slicings. so the geometrical space of wavefunctionals can be characterized without mention of the dynamical EFE. The EFE (again, in terms of the Einstein-Hilbert action) can then pick out the solutions.

So that’s how I understand the overall situation. If it is correct, them we have to be very careful about what is conveyed by the term “gauge equivalence” between two states, since it might mean that we have two different co-ordinate dependent descriptions of the same physical state, or it could mean that we have two Cauchy slices taken from the same 4-dimensional solution to the field equations. Making this distinction is essential for discussing the information loss problem.

Now let’s assume we can make the distinction between some initial space of wavefunctionals, a subset of that space that are the geometrical wavefunctionals, and a further subset of that space that are the solutions. I assert again that the criterion of identity for two operators cannot be that the have the same action on the solution space: the zero operator example shows that. They must at least have the same action on the geometrical space. And to be honest, I don’t see why one should not demand that they have the same action even on the total initial space that includes non-geometrical wavefunctionals. If they act differently anywhere, why aren’t they different operators?

I personally think this can be pushed even further. Are two functions on the positive integers the same if they both assign the same value to each positive integer? That sounds unassailable. But consider these two functions. One is, again, the zero function: it assigns zero to every positive integer. The other function is defined as follows: it assigns zero to every odd integer, and to 2, and to every other even integer that is the sum of two primes. It assigns 1 to all other integers. Now if Goldbach’s conjecture is true, this second function assigns zero to all positive integers. But is it really “the same” function as the zero function? I think there is a decent sense in which the functions are tremendously different, and just happen to assign the same values to all positive integers. But this question gets us deep into philosophy of mathematics. I don’t want to insist on it, but leave it for your consideration.

DONE!

Now one can read that while the WdW equation is essentially impossible to solve in general (and in some cases makes no sense), it can be solved in certain cases of cosmology. One reason is that the cosmologists impose a condition of perfect spherical symmetry on their problems, which kills off a couple of degrees of freedom. And with the remaining 2 degrees of freedom there is even more help: in these special cases there is typically a unique York slicing that has constant curvature on the Cauchy slices. In other words, in this setting (but not in general) we can essentially fix a gauge to cut out the degrees of freedom that go along with alternative slicings. So the geometrical space of wavefunctionals can be characterized without mention of the dynamical EFE. The EFE (again, in terms of the Einstein-Hilbert action) can then pick out the solutions.

So that’s how I understand the overall situation. If it is correct, them we have to be very careful about what is conveyed by the term “gauge equivalence” between two states, since it might mean that we have two different co-ordinate dependent descriptions of the same physical state, or it could mean that we have two Cauchy slices taken from the same 4-dimensional solution to the field equations. Making this distinction is essential for discussing the information loss problem.

Now let’s assume we can make the distinction between some initial space of wavefunctionals, a subset of that space that are the geometrical wavefunctionals, and a further subset of that space that are the solutions. I assert again that the criterion of identity for two operators cannot be that the have the same action on the solution space: the zero operator example shows that. They must at least have the same action on the geometrical space. And to be honest, I don’t see why one should not demand that they have the same action even on the total initial space that includes non-geometrical wavefunctionals. If they act differently anywhere, why aren’t they different operators?

I personally think this can be pushed even further. Are two functions on the positive integers the same if they both assign the same value to each positive integer? That sounds unassailable. But consider these two functions. One is, again, the zero function: it assigns zero to every positive integer. The other function is defined as follows: it assigns zero to every odd integer, and to 2, and to every other even integer that is the sum of two primes. It assigns 1 to all other integers. Now if Goldbach’s conjecture is true, this second function assigns zero to all positive integers. But is it really “the same” function as the zero function? I think there is a decent sense in which the functions are tremendously different, and just happen to assign the same values to all positive integers. But this question gets us deep into philosophy of mathematics. I don’t want to insist on it, but leave it for your consideration.

DONE

Tim,

I essentially agree with most of these points, but would like to clarify some things

1) I have a suspicion that you aren't familiar with the Penrose diagram for AdS (based on your previous reference to a non-existent notion of Bondi mass in AdS). Going forward this will be important, so let me state that the Penrose diagram for AdS is a solid cylinder, and so the conformal boundary is a timelike surface. For example, it should then be clear what I mean when I say that in the standard Penrose diagram for an evaporating black hole in AdS there is no connected Cauchy surface that attaches to the boundary past a critical boundary time. This is to be contrasted with the asymptotically flat case.

2) In either asymp. AdS or flat spacetime, two spatial slices that hit the boundary at the *same* time can be thought of as "gauge equivalent" in the sense that one can evolve from one to the other under the action of the Hamiltonian constraint. But this is definitely not the case for two slices that hit the boundary at *different* times. The operator that moves slices along in boundary time is M_ADM, which is not a generator of gauge transformations since it does not annihilate physical states.

3) Related to this, time at infinity has a clear physical meaning, and there is no "problem of time" as far as it is concerned. The asymptotic boundary conditions one imposes define what is meant by time at infinity. You can think of there as being a gigantic clock at infinity that displays the time -- its large size and mass implying that it behaves classically. Two different readings on this clock are in no sense gauge equivalent.

4) There are no significant conceptual puzzles in studying canonical gravity in the context of perturbation theory around AdS or flat spacetime. We can first choose a gauge, solve the constraint equations, and then plug the solutions into the M_ADM operator. This gives a perfectly well defined Hamiltonian that generates time evolution on a sensible Hilbert space. This is basically what ADM originally did. Conceptual issues arise when we want to go beyond small fluctuations around AdS or flat spacetime, or consider a spacetime with no boundary, although even here we more-or-less know how to proceed in the semiclassical regime (see the Banks reference),

cont

cont

With these comments in mind, let's come back the meaning of the WdW wavefunction and the H_t \Psi = 0 constraint, which I have now understood somewhat better. We can separate out the gauge vs dynamical aspects of this equation as follows. Let us consider the case of gravity restricted to spherical symmetry with no matter or cosmological constant. Then the theory has no dynamics and the only classical solution is pure Minkowski space. It is very instructive to consider H_t \Psi = 0 in this case (see the 1990 paper by Fischler, Morgan and Polchinksi for details and much interesting commentary). The metric on a spatial slice can be written ds^2 = L^2(r) dr^2 + R^2(r)d\Omega^2, so that we have two degrees of freedom corresponding to the functions (L(r), R(r)), and our wavefunctional depends on these two functions. The H_i constraints just say that \Psi should be invariant under changes in L and R that corresponds to a reparametrization of r. We can attack the equation H_t \Psi = 0 in the WKB "approximation" (quote marks because I'm not sure this is really a controlled approximation, but let's assume that it is at least qualitatively valid). In the WKB approximation the wavefunction can either be oscillatory with modulus 1 (the classically allowed region) or be decaying and have modulus less than 1 (the classically forbidden region). So the space of (L,R) functions is thus separated into classically allowed and forbidden regions. What does this mean? Well, consider the full Minkowski metric, and think of the space of all spherical 3-geometries that can be embedded in it, allowing for the Minkowski time coordinate to vary along the slice. This defines a space of (L,R) functions that turns out to be precisely the same as the classically allowed space. So we see that the solutions of H_t \Psi = 0 are (up to the exponentially small tails) restricting \Psi to the (L,R) functions that have an interpretation in terms of 4-geometries. Non-solutions of H_t \Psi = 0 will lack any such interpretation.

I should note that the physical meaning of this wavefunction in term of measurement probabilities is very unclear. One would have to carefully develop the theory of what it means to measure a 3-geometry.

Now, in the case where we have matter and dynamics, I think something similar will occur (though I haven't seen this worked out anywhere). Say we include a scalar field, but maintain spherical symmetry. I believe that solving H_t \Psi = 0 in WKB will yield a classically allowed region corresponding to all possible embeddings of 3-geometries into solutions of Einstein's equations with the matter content.

Finally, let me add that to the extent that we understand it, AdS/CFT describes gravity in the bulk in the gauge fixed form in the sense that the map is between CFT states on the one hand and bulk states that are solutions of H_i \Psi = H_t \Psi = 0 on the other. There is no CFT version of the H_i \Psi = H_t \Psi = 0 equations, so they have apparently "already been solved". Somehow, in going from the bulk to the CFT one has solved these equation leaving just the physical Hilbert space and dynamical time evolution as governed by the physical Hamiltonian, H_CFT = H_ADM. This brings us to the frontier of what is understood.

TIm,

In case it is not clear, my last message was written before your final one appeared, so let me comment on that, since I definitely do not agree with your statement

". I assert again that the criterion of identity for two operators cannot be that the have the same action on the solution space: the zero operator example shows that. They must at least have the same action on the geometrical space. And to be honest, I don’t see why one should not demand that they have the same action even on the total initial space that includes non-geometrical wavefunctionals. If they act differently anywhere, why aren’t they different operators?"The biggest problem with this is that it defines a rule for equating operators that depends on the particulars of how the theory in question was constructed and not purely on the theory itself. At the end of the day we have a particular quantum theory with a particular physical Hilbert space, and we only want to make statements that refer to that. The whole point of AdS/CFT is to arrive at the same quantum theory via two totally different starting points (quantum gravity in the bulk or CFT on the boundary). If you insist on sticking to your criterion then you can forget about ever making sense of AdS/CFT, since you will be led to operator distinctions on one side of the duality that have no counterpart on the other. To follow up on your integer example, consider the following analogy. Suppose we have two "theories" we call "AdS" and "CFT" that each produce a function on the integers via two different routes, but that are supposed to agree if the duality is correct. They each do so by first producing a function on the real line and then restricting it to the integers. Suppose "AdS" produces f(x) =1, which thus reduces to f(n) =1, while "CFT" produces f(x) = 1+ sin(pi x) which again reduces to f(n)=1. You would be saying that the duality has failed here because the functions disagree on non-integer x, even though only integer x have physical meaning.

One of the biggest themes in modern physics is the idea of duality: that the same quantum theory can be obtained by starting from different classical systems. This is especially prevalent in quantum field theory, and there are many examples where you start with two different classical field theories with different gauge symmetries and after quantization they are the same. One of the theories could even have no gauge symmetry at all while the other does. The statement that you end up with the same quantum theory is that there is a unitary equivalence between the physical Hilbert spaces and the operators that act on them. The modern point of view is that gauge symmetry is just a human invention that is useful in formulating theories, and its presence or absence is convention dependent. Your rule for equating two operators will depend on these arbitrary conventions.

Black Hole Guy,

Yes, it is becoming more and more obvious that the issue of identity of operators is lying at the center of a lot of our disagreement. I think it kept popping up before, and now we have brought it out into the open, which is really helpful.

It may be helpful to talking about the meaning of "duality" as well. Clearly the term as it is used in physics has a much broader application than the one you mention. Take string theory. for example. According to Witten anyway (sorry for the appeal to authority, but I don't know the theory really well) string theory is an intrinsically quantum theory, that is, it is not the quantization of any classical theory. (Even if this isn't right, I see no reason why one should always have to start with a classical theory.) But string theory is supposed to exhibit dualities. In general, a duality (as I understand it) is an isomorphism of structure, a mapping from the operators in one theory to the operators in another (or itself!) and the states in the first theory to the states in the other (or itself) such that the probabilities derived from the corresponding operators operating on the corresponding states are the same. In the limit, such a duality could cover the entire theories which, as I understand it, is the Holographic Hypothesis.In the case of AdS/CFT I find the claim that there is such a complete duality frankly incredible. There might be a duality that maps the CFT to a part of the bulk theory, and that might be of interest, but then we need to get clear about what maps to what.

con't

In any case, I really do want to highlight this sentence from your post: "The modern point of view is that gauge symmetry is just a human invention that is useful in formulating theories, and its presence or absence is convention dependent." There are gauge symmetries that I understand in this way, such as the gauge symmetry in classical E and M stated in terms of the scalar and vector potentials. It is easy to understand that theory as one where the potentials are human inventions that, for purely mathematical reasons, can be useful in describing electro-magnetic systems. Choice of a gauge for the potentials is then just a convention. And the choice of a co-ordinate system to describe a system is obviously merely conventional. But as we have seen, the constraints that appear in WdW are not merely the reflection of gauge symmetries: the dynamics of the gravitational theory is also encoded in the constraints, and that is not merely conventional. When someone says that the time development on WdW is "pure gauge", I think that is a wildly misleading claim. It arises from the fact that the choice of a Cauchy foliation is a free choice: there are lots of foliations and none is physically preferred. But the dynamics itself is not somehow "just a human invention". Assimilating the freedom to slice up the space-time with the gauge freedom in E & M potentials gives one the wrong idea. I'm sure we will come back to this.

As for the operators: what is the physical import of the Hamiltonian operator? In many cases, it is the generator of time translation, i.e. of how the state of the universe changes with time. Since "changes with time" is not univocal in GR (because of the alternative possible foliations), it is clear that the Hamiltonian must take a somewhat unfamiliar form. But it still plays a central role in implementing the physics: by annihilating *only a certain subset* of states on the kinematical Hilbert space the Hamiltonian sorts out the space-times that obey the GR dynamics from those that don't. The zero operator clearly does no such thing. It can play no role in distinguishing physical possibilities from physical impossibilities. For some reason I can't grasp, you seem to want to downplay or even ignore the programmatic role that the Hamiltonian plays in defining the physical state space, and just act as if we were given the physical state space from on high. Then the Hamiltonian operator has no real work to do: the work was already done in arriving at the physical state space. If you see the work that the Hamiltonian is doing, it is clear that the zero operator can't do that work.

I'm not quite sure how to answer your question about the functions because the meaning of "duality" is not clear to me there. But certainly the *functions* are different *functions* if they take different values for non-integer arguments! Maybe the duality does not require that the functions be the same. And maybe the duality in physics does not require that the operators be the same. But the operators are certainly not the same.

We do need to distinguish "merely gauge" differences in descriptions of physical situations from physically contentful ones. If the differences are merely gauge, then the situations are physically identical. But, in addition, we have to distinguish contentful physical situations that obey the dynamical lows from those that don't. The zero operator plays no role in doing either of these things, The Hamiltonian does them both. Hence they are not the same operator.

Black Hole Guy,

Thanks for your last comments. I think we are moving in a very constructive direction at this point, even though we still have some major disagreements. Let me push a little further on ADS.

I have indeed found the Penrose diagram for AdS, and even with a little digging a Penrose diagram for AdS with an evaporating black hole, although I am not sure if it is standard. I hope that I didn't say that there is no Bondi mass for AdS (not so easy to search through all these posts!) but my understanding is the opposite: in AdS one wants to use Bondi rather than ADM. I'm not claiming this makes a substantial difference, but it was what I had gathered.

As you say the boundary in ADS is timelike, which makes for an important difference from the standard Penrose diagram for an evaporating black hole, which is asymptotically flat. So it will be very. very important to keep separate the results we may get for the AdS case from other possible cases. It would, for example, be fascinating if one could solve the information loss problem in AdS, but only using resources available there, and not in what we take to be the realistic case!

So on the boundary of AdS there is a nice lightcone structure in terms of which a dynamics can be defined. And the boundary of a Cauchy surface in the bulk is going to be a Cauchy surface of the CFT. The CFT dynamics can then evolve that boundary forward, and we can consider a sequence of Cauchy surfaces in the bulk that have the corresponding boundaries.

But note also: exactly specifying a sequence of boundary conditions for the Cauchy surfaces in the bulk does not (unless some other constraint has been imposed) specify a unique foliation into Cauchy surfaces in the bulk. Lots of quite different Cauchy surfaces in the bulk will have the same asymptotics (indeed will overlap beyond some radius R).

When you write "In either asymp. AdS or flat spacetime, two spatial slices that hit the boundary at the *same* time can be thought of as "gauge equivalent" in the sense that one can evolve from one to the other under the action of the Hamiltonian constraint", this is exactly the sense of "gauge equivalent that I want to warn against! Take two spatial slices that overlap beyond some radius R but diverge within R: one curving "up" and the other "down" so that every point on the first where they diverge is to the future of points on the second. The physical states on these surfaces are not "gauge equivalent" in the sense of being physically identical! Indeed, the first might cut above the evaporation event and the other below it (maybe even below the event horizon of the black hole). If one thought that they are "gauge equivalent" in the normal sense then trivially they contain the same "information": they are physically identical. But whether they contain the same information is part of the basic issue we are discussing, and is not a triviality.

My claim all along, or course, is that the relevant spacelike surfaces in the bulk that meet the boundary must be Cauchy, so on a baby universe scenario they change from connected to disconnected. And the reason for this is that they are generated by the action of the Hamiltonian, and the Hamiltonian generates states on Cauchy surfaces from other states on Cauchy surfaces. If this is right, then there are space-like surfaces in the bulk that have the right boundary behavior but still are not "gauge equivalent" to each other in any sense: for example just the piece of a disconnected Cauchy slice that reaches the boundary 9Sigma2out) has the right boundary behavior but it is not equivalent in any sense to the union of that piece with Sigma2in.

Do you agree with these comments?

Tim,

Let's leave duality and string theory aside for the moment. Do you agree with the following statement. Suppose we have two operators, A and B, that act the same on the physical Hilbert space but differently on the kinematic Hilbert space. Then there is no conceivable experiment that can distinguish between A and B. Agreed?

Here is a parable. Suppose we have a civilization that lives on a spatial lattice. There are only the discrete lattice points, and there is no meaning to the "space between the points". The physicists theories produce predictions for various observable phenomena, and let's suppose that one such prediction is a function A(n), where n labels the lattice points. And suppose that the equations that give rise to this prediction involve only the discrete lattice points. Now, some other physicist comes up with a new formulation that involves extending the lattice to the real line. His equation produce a function B(x) on the real line, and then he projects it down to the lattice, B(x) -> B(n). And some third physicist has another formulation that produces C(x) which projects down to C(n). When they test their theories they all get agreement, since A(n) = B(n) = C(n). They can argue forever about whether the real line is "real" or not, but the point I want to make is that they all arrive at the same observable predictions at the end of the day.

We need to accept that a given physical theory can be formulated in many different ways, some involving non-physical degrees of freedom appearing at an intermediate stage. If you insist on focusing on the ways in which they differ in terms of their treatment of non-physical degrees you will miss the essential point, which is that they agree on all conceivable experiments, and so should be said to be the same theory, just formulated in different ways. AdS/CFT is telling us that quantum gravity can be formulated in two very different ways: as gravitons etc. in AdS, or as conformal field theory on the boundary. The only claim here is that they make the same physical predictions. Of course, there is much subtlety in defining precisely what the words "physical prediction" means in this context, and this connects on to current research.

Tim,

"my understanding is the opposite: in AdS one wants to use Bondi rather than ADM."This is partly semantics, but my point is that there is no analog of Bondi mass in AdS. Bondi mass is associated with null infinity, which does not exist in AdS since the boundary is timelike. The energy that is defined in AdS is akin to the ADM energy in flat space: it is defined at spatial infinity, and is conserved in time.

"When you write "In either asymp. AdS or flat spacetime, two spatial slices that hit the boundary at the *same* time can be thought of as "gauge equivalent" in the sense that one can evolve from one to the other under the action of the Hamiltonian constraint", this is exactly the sense of "gauge equivalent that I want to warn against!"This is reasonable. So let us agree to restrict usage of "gauge equivalent" to those bulk slices which differ merely by a reparametrization of the spatial coordinates on the slice.

I agree with your other comments, but let me note the following.

In this canonical setup there is an important structural difference between gravity and E&M. In E&M we can take any wavefunction that is annihilated by the Gauss law constraint -- and hence takes the same value on gauge equivalent configurations -- and then use this as an initial condition for the time dependent Schrodinger equation. But in the gravity case we cannot start from *any* wavefunction with the analogous property (in this case being annihilated by the H_i constraints). In addition, we need to demand that the wavefunction is annihilated by the H_t constraint. Only then can we use this in the time dependent Schrodinger equation, which in this case is the equation that moves the wavefunction forward in boundary time according to id/dt Psi = M_ADM Psi.

This distinction between gravity and E&M would largely disappear if we gauge fix by "choosing a time" in the bulk. However, one can run into problems here since it is not obvious a priori ( except in perturbation theory around a fixed background) what are good and bad gauge choices. So let us refrain from doing this.

One needs to be careful about statements like

""Hamiltonian generates states on Cauchy surfaces from other states on Cauchy surfaces". The notion of a state being defined on a Cauchy surface is the appropriate one if we are thinking about quantum matter on a fixed (as in classical) spacetime background. But not if we are talking about the full WdW wavefunction, since here we have that the state is a wavefunction whose arguments are 3-geometries and matter configurations on them. In the semiclassical limit, the former picture should emerge from the latter, but when we talk about things like connected surfaces breaking up into disconnected components we are outside this regime.Now, suppose we have managed to solve for a Psi obeying H_i Psi = H_t Psi = 0. How do we interpret this object? We would like to think that it computes probabilities in some way, but how? Mathematically, the analogous issue is that in general there is no obvious way to define a scalar product on the space of wavefunctions that would give it a Hilbert space structure. Outside of perturbation theory or the semiclassical limit this is poorly understood as far as I know (though a substantial literature exists).

I bring this up because one needs to understand this issue to give meaning to the idea that before the black hole was formed there were connected Cauchy surfaces but after it evaporated there are only disconnected Cauchy surfaces. Obviously this notion is apparent in the usual Penrose diagram for this process, but what about in the WdW wavefunction? One wants a statement along the lines that at late boundary time the 3-geometry is disconnected with high probability.

BHG,

In reply to your parable above, another one. Our system is a spin-½ particle. Now we consider 2 operators. One is the total spin operator S - ½, and the other is our old friend the zero operator. I ask an experimentalist to measure each. In the first case she sets up some Stern-Gerlach apparatuses and does a series of experiments, calculates a number, subtracts ½ and returns the number 0. In the second case she stares at the operator, never leaves her office for the lab, writes down the number 0 on a piece of paper and returns it.

Are these, in the sense relevant for physics, the same operator?

Tim,

Your parable indeed illustrates the issue well. The Stern-Gerlach apparatus measures S-1/2 of whatever particle is passed through it. So I can use it to measure the spin of a spin-0 particle, a spin-1 particle, whatever. In other words, S-1/2 is a operator that acts nontrivially on the physical Hilbert space, but just happens to annihilate a specific type of state. So I can do experiments to distinguish it from the 0 operator. Even if only spin-1/2 particles are present in this world, I can pass two of them through the apparatus together, and the same result entails. Contrast this with the H_t constraint in gravity: it is zero on all physical states, and no experiment can produce a nonzero eigenvalue for it.

BHG,

I owe you an answer to your parable about operators. I would say this.

A physical theory uses mathematics as a means of representing the physical world. But the mathematical structure is not self-interpreting. The structure comes (or should come) with a commentary that explains which parts of the mathematics are meant to represent physical reality and which are not. For example, someone doing classical E & M could comment that of course the vector and scalar potentials are not supposed to represent physical things: gauge-equivalent states represent the same physical reality.

So it depends on the commentary from A, B and C. If they all declare that they in fact live on a lattice, then I might well accept that they have different presentations of the same physical theory. Maybe one is more mathematically tractable, and hence the obvious one to use for practical reasons. But if B insists that his theory postulates the existence of a continuum but only the discrete locations are observable, then I would insist that B has proposed an alternative physical theory to A. There may or may not be—even in principled—an empirical test to decide between them. Bad luck if there isn't one. Maybe there still are some compelling reasons to prefer one of the two. If not, and the empirical predictions hold up, maybe we will never know which theory is correct. But they are still different theories.

The commentary can declare a mathematical degree freedom to be non-physical. That relieves one of the obligation to make physical sense of it. But not all the mathematical degrees of freedom can be non-physical! Sorting out which mathematical degrees of freedom correspond to physical degrees and which do not is something that has never been agreed upon by physicists about quantum theory. So the project of becoming clear about this now for AdS/CFT is tricky.

Tim,

Allow me to play amateur Philosopher of Science for a moment. Suppose an alien spaceship left us with black box that can provide the precise numerical outcome of any experiment. Would science then be done? No: we humans are less interested in the answers than in the story of how the outcome comes to be. E.g., I personally don't care about the precise value of the neutron lifetime per se, but I am very interested in the story of how the neutron decays. Now, there can of course be many different stories that can describe all the output data of the black box, and the stories may differ wildly in their account of what is "real" versus what is just "mathematical fiction". AdS/CFT is a vivid example of this. They represent two wildly different accounts, even "living" in spacetimes of different dimensionality, that purport to explain the same measured outcomes. The goal now is to understand how these wildly different stories can in fact agree on outcomes when we consider such things as black hole evaporation. This all to say that we should not demand that the two stories match up with each other at intermediate stages, but rather that they agree on the final outcome.

I believe that these comments are in basic harmony with your recent post.

None of my business, but I think most scientists would say the part of B's theory that is not observable is not science and therefore that part is not a scientific theory.

The previous example using Goldbach's Conjecture is similar. As science the two models agree for all observed values (so far), so they are scientifically the same - until proved otherwise by observation or some process of induction.

The whole point of science, as I understand it, is to ground practical knowledge in observation, not in philosophical speculation.

(This comment is not intended as some useful or necessary insight, but to show how the discussion seems from the perspective of the Peanut Gallery.)

BHG,

Let me go back to my parable. The idea was that the whole physical system is just a spin-½ particle, so all of the physical states for that system will yield ½ for a total spin measurement. So if we subtract off the ½ we get an operator that will annihilate every physical state. If you are worried about treating a single particle as the whole system, let me try again. One idea that is floated from time to time, and which I think I understand, is that in some sense the net energy of the entire universe should be zero. So imagine constructing an operator that has pieces that reflect all of the different contributions to the net energy—a kinetic piece and various potentials, etc.—and then making it a condition for physical acceptability that as universal state be an eigenstate of this operator with eigenvalue zero. This seems perfectly coherent, right? And in this case, by definition, the action of the complicated operator on every physical state is identical to the operation of the zero operator. But they are mathematically different operators, of course, and I would say physically different. After all, the one contains lots of physical information about the various sources of energy and the other doesn't. More importantly, the first is doing essential work in the theory—distinguishing the physically possible states from the kinematically but not physically possible ones—which is work that the zero operator obviously cannot possibly do. As my first parable illustrated, if it were possible to set up an actual experiment to measure the total energy of the universe (and maybe it is not physically possible to do this), then the physical structure of the experiment will be dictated by the mathematical structure of the operator. But the zero operator obviously dictates no such experimental conditions at all. I can't see that anything I have just said is at all controversial, and we can say the same thing about any operator that every physical state is an eigenstate of with the same eigenstate. I honestly don't understand how you can maintain that for the purposes of physics these operators are "the same" as the zero operator.

I do understand that some of the constraint operators in the Hamiltonian are not so much physical in this sense, but rather are conditions of a more general sort of coherence, the coherence required to interpret a solution as specifying a particular 4-dimensional space-time. I grant that only such states ought to count as in the kinematical space. But there are certainly states that are coherent in this sense but not physical. And the Hamiltonian does, in addition to the work of weeding out the incoherent states, the additional job of sorting the physical from the non-physical coherent states that are in the kinematical space.. And the zero operator can do none of this work. So your proposed condition for physical identity of operators just seems indefensible. Can you explain how you would solve the issues I have raised?

Tim,

It is worth recalling how this line of discussion originated, which is in the use of AdS/CFT in analyzing black hole evaporation. A point I want to emphasize is that AdS/CFT duality is a statement about the equivalence of physical Hilbert spaces and operators acting on them. Statements on the AdS side regarding unphysical states or operators will likely not have any counterpart on the CFT side.

My criteria that two operators be equivalent is that no conceivable, but physically realizable in principle, experiment can distinguish them. Mathematically this is expressed by saying that two operators are equivalent if they have the same matrix elements between all physical states. You want to enlarge the criterion by considering how the operator acts on some larger Hilbert space in which the physical Hilbert space is embedded. You are free to do this, as long as you keep in mind that in so doing you are making distinctions that have no counterpart in terms of physical measurements; I think of these as statements about how we choose to formulate a theory and as such are convention dependent. But if it will make you feel better I am happy to replace my statement "the Hamiltonian in GR is a boundary operator" by "the Hamiltonian in GR has the same matrix elements between physical states as a boundary operator". All my arguments using AdS/CFT will only use this weaker form of operator equivalence. The AdS and CFT descriptions, in their standard formulations, have wildly different unphysical Hilbert spaces (e.g. in terms of gravitons and gluons respectively) and operators that act on them, but the "miracle" is that they become (conjecturally) equivalent once restricted to the physical Hilbert space of each.

Regarding your comments about the total energy of the universe (assumed to be closed): the statement that the total energy is zero is the statement that H_t = 0 on physical states. As you say, H_t is a sum of terms, and one could think of the terms as being kinetic energy, potential energy and so on, according to some rule for splitting up the operator, such that they all sum up to zero. Bear in mind though that the individual terms are not themselves physical operators in that they do not in general take physical states to physical states. For sure, acting on the full kinematic Hilbert space the operators H_t and 0 are different operators, and in this formulation H_t does the work of defining what the physical Hilbert space is. I have no disagreement with that. What I am saying is that you shouldn't expect AdS/CFT to hold at the level of kinematic Hilbert spaces, so this distinction between H_t and 0 need not have any counterpart in CFT. We can avoid confusion by clarifying terminology as indicated in the previous paragraph, making clear when we are only making statements about the action of some operator on the physical Hilbert space.

Jim V,

The idea that the content of a scientific theory is restricted to what is observable has been tried out (this is a mantra of various stripes of positivists) and has been shown to be unworkable over and over. Einstein expressed one key observation when he said that it is the theory itself that determines what is observable. That is, a clear theory specifies an ontology (what the theory postulates to exist) and a dynamics (equations that specify, either deterministically or probabilistically, how the ontology behave through time). All of this gets specified first, without regard to observability. Then if one wants to figure out what is observable in the theory one uses the theory itself: model the observer physically and then see under what conditions the observer can get information about a target system, and what information it can acquire. Observing—getting information about a distal object—requires the right sort of dynamical connection between the observer and the observed. Whether of not that connection exists, or even can exist, is determined by the laws of the theory.

One can ask what a theory postulates, and what evidence there is for various parts of the theory. The evidence can be weaker or stronger, and it is important to ask what evidence there is and which parts of the theory it bears on, and how strongly. But drawing a line between the "scientific" and "unscientific" part is not useful: the whole process of theory articulation and test by evidence is part of scientific inquiry.

Black Hole Guy,

I think we have managed to make some conceptual and terminological distinctions that will allow us to avoid talking past each other. And I think that we have agreement on many of the main points. So let's try to get back to the bearing of AdS/CFT on our original discussion, and maybe we can make some progress. Let me start with a few observations to see if we agree.

1) At the end of the day, what we are interested in is our original evaporating black hole scenario, which is not in an asymptotically AdS space-time. So as we consider the AdS case we should keep especially close tabs on whether some result or argument relies on structure that exists in AdS but is absent in the evaporating black hole case.

2) As an example, in the original case, the fictive spatial infinity is a single point. In order for it to be possible to enlarge the original space-time by such a point (together with null and timelike infinities) , the asymptotic behavior of the original space-time has to be very special (much more than just that the metric approach flatness as the spatial coordinate for to infinity). So one thing to discuss is whether the evaporating black hole case has (or plausibly has) the right structure to allow for the addition of these fictive "boundary points".

3) I am still not clear about the extent to which the original case and its spatial boundary is analogous to the AdS case. As you mention, AdS has a timelike boundary, which means that there can be a Hamiltonian on the boundary that generates boundary states from each other. Any Cauchy surface in the bulk limits to a Cauchy surface on the surface, so any sequence of Cauchy surfaces on the boundary corresponds to a sequence of sets of Cauchy surfaces in the bulk. Sets because different Cauchy surfaces in the bulk can have the same asymptotics (indeed two different Cauchy surfaces in the bulk can overlap as they approach the boundary). So is it important for the correspondence that there be these sequential sets of Cauchy surfaces? If so, then it looks like bad news for hoping to find an analog in the original case, since there is only one point at spatial infinity. If not, then the dynamics on the boundary is really playing no essential role in the case of interest.

4) Finally, I need to recur to something I have already brought up: what exactly is the "correspondence" in the AdS/CFT correspondence supposed to be? In order of increasing interest for our case:

a) There are some operators and states on the boundary and some operators and states in the bulk such that there is an isomorphism between them. (These might be interesting operators and states, and the isomorphism might simplify certain calculations.

b) There is a complete set of operators on the boundary and basis set of states which are isomorphic to some set of operators and states in the bulk

c) There is a complete set of operators on the boundary and basis set of states which are isomorphic to a complete set of operators and some set of states in the bulk

4) There is a complete set of operators on the boundary and basis set of states which are isomorphic to a complete set of operators and basis set of states in the bulk.

Is it a), b), c), d) or none of the above?

Cheers,

Tim .

Tim,

"the asymptotic behavior of the original space-time has to be very special (much more than just that the metric approach flatness as the spatial coordinate for to infinity). So one thing to discuss is whether the evaporating black hole case has (or plausibly has) the right structure to allow for the addition of these fictive "boundary points"."I'm not sure what you're getting at here. We can just use the standard definition of asymptotic flatness which has built into it the existence of a conformal compactification. This certainly allows for processes like black hole evaporation. What are you worried about?

"So is it important for the correspondence that there be these sequential sets of Cauchy surfaces? If so, then it looks like bad news for hoping to find an analog in the original case, since there is only one point at spatial infinity."Again, I don't follow. Note that in the asymptotically flat context spatial infinity is not a point, it's only a point in the unphysical conformally compactified spacetime. In asymptotically flat spacetime we can certainly have a sequence of Cauchy surfaces and a Hamiltonian which generates evolution among these. So in both AdS and flat space there are "sequential sets of Cauchy surfaces". As to whether this is "important for the correspondence", all I can say is that it is a fact, so in that sense yes it is important.

"what exactly is the "correspondence" in the AdS/CFT correspondence supposed to be? "Here we have to cognizant of the fact that while we do understand pretty much everything about the basic structure of the CFT (even if we can't actually compute everything) the same is not true of the AdS side. There is no complete definition of quantum gravity in AdS, and indeed part of what we want to use AdS/CFT for is to arrive at such a definition. Of course, we do understand a lot about quantum gravity in AdS in various regimes (e.g. perturbation theory around AdS and around various other classical solutions) and we know how the physics in these regimes matches up with the CFT side. It is always possible that an eventual formulation of QG in AdS will involve a larger set of states and operators than is present on the CFT side. However (assuming some version of AdS/CFT is correct) there will have to be a consistent truncation of that theory down to the smaller space that the CFT sees, since the CFT is a self-consistent theory. The key thing to note here is that we know that the scope of AdS/CFT is broad enough to include the process of black hole formation and evaporation. We know how to prepare a CFT state that corresponds to a collapsing ball of matter, and we know that the CFT describes the resulting evolution as a process of thermalization, and we know that the state after a long time is a pure state that maps to a bulk state that is a gas of quanta in AdS. What we don't understand are questions like: what is going inside the black hole horizon during the evaporation process.

So as a working hypothesis I think we can assume your strongest option (4) for the correspondence, provided that operators and states are all taken to be physical, in the sense we have discussed. Bulk operators that act outside of the physical Hilbert space will not have any analog on the CFT side.

Finally, while it is logically possible that there is some important distinction between asymptotically AdS and flat spacetimes when it comes to black hole evaporation, this seems very unlikely to me. The AdS radius can be taken to be arbitrarily large compared to the black hole size, such that any reasonable observer near the black hole would be unable to see the distinction. If the relevant physics is at all local, it shouldn't matter what the spacetime metric is doing 100 megaparsecs away.

There is no Hawking-Page phase transition in asymptotically flat space time; nor a minimum blackhole temperature. Yes, it is very annoying and puzzling to me, but it seems that the global structure of spacetime is important, even though "physics is local".

If there was some kind of conserved or mostly-conserved unitarity or information current, so that we could calculate potential unitarity leakage (or absence of loss of unitarity; or information loss) that would be much more satisfying, and goes in hand with "physics is local". We don't seem to have that. It seems to me that "information" and "unitarity" are global, not localizable.

It is also worth remembering: "The bizarre anti{de Sitter spacetime"

https://arxiv.org/pdf/1611.01118.pdf

"In this spacetime almost everything is bizarre including its name.

(CAdS is the covering AdS space).

"Since the in nity J is actually timelike, the e ect is that far future cannot

be predicted in CAdS space....Physics in CAdS is unpredictable."

"In any field theory the ground state solution must be stable against small perturbations, otherwise the theory is unphysical. For Minkowski space it has been proven after long and sophisticated investigations that the space is stable since sufficiently small initial perturbations vanish in distant future due to radiating off their energy to infinity. The spatial infinity J of CAdS space actually is a timelike hypersurface and any radiation may either enter the space through J or escape through it. It is therefore crucial for the question of stability to correctly choose a boundary condition at infinity. Most researchers assume reflective boundary conditions: there is no energy flux across the conformal boundary J , in other terms the boundary acts like a mirror at which outgoing fields

(perturbations) bounce off and return to the interior of the spacetime. Under

this assumption P. Bizon recently received a renowned result: CAdS space is unstable against formation of a black hole for a large class of arbitrarily small perturbations"

"The geodesics have in CAdS space infinite extension, yet their relationships cannot be altered in comparison to these in AdS space. Two geodesics having a common initial point must intersect ... and the intersections will repeat infinitely many times, always after the same interval of the proper time."

"The fact that in CAdS space all timelike geodesics starting from a common

point can only recede from each other to a finite distance and then

must intersect infinite many times, has two important consequences. First, a

timelike geodesic cannot reach the spatial infinity J ." "Second, there are points inside the future light cone of any P0 that cannot be reached from P0 by any timelike geodesic."

"The conclusion, therefore, is unambiguous: this spacetime is unphysical and cannot describe a physical world."

TM, thanks for the reply, although I was satisfied to express an outsider position without being part of the discussion.

We disagree; as I see it what has proven unworkable is pure philosophy without a strong connection to observation, which has led to theology, astrology, alchemy, homeopathy, Republicanism, and so on.

I don't mean to restrict how theories, scientific or otherwise, are conceived. Whatever works for someone is fine with me, but what, as I understand it, the scientific method demands is that hypotheses be tested against observation. Therefore, it seems to me, any theory which can't be tested by observations cannot be called scientific. Also, any theory which predicts all relevant observations over a significant period of time is scientific (until it ceases to do so), regardless of philosophical objections. The universe is probably too complex for human brainpower to ever completely comprehend it. In science, we are simply doing the best we can.

Jim V

Of course, by your criterion it is arguable that none of the work on string theory and M theory has ever been scientific, despite dominating theoretical physics for decades. One can embrace such a result, but it might give one pause.

BHG,

Let me explain some of the worries I have. Maybe you can clear them up.

One worry has to do with how restrictive the conditions for an asymptotically flat space-time are. Wald, in his General Relativity, lists five conditions, and the initial account only applies to vacuum solutions. He loosens the conditions somewhat, but it is not obvious that a space-time with an evaporating black hole could be asymptotically flat. And that is required in order to append a boundary to the space-time.

"Note that in the asymptotically flat context spatial infinity is not a point, it's only a point in the unphysical conformally compactified spacetime. In asymptotically flat spacetime we can certainly have a sequence of Cauchy surfaces and a Hamiltonian which generates evolution among these." Isn't the "unphysical conformally compactified space-time" exactly the space-time on whose boundary the CFT lives? If all of the Cauchy surfaces limit to spatial infinity and spatial infinity is just a point, then wouldn't the CFT have to be defined on that point? (Things may look more reasonable in an asymptotically AdS space-time, but that would rule out the main case that we were interested in.

It appears that there is still a lot of guessing about AdS/CFT, and, as you say, no understanding of how to recover what is going on behind the event horizon from the CFT. If that is right, then how do we know that what happens behind isn't the production of a baby universe? We got into all this because you wanted to claim that the non-degeneracy of the state in the CFT would be incompatible with baby universes. I actually found a place in Maldacena's original paper where he claims that there is only a correspondence between a *sector* of the CFT and the bulk theory, so the CFT actually has *more* states than the bulk theory does.

Maybe we will do best to try to avoid general issues abut AdS/CFT and focus on our issue. You said that the structure of the CFT is incompatible with the bulk theory producing baby universes, and the issue has to do with degeneracies. Can you state the argument as completely as possible, and then we will have a better sense of what properties the AdS/CFT correspondence has to have for the argument to go through.

Tim,

Let me leave aside the asymptotically flat case, except for the following comments. First, standard definitions of asymptotic flatness certainly allow for evaporating black holes: the radiation from a black hole is thermal, and not qualitatively different than the radiation from an ordinary star, and the definitions of asymptotic flatness would be pretty lame if they didn't allow for the presence of stars. Second, there is no well understood analog of AdS/CFT in asymptotically flat space. On the other hand, for a solar mass black hole in an AdS space with curvature radius of 100 megaparsecs, no local observer would be able to distinguish this from a black hole in asymptotically flat space, so it seems plausible that the mechanism of information loss/preservation will be essentially the same in the two cases.

". I actually found a place in Maldacena's original paper where he claims that there is only a correspondence between a *sector* of the CFT and the bulk theory, so the CFT actually has *more* states than the bulk theory does."I'm quite sure you are misunderstanding something here, but you'll have to point me to the place in the paper where you read this before I comment further.

With that out of the way, I return to the AdS/CFT based argument for no baby universes. Let's start with some standard assumptions about the CFT. Note that the CFTs that appears in the standard examples of AdS/CFT are very well understood, having been studied for decades, going back to the 1970s. This doesn't mean we can always compute everything analytically of course. I will assume that the theory has a Hamiltonian with a non-degenerate spectrum (this is not actually quite true, since there are small degeneracies due to global symmetries, but this is not germane, as I can explain if necessary). There is a unique vacuum state invariant under the conformal group. If I act on the vacuum with some operator to produce an excited state, the system will evolve in time, eventually settling down to a state which looks thermal, in the sense of being nearly indistinguishable from a thermal ensemble.

Putting the CFT aside for the moment, let's consider black hole formation in AdS. We start in the vacuum state, given by empty AdS. At some time we inject an infalling pulse of matter from the boundary. This is described by some pure state. It then gravitationally collapses to form a black hole, which subsequently evaporates by Hawking radiation. Viewed from the outside it appears as if the black hole fully decays into radiation. However, examining the Penrose diagram for this process it seems that what has happened is that the original pure state has evolved to a pure state defined on a disconnected Cauchy surface, with one component inside the horizon and another component outside. The baby universe scenario is the statement that the bulk Hilbert space is therefore a tensor product and the pure state is highly entangled among the two factors. We conclude that no information is lost overall, but any observer who can only make measurements on the tensor factor outside the horizon sees a mixed state.

cont

cont

What is wrong with this? I now invoke that the bulk Hamiltonian is a boundary operator in the sense I have defined it previously (its matrix elements between physical states can be extracted solely from measurements at the boundary). So, by definition it follows that operators that act on the tensor factor inside the horizon commute with the Hamiltonian (more precisely, the physical state matrix elements of the commutator vanish). This implies that the spectrum of the bulk Hamiltonian is highly degenerate, since given any energy eigenstate we can act with any of the large number of these behind the horizon operators without changing the energy.

This conflicts with the CFT. We know how to prepare the CFT state that corresponds to the initial collapsing matter, and we know that it evolves forward in time by Hamiltonian evolution. The final state it evolves to is not highly degenerate. In fact, the final CFT state is a pure state, and since it is essentially thermal it can be mapped to a thermal gas of quanta in AdS. So the CFT tells us that in the bulk the final state of the Hawking radiation by itself must be pure.

There are various issues here that I have skirted over in the service of brevity. But the main point is that the proposed bulk scenario is one in which the Hilbert space breaks up into tensor factors, with operators from one factor commuting with the Hamiltonian. This is not possible in the CFT.

Black Hole Guy,

You gave a perfect summary of the situation. So let's stay focussed in this until we reach an agreement: there are only a couple of paragraphs to go through. We can see in those paragraphs where your definition of "the same" operator and mine will disagree, and hopefully we will see how that disagreement has brought us to different conclusions.

Let me quote what I take to be the key steps in your argument. The first is this:

"The baby universe scenario is the statement that the bulk Hilbert space is therefore a tensor product and the pure state is highly entangled among the two factors. We conclude that no information is lost overall, but any observer who can only make measurements on the tensor factor outside the horizon sees a mixed state."

The second is this:

"What is wrong with this? I now invoke that the bulk Hamiltonian is a boundary operator in the sense I have defined it previously (its matrix elements between physical states can be extracted solely from measurements at the boundary). So, by definition it follows that operators that act on the tensor factor inside the horizon commute with the Hamiltonian (more precisely, the physical state matrix elements of the commutator vanish). This implies that the spectrum of the bulk Hamiltonian is highly degenerate, since given any energy eigenstate we can act with any of the large number of these behind the horizon operators without changing the energy."

Here is my question. If one accepts this reasoning, why doesn't it apply just as well to *any* solution? Surely in any account the Hilbert space of the bulk can be written as the tensor product of Hilbert spaces in many ways, where one Hilbert space contains the pure boundary states. In other words, why does it matter here that the Cauchy surface becomes disconnected? Let the Cauchy surfaces all stay connected, but divide the bulk into two regions by a surface that is R X S^n-2, with the event horizon of the black hole contained inside this surface. Every Cauchy surface gets divided into 2 parts: one in the interior of R X S^n-2 and the other in the exterior + the intersection with R X S^n-2, Why can't the Hilbert space of the bulk now be written as a tensor product of the Hilbert space of the interior of R X S^n-2 and the Hilbert space of the exterior + R X S^n-2? The boundary operator will not act on the interior part, just as it does not act on the baby universe part. If we regard the Hamiltonian of the interior of the bulk as the zero operator, as you wish to do, then of course the Hamiltonian of the interior of R X S^n-2 will commute with all other operators that act inside R x S^n-2. The rest of your argument then goes through verbatim.

What am I missing here? I just can't see any difference at all in the two cases.

If we can clear this up, that will be real progress.

Tim,

"Here is my question. If one accepts this reasoning, why doesn't it apply just as well to *any* solution? Surely in any account the Hilbert space of the bulk can be written as the tensor product of Hilbert spaces in many ways, where one Hilbert space contains the pure boundary states. In other words, why does it matter here that the Cauchy surface becomes disconnected?"Great, we have arrived at the key point, which in fact lies at the heart of how a holographic duality like AdS/CFT is possible in the first place. I want to convince that you in a gravitational theory the Hilbert space cannot be written as a tensor product in the way you indicate above.

Let's consider a region of empty Minkowski space contained within some spatial sphere of radius R_big. Inside that let there be a smaller sphere of radius R_small. First consider a non-gravitational theory consisting of just some scalar fields, say. Then it is indeed the case that the Hilbert space can be written as a tensor product, with one factor describing the region r < R_small and the other factor describing the region R_small < r < R_big. In particular, this means that I can act on the vacuum state with an operator localized in r < R_small, such that the new state so produced is indistinguishable from the vacuum state insofar as any measurement performed in the region r > R_small is concerned. (This statement holds at sufficiently early times, since eventually the excitation will propagate outside R_small. ) So we can talk about excitations that are entirely localized in R_small, and so it is sensible to think of the HIlbert space as a tensor product.

Now turn on gravity. Quantum states then involve both the scalar field and the gravitational field. The big difference from the previous case comes from the fact that energy gravitates universally. Any excitation inside R_small inevitably has an associated gravitational field that "leaks" out into the region r > R_small. Indeed, I can measure the energy of the excitation by sitting at R_big and measuring the gravitational field there. At the classical level, the statement is that there does not exist any solution of Einstein's equations corresponding to non-Minkowski space for r< R_small and Minkowski space for r>R_small, even for a small amount of time. At the quantum level the statement is that the physical Hilbert space is not a tensor product, since there is no way to create excitations in rR_small region and find that they are indistinguishable from what I would get in the vacuum state, then I can conclude with certainty that we have the vacuum in r< R_small. Obviously, such a statement would be impossible in a non-gravitational theory.

I hope this makes it clear why it is that if we have a situation with connected Cauchy surfaces then the Hilbert space in quantum gravity cannot be a tensor product. We can then turn to the case of disconnected Cauchy surfaces, which is where AdS/CFT comes in.

This is one nice post on the "Naturalness" conjecture which claims that parameters of sensible theories should differ not more than an order of magnitude from a dimensionless "characteristic constant" of the theory.

In cosmology and particle physics this characteristic has been chosen as the Planck mass, and now everybody wonders why the Higgs is so light.

As a former experimental nuclear physicist I always marvel that an elementary particle is as massive as an Iodine atom ! "Lightness is in the eye of the beholder.

The numerological games a la Eddington, Dirac etc. are entertaining, as for a column in Scientific American, but in my view they should not guide the advancement of science

Black Hole Guy,

Sorry to be taking so long: I am at a conference that takes up all the day.

It is interesting if the Hilbert Space of solutions to WdW cannot be written as a tensor product space but as far as I can tell, seeing the direction this is taking, the talk of connected and disconnected Cauchy surfaces is a red herring here. The feature of gravity that you seem to be appealing to is this: the existence of the space-like constraints that arise from the shift operator means that if I act at one point of a connected Cauchy surface with an excitation, there will have to be changes across the whole Cauchy surface, out to spatial infinity. If I am following, you then want to conclude that if the Cauchy surface is disconnected, an excitation in one piece need only change the state on that piece, and if it is the piece not connected to the boundary the excitation cannot register at the boundary. Hence if the Cauchy surface is disconnected, we get degeneracy at the boundary.

But (assuming this is where you are going) this whole argument has taken a wrong turn. Note that in many cases there is no objective fact about whether an event lies on a connected or disconnected Cauchy surface. In many cases, it depends on the foliation. More particularly, every event inside the event horizon is on a connected Cauchy surface in some foliations and a disconnected Cauchy surface in other foliations. So if it suffices for an excitation to alter the boundary condition that it sits on a connected Cauchy surface, that works for every event inside the event horizon.

Indeed, the only events that objectively sit on disconnected Cauchy slices (i.e., the slices are disconnected in all foliations) are events in the future light-cone of the Evaporation Event. And in their case, they all sit on the piece of the disconnected surface that connects to the boundary. So by this criterion every event is connected to the boundary in the sense that every excitation anywhere has effects at the boundary.

If I have gotten the drift of the argument wrong, then please correct. But since that seems to be the direction it was going, I thought I would point out that it is not headed towards the conclusion that you have claimed.

I am not sure how to take this statement in Hawking & Ellis "The large scale structure of space-time":

One can think of a singularity as a place where our present laws of physics break down. Alternatively, one can think of it as representing part of the edge of space-time, but a part which is at a finite distance instead of at infinity. On this view, singularities are not so bad, but one still has the problem of the boundary conditions. In other words, one does not know what will come out of the singularity."If a singularity is a boundary at a finite distance, then a spacetime that is supposedly asymptotically AdS is technically no longer purely asymptotically AdS when a black hole is formed with a singularity, because there is this piece of boundary that is the singularity. The CFT in such an AdS/CFT is an incomplete specification of boundary conditions.

Of course, maybe I'm taking the above quoted statement too literally.

Tim,

I don't disagree with what you are saying, but it doesn't really address my point. Let me attempt to boil the argument down even more. I believe that you want to claim that there is a sense in which at "late times" the bulk Hilbert takes the form of a tensor product, with one factor describing degrees of freedom inside the horizon (or baby universe) and the other describing the region connected to the AdS boundary. Now, since the Hamiltonian is a boundary operator in the precise sense we have discussed, it follows that the Hamiltonian acts purely on the second tensor factor. My point is simply that the CFT doesn't have this structure: the CFT Hilbert space does not take the form of a tensor product, with the Hamiltonian acting on only one factor. (Of course, here I mean that both factors have dimension greater than one, and I am referring to the types of CFTs that arise in actual realizations of AdS/CFT. )

Arun,

That is an interesting observation. I would be interested in BHG's response.

Black Hole Guy,

For the moment, let's accept what you say. I can't see how it is relevant. The difference in structure of the respective Hilbert spaces would violate the Holographic Hypothesis in its strongest form, but we have never agreed that AdS/CFT is even an instance of the Holographic Hypotheis (we never ever got a sharp statement of the content of AdS/CFT). Are you saying one must accept Holography to make the argument you are making?

Tim,

No, you don't have to simply accept that. Let me try again. Based on many, many, explicit computations, we know that the relevant CFTs define a theory of quantum gravity in AdS in the sense that: a) they are quantum theories, that b) reduce to ordinary bulk semi-classical gravity in appropriate regimes. That is, Einstein's equations, Hawking radiation, etc. emerge. My statements regard *this* theory of quantum gravity, allowing for the logical possibility for there to exist other theories of quantum gravity where the physics is different. Now, in this theory, we know that pure states evolve to pure states in the process of black hole formation and evaporation, since the CFT manifestly has this property. At this point, the retort is "yes, the final state of the CFT is pure, but maybe the bulk description of this state is as a tensor product, with one factor behind the horizon and the other outside, in which case information is effectively lost to an observer who only has access to the second factor". My point is that this scenario is impossible, because the tensor product assumption combined with the fact that the Hamiltonian is a boundary operator is in conflict with basic CFT structure, as I explained in previous posts.

As I said, there is always the logical possibility that the theory of quantum gravity relevant to our world is fundamentally different from the one described by AdS/CFT. And even in the AdS/CFT context, nobody is really satisfied in the sense that we don't have any fully satisfactory story in the bulk about how the information can come out in the Hawking radiation, even if we are confident that this is what happens.

Arun,

The singularity inside a black hole is a spacelike surface, so it occurs "in the future" not at a finite spatial distance like you seem to be imagining. The statement that a spacetime is asymptotically AdS is a statement about how fields behave at large spatial distance, so it is perfectly compatible with the existence of a singular spacelike surface inside the event horizon. Also, the singularity is just a breakdown of classical GR, but everyone expects that the equations of quantum gravity will tell us how to evolve "through" the singularity without having to invoke additional boundary conditions. Finally, the CFT clearly has all the boundary conditions it needs to define a unique evolution, and it has no room or need for additional boundary conditions corresponding to those at a singularity inside AdS.

Thanks, BHG!

Black Hole Guy!

I think I have got this worked out now. We have both been missing a key fact that was sitting in front of our noses. Before coming to that, let's review the bidding.

At this point we have a few outstanding disputes. One is whether it is legitimate to think of the Hamiltonian as just a boundary term or as a boundary term together with a bulk term that includes the Hamiltonian constraint and the spatial diffeomorphism constraint. I remain unconvinced on your view here. Another is whether the solution space is a Hilbert space at all (Thiemann says not). Yet another is whether the solution space—granting that it is a Hilbert space—has a tensor product space structure. I have claimed that it does, although that was really off the top of my head, having not thought about the matter. I'm open to the claim that it is not a tensor product. Maybe the observation I am about to make will shed light on this. I'm not sure.

Here is the observation. Both of us have been presuming that the disconnection of the Cauchy surfaces in the Penrose diagram reflects the disconnection of the baby universe from the bulk. Such a disconnection means—according to your critique—that one can act with creation or annihilation operators in the baby universe without creating any change in the physical state outside the event horizon. Hence multiple interior solutions are compatible with the same exterior solution, and the exterior solutions are degenerate. But the exterior solution includes the boundary, so the boundary solutions would have to be degenerate. The CFT, however is not degenerate in this way, so something must have gone wrong.

What we have both been missing is that the disconnection of the Cauchy surface does not imply that operators inside the event horizon have no effect outside. In reality, it is just the opposite: the disconnection indicates that acting with operators outside the event horizon can have no effect inside the horizon. The disconnection does not isolate the exterior from the interior, is isolates the interior from the exterior!

The key is the observation I made a few days ago. Since there is no preferred foliation, one must be cautious in reading any significance into the behavior of a single foliation. Suppose, for example, that everything is OK so long as the Cauchy surface is connected. Then things should be OK at any event that sits on a connected Cauchy surface, even if there are other Cauchy surfaces on which it is disconnected. But every event inside the event horizon sits on a connected Cauchy surface. The only events which *objectively* sit on disconnected Cauchy surfaces, i.e. which *only* sit on disconnected Cauchy surfaces, are the event in or on the future light cone of the Evaporation Event. It is *these* events that display a novel sort of behavior, not the events inside the event horizon, i.e. the events in the "baby universe".

What is novel about space-time point in the future of the Evaporation Event is that excitations there have no effect in the interior of the event horizon even though they are space-like separated from the interior. Of course, the fact that an excitation can have any effect at space-like separation might sound strange: shouldn’t an excitation only have effects in its future light-cone? But that isn’t right: in the gravitational case the spatial diffeomorphism constraint means that acting on any point with a excitation can require changes at space-like separation. More concretely, an excitation inside the event horizon that changes the mass there must also alter the event horizon itself, which has knock-on effects outside the horizon, all the way out to the boundary. But an excitation to the future of the Evaporation Event will only alter the physics outside the event horizon: it will leave no trace inside the horizon, in the baby universe.

Con't

Think of it this way: the spatial diffeomorphism constraint acts within a Cauchy surface. If a Cauchy surface is disconnected at an event, then the action is confined to the piece connected to the excitation. If we consider all of the Cauchy surfaces through the event, we get the range of the effect of the excitation produced by the spatial diffeomorphism constraint (along with the Hamiltonian constraint). As we have seen, by this criterion all the excitations inside the event horizon have effects out to spatial infinity where the boundary term resides. But excitations in the future of the Evaporation Event have no route to the interior of the event horizon since no connected Cauchy surface leads from one to the other. The spatial diffeomorphism constraint cannot, as it were, jump over the disconnection.

If this is the right story, then the disconnection of the Cauchy surfaces does not indicate an isolation of any part of the space-time from the boundary: every event lies on a Cauchy surface connected to the boundary. It does indicate an isolation of the interior of the event horizon from the future of the Evaporation Event, but that is no problem for the CFT duality since the boundary term is not inside the event horizon. So the grounds for your worry about degeneracy evaporate. Everything hangs together without a problem.

Tim,

I am not seeing how your comments address the main issue. To clarify, I am not claiming there is any issue at early boundary times, such that there exists a Cauchy surface that contains both the boundary slice and the interior of the horizon. I believe this is what your comments pertain to. Rather, I want to focus on a boundary time well after the black hole has evaporated, so that no such Cauchy surface exists. The CFT state at this (or any other) time is pure. In some shape or another, you are proposing that this CFT state corresponds to a bulk state which is also pure, but such that an observer who can only make measurements on the part of the Cauchy surface connected to this late time boundary point sees a mixed state. What are you proposing for a mathematical description of this state? Is it a tensor product, with one factor describing the degrees of freedom inside the horizon and the other outside? (that's what I would think is the mathematical characterization of saying that a baby universe has "split off" from the rest of the universe). If yes, then my argument addresses your scenario. If not, then I guess I don't know what you are proposing. To reiterate, as far as the region outside the horizon is concerned, I only want to make reference to very late times such that the only Cauchy surfaces are disconnected.

BHG,

According to the Penrose diagram, as I have shown, there just is no "boundary time well after the black hole has evaporated, so that no such Cauchy surface exists." All such late time Cauchy surfaces have a disconnected piece inside the horizon. That's the whole point.

You ask what the mathematical description of the state is. On the whole surface, it is pure, which answers the problem about unitarity. On each of the disconnected pieces it is mixed, since the pieces are entangled. So it is certainly not a product state of two states, one on each piece.

When you ask "Is it a tensor product, with one factor describing the degrees of freedom inside the horizon and the other outside? (that's what I would think is the mathematical characterization of saying that a baby universe has "split off" from the rest of the universe)." This is obviously not a question about the mathematical structure of the *state* but the mathematical structure of a *space of states*. Which raises the question of which space we are talking about. I had assumed you meant the *kinematic* space, and made a guess that it is a product state. That guess may be wrong: I have no investment in it in any case. But you apparently meant the space of *solutions*, which I never had in mind. As I have repeatedly said, I have no reason to think that the space of *solutions* is even a Hilbert space, much less a tensor product of two Hilbert spaces. Indeed, I have reason to think it isn't. But if it is, then I agree that it does not have the structure of a product space such as you describe. Certainly, acting with an excitation inside the event horizon is *not* compatible with leaving the state outside unchanged: changing the interior mass changes the horizon structure and hence the exterior state.

Now having settled this, what is the argument that anything about the CFT is incompatible with this solution?

Tim,

I seem to have made several unfortunate typos etc. in my last message which garbled my point. I first of all meant to say that there is no issue at sufficiently early times such that there exists a connected Cauchy surface. I want to focus instead on a late boundary time such that there is no connected Cauchy surface, only a disconnected one. Then I meant to ask whether the bulk Hilbert space that is dual to the CFT Hilbert space at this boundary time takes the form of a tensor product, with one factor describing the interior of the event horizon. Note here that I will only ever refer to the physical Hilbert space -- the larger kinematical Hilbert space contains all sorts of unphysical states that can never be created and are convention dependent. If you want to claim that that there simply is no physical Hilbert space then we may as well end the discussion here, since you are then proposing that the basic structure of quantum mechanics breaks down after all, not to mention AdS/CFT.

Assuming you do claim that there is a tensor product Hilbert space in this disconnected Cauchy surface case, would you not agree that the action of an operator in the region connected to the boundary (again, always at late times) has no observable effect inside the event horizon? If yes, then it follows that the action of an operator inside the event horizon has no observable effect on the exterior late time region, since in both cases what we are asking here is whether commutators [O_in, O_out] vanish or not. But then it follows that the action of O_in does not change the energy, since the latter is measured at the boundary. This leads to the contradiction with what we know from the CFT about the spectrum of the energy.

BHG,

I think I have answered all this, but one more time:

There is a kinematical Hilbert space, that really is a Hilbert space. Then, in the case of WdW, there is a solution space, the elements of the kinematical Hilbert space that are annihilated by the Hamiltonian operator. At this point in the process of constructing solutions, it is evidently absolutely essential that the Hamiltonian operator *not* be the zero operator, or the zero operator together with a boundary term, or just a boundary term. If it were the zero operator all the states in the kinematical Hilbert space would count as solutions. If it were just a boundary operator, then being a solution would put no constraints on the interior. I assume there is nothing in this paragraph to dispute.

OK, so we operate with the Hamiltonian on the kinematic Hilbert space to get a space of solutions. Is this space itself a Hilbert space? I have no dog in this fight one way or the other. You assert that if it isn't a Hilbert space then "the basic structure of quantum mechanics breaks down". I don't see why that should be true, but maybe it is. I do have the explicit claim of Thiemann that the solution space in one case, the set of physical states, is *not* a Hilbert space. As far as I can tell, there is nothing wrong with that result at all. In particular, it does not suggest that quantum mechanics breaks down. I have asked around a bit whether the solution space of a theory must be a Hilbert space, and have gotten the reaction that it might or might no be—the issue has to do with limits of Cauchy sequences—but nobody thought that it mattered much one way or the other. Everyone does think that the kinematical space is a Hilbert space. So that's where I am: actually indifferent, but puzzled about why you seem to be so sure that the solution space is a Hilbert space.

So let's just grant it is. Fine. Next question: does it have the structure of a tensor product of Hilbert spaces? Since I am not committed to it being a Hilbert space at all, I am obviously not committed to it being a tensor product of Hilbert spaces. And seeing your argument that it isn't—namely the universality of gravity implies that a change anywhere implies a change everywhere—let's just grant that it *isn't* a tensor product of Hilbert spaces. What follows?

As far as I can see, nothing follows that creates any problems for my solution. Certainly, if I produce an excitation inside the event horizon, it ought to have effects everywhere. No problem with effects inside the horizon, and outside the horizon the state changes due to the effect of the extra mass inside. So this is not a case where a change inside leaves some other region unchanged.

What about an excitation outside the event horizon: will it always lead to changes inside? In particular, what about an excitation in the future light-cone of the Evaporation Event? This is trickier: the change would require a change along the outside part of the Cauchy surface out to spatial infinity, but would not in any obvious way affect the interior of the Event Horizon.Maybe there would have to be some other effect as the changed physical situation must be changed in the back light cone of the excitation. Maybe that excitation must change the interior somehow. But let's suppose the worst: suppose it doesn't. Suppose acting on the future of the Evaporation Event does not influence the interior of the event horizon. Well...so what? It is still the case that acting *anywhere* with an excitation alters the state at spatial infinity where the boundary is. There is no degeneracy at the boundary.So why is the observation that there is no degeneracy in the CFT a problem?

As far as I can tell, none of this requires a product space structure of the solution space. So what's the problem?

BHG,

One more comment. I don't think that your argument about the action of O_in not changing the energy makes sense. Take any two localized operators at different points on a Cauchy surface (connected or disconnected). They commute. That's just the ETCRs, right? You seem to want to conclude from this common commutator structure that there will be degeneracies in the spectrum. But it can't be that simple, or QFT would always have degeneracies since it imposes the ETCRs.

Also: If acting inside the event horizon changes the energy there, then it changes the ADM mass at the boundary. And if such a change registered at an earlier time normal unary evolution will make changes at later times as well. So acting inside the horizon will change the late time boundary.

Tim,

On the kinematical vs. physical Hilbert space question, the point is very simple. Suppose that the space of physical states could not be made into a Hilbert space. How do you propose to do quantum mechanics then? Everyone agrees that the only states that can ever occur are physical states (as the name implies), so the basic questions we want to ask are things like: what is the probability for one physical state to evolve into another. The rules of quantum mechanics say that you compute the inner product between two states and then square it to get the probability. The probabilities have to be positive and add up to one. This is precisely the structure implied by having a Hilbert space. If you don't have a Hilbert space defined on the space of physical states then you can't compute inner products among these states, and so can't compute any probabilities. Perhaps what has caused confusion here is the technical question of whether the required inner product can be obtained by starting with an inner product on the kinematical space and then restricting it to the physical space. That procedure can indeed fail due to convergence issues (which is what Thiemann is referring to), but it no way relieves one of the need to define some inner product, by hook or crook, on the physical Hilbert space if you want to do quantum mechanics.

"Take any two localized operators at different points on a Cauchy surface (connected or disconnected). They commute. That's just the ETCRs, right? You seem to want to conclude from this common commutator structure that there will be degeneracies in the spectrum. But it can't be that simple, or QFT would always have degeneracies since it imposes the ETCRs".In ordinary QFT without gravity the Hamiltonian is of course not a boundary term, so there is no argument in this case. There is also no issue to resolve in the case of gravity and a connected Cauchy surface, as I have explained. The issue just concerns gravity and a disconnected Cauchy surface.

I really don't understand what your position is now, so let me ask the following, where we again focus on late boundary times such that the Cauchy surface is disconnected. You are proposing that information is lost as far as an observer confined to the exterior region of the Cauchy surface is concerned. This means there are lots of different pure states defined on the full Cauchy surface that look the same (technically, define the same reduced density matrix) as far as the exterior region is concerned. So in particular they all have the same energy, since energy is defined in the exterior region. I don't how you can escape that conclusion.

So it seems to me like you want to have it both ways: you want to say that there are lots ways to change the state inside the horizon that are undetectable outside, yet object to my claim that this implies a degeneracy in the energy spectrum.

BHG,

Now I see why you have been reacting about the Hilbert space comments the way you have. Essentially, you are thinking of a Hilbert space as a vector space with an inner product, so without that you can't do quantum theory. I was actually concerned about the completeness requirement for a Hilbert space, not the inner product requirement. If you just said that the solution space was an inner product space...sure, no problem, it can inherit the inner product from the kinematical space. But it does not obviously inherit completeness. If I read Thiemann right, in his case it doesn't inherit completeness: you need to complete it by adding some non-physical states.

The technical point about the solution space not being a Hilbert space (in the exact sense) is not important in itself. I was just using the example in Thiemann to show that what is meant by "the Hilbert space"there is the kinematical space, not the solution space. And that was important in our debate about the identity conditions of the Hamiltonian: whether it is identified by its action on the kinematical space or the solution space. I still maintain that it is the action on the kinematical space, so the Hamiltonian in WdW just is not the zero operator. This seems so obvious to me that I can't express it more clearly: the Hamiltonian mathematically expresses the constraints that determine which kinematical states are solutions. The zero operator does not. They are not the same and should not be used interchangeably.So the main issue was not whether the solution space really is a Hilbert space in the sense of being complete—nothing hangs on that—but which Hilbert space is relevant for identifying operators. Since in Thiemann the solution space is not complete, it was just a clear indication of what he meant by "the Hilbert space".

Moving on...

Answering your question: I can't see that your question has anything to do with the Cauchy surface being disconnected. Let's take an early. connected surface that has a piece inside the event horizon. Now I can act with an excitation at different points inside the event horizon yielding different states. If each excitation increases the energy inside the event horizon the same amount, they will have exactly the same effect outside the event horizon (by the no-hair "theorem"), and the external observer (outside the event horizon) cannot tell exactly which excitation occurred. A fortiori, the change in the boundary condition (which will change, because the ADM mass changes and the event horizon changes) cannot distinguish between the two excitations. So there is degeneracy at the boundary. To quote you: "This means there are lots of different pure states defined on the full Cauchy surface that look the same (technically, define the same reduced density matrix) as far as the exterior region is concerned."

All of that is correct, and follows not from any Cauchy surface being disconnected but from the existence of an event horizon in the first place, i.e. the presence of a black hole. The no-hair "theorem" implies exactly what you wrote about "the exterior region" where that means "exterior to the event horizon". So if I have a problem, you have the identical problem, as far as I can see.

Con't

In the case of the disconnected surface, acting on the "in" piece will have some effect on the "out" state: if I add mass to the interior, there will be correspondingly more Hawking radiation that reaches the "out" piece. This will carry information about the additional mass/energy inside. But it will carry no more information than that. So the case of the connected Cauchy surface with a piece inside the event horizon and that of the disconnected Cauchy surface with a piece inside the event horizon are strictly parallel. That can't possibly be an objection to my solution. It is an objection to the Holographic Hypothesis, which I think we have no grounds at all to accept, and lot's of grounds (like this one) to reject. Since I am still not clear whether in your view AdS/CFT is a holographic conjecture, and can't say whether it is reason to reject AdS/CFT. It might be.

Tim,

I think we are making some progress here. I hope we can agree on the following: 1) There exists a theory of quantum gravity in the bulk. 2) This theory obeys the axioms of quantum mechanics; namely, physical states are in correspondence with rays in Hilbert space. 3) If there is an asymptotic boundary, there exists a Hamiltonian that generates time translation, where time is defined on the boundary. 4) The Hamiltonian is a boundary operator, in the sense that its matrix elements between physical states can be measured by operations at the boundary.

If necessary, I can defend each of these statements and verify them explicitly in perturbation theory around some background. Note that I am only referring to *physical * states here, since at the end of the day they are all that matter.

You ask:

" I am still not clear whether in your view AdS/CFT is a holographic conjecture,"I am happy to answer this if you state the question more precisely, defining what you mean by a "holographic conjecture"Now, let's get back to the main issues. You write

" Let's take an early. connected surface that has a piece inside the event horizon. Now I can act with an excitation at different points inside the event horizon yielding different states... they will have exactly the same effect outside the event horizon (by the no-hair "theorem"), .... So there is degeneracy at the boundary." <\i>

First of all, the no-hair theorem is a statement about classical General Relativity, while here we are talking about the quantum theory, In any case, we are getting to the heart of the matter. Namely, if in the the quantum theory it is possible in this setup to exhibit a large class of physical quantum states that look different inside the horizon but are all indistinguishable to measurement outside the horizon, then this would contradict AdS/CFT. I realize this sounds counterintuitive, but AdS/CFT says you can always reconstruct the state from measurements at the boundary. If you present to me a controlled setup where we can do reliable computations (like in perturbation theory about some background) then I can show you precisely how this works. These are not just idle words: there is a huge body of careful work on this question

BHG,

Let's start with your four propositions. 3) is ambiguous. It could mean that H generates time translation *for the asymptotic boundary state*, or it could mean that H generates time translation *in the bulk*. If you mean the latter, then I do not even know what it would be to accept it. The very notion of "time translation" presupposes a global time function, i.e. a foliation into Cauchy slices. Since that can be done many ways that completely agree at the boundary, having a time defined on the boundary does not automatically define a time in the bulk. So: do you mean the former or the latter? If the latter, how is the time function on the boundary extended into the bulk?

4) raises again the question of what counts as "the Hamiltonian" and also what counts as a "boundary operator". What the term suggests is an operator that is only sensitive to the state at the boundary, which cannot be right. If it were, then changes in the interior of the bulk would make no difference to the effect of the Hamiltonian, so the Hamiltonian could not determine the interior state. Maybe your characterization in terms of matrix elements does not imply this insensitivity to the bulk, but if so then the term "boundary operator" is misleading.

By a "Holographic hypothesis" I mean the assertion that there is a 1-to-1 map from states in the bulk to states in the boundary and operators on the bulk to operators on the boundary that is an isomorphism: the probabilities assigned to corresponding operators on corresponding states in the two theories are always the same. In that sense, they are "the same theory". Again we face the question of this means for all kinematical states, or all physical states, I guess I mean the kinematical states. If you think it holds for the physical states but not the kinematical states, then just say that.

About AdS/CFT: yes, it does sound counterintuitive. But there is something more than that. The original Maldacena paper is very, very particular: it concerns a string theory with branes, and the limits as N goes to infinity. None of this specific detail has been any part of our discussions, and indeed bringing up AdS/CFT in the context of my paper (which does not mention black holes in AdS) suggests that the term "AdS/CFT" is now being used as code for some much, much more general principle than is defended by Maldacena. So can you state exactly what AdS/CFT means as you use it?

One more question: as you use it, does "AdS/CFT" only refer to cases where the bulk theory is pure gravity? For example, does the characterization of the bulk contain electromagnetic degrees of freedom? If not, why think it tell us anything about realistic cases?

Let's clear up these questions before going on

Tim,

Let me once again stress that my arguments make reference only to physical states. Any argument that relies on unphysical kinematic states is inherently flawed, since such states never occur and are just part of the formalism used in one specific approach to arriving at the quantum theory (in other approaches, such states are absent from the outset).

" 3) is ambiguous."I explained this earlier and provided a reference, so was relying on that discussion. I will briefly recap. Saying that we are considering spacetimes that are asymptotically such-and-such means that the metric behaves in a prescribed way asymptotically, and using this one can give physical meaning to time at infinity, which you can think of as being the reading on a classical clock at infinity. Physical state wavefunctions will take the form \Psi[ g_{ij}, t], where t is this time at infinity, and g_{ij} is the spatial metric on a 3-dimensional surface, which you should think of as being attached to infinity at the time t. So the wavefunction is a function of all possible 3-geometries that asymptote to a given boundary time. If you compute these physical state wavefunctions in the classical limit you can see how you get back the notion of classical 4-geometries complete with local time evolution. Essentially, the wavefunctions in this limit are peaked at 3-geometries that can be embedded in a 4-geometry that is a solution of Einstein's equations. You can see this worked out explicitly in the reference I mentioned, Phys.Rev. D42 (1990) 4042-4055, by Fischler, Morgan, and Polchinski.

"4) raises again the question of what counts as "the Hamiltonian" and also what counts as a "boundary operator" "The Hamiltonian is literally built out of the metric near the boundary, so it's manifest that its matrix elements can be computed by measurements at the boundary. Again, this is for matrix elements of physical states. If you insist on embedding this in the larger kinematical space, then we can say that the Hamiltonian also has a bulk piece given by integrals over the constraints; these annihilate physical states (by definition) so do not contribute to any physical matrix element.

" What the term suggests is an operator that is only sensitive to the state at the boundary, which cannot be right. If it were, then changes in the interior of the bulk would make no difference to the effect of the Hamiltonian, so the Hamiltonian could not determine the interior state. "Again, you simply cannot make a change confined solely to the interior, since physical operators must include the gravitational field of the excitation, and that field stretches out to infinity where it can be measured. If there happen to exist two states with precisely the same energy, then indeed the Hamiltonian cannot distinguish them. We can discuss this case more if you'd like, but for now I will just say that in any AdS/CFT example this only occurs if the states are related by a symmetry, and this is not relevant for the issues we are discussing.

cont

cont

"By a "Holographic hypothesis" I mean the assertion that there is a 1-to-1 map from states in the bulk to states in the boundary and operators on the bulk to operators on the boundary that is an isomorphism: the probabilities assigned to corresponding operators on corresponding states in the two theories are always the same. In that sense, they are "the same theory". Again we face the question of this means for all kinematical states, or all physical states, I guess I mean the kinematical states. If you think it holds for the physical states but not the kinematical states, then just say that."If you replace "the boundary" with "CFT", then this is a correct characterization. And it definitely refers only to physical states, as the CFT has no notion of kinematical states to begin with. For a large class of operators, including all those we have been discussing, CFT operators can be identified with boundary limits of bulk operators. It might be true in complete generality, but I think it's fair to say that this is an open question (to me anyway). I don't think this subtlety is directly relevant to our discussion so far.

About realizations of AdS/CFT. Yes, Maldacena's original examples came from explicit string constructions, and that is still the route we typically go by if we want an example that is completely well defined at all energy scales. At low energies, these examples look like General Relativity coupled to some matter fields, including some mix of scalar fields, fermions, electromagnetic fields and so on. For many purposes, arguments can be made within this low energy effective theory, and that is the spirit I am adopting. Generically matter fields are present, and in fact there is no known example coming from a string construction that reduces to General Relativity without matter at low energy.

" my paper (which does not mention black holes in AdS)"Of course it does, on p. 23. There you make the rather unfortunate remark "in the absence of any understanding of how the space-time geometry of the bulk is “encoded” in the degrees of freedom at the surface". It's unfortunate because anyone who reads this and is familiar with all of the detailed understanding of this topic will likely dismiss your paper out of hand. I don't think it helps your case.

Tim Maudlin wrote:

By a "Holographic hypothesis" I mean the assertion that there is a 1-to-1 map from states in the bulk to states in the boundary and operators on the bulk to operators on the boundary that is an isomorphism: the probabilities assigned to corresponding operators on corresponding states in the two theories are always the same. In that sense, they are "the same theory". Again we face the question of this means for all kinematical states, or all physical states, I guess I mean the kinematical states. If you think it holds for the physical states but not the kinematical states, then just say that.IMO, if the holographic hypothesis holds for reasons related to physics, and not purely mathematical reasons quite unrelated to physics, then the holographic hypothesis should hold for physical states (and IMO, for those physical states in the bulk that can be constructed by shooting in and out stuff through the boundary + the evolution of that stuff in the bulk).

BHG;

I read through the piece you mention by Fischler, Morgan, and Polchinski. I do not have command of the details, but there are some things that are clear.

1) Like everyone else except you, the specify the Hamiltonian in terms of specific constraint equations. They could not do anything if the Hamiltonian were the zero operator, as you oddly insist it is equivalent to. Your insistence on this cannot be reconciled with any of the work I have ever seen.

2) They explicitly discuss topology change of the Cauchy surfaces in a connected space-time. Just the think I am arguing for.

3) They never suggest that the Hamiltonian is just a boundary term.

4) They mostly restrict discussion to spherically symmetric situations, cutting down the degrees of freedom drastically.

In sum, there is nothing in this paper I can see that backs up any of the claims that you make that I have been disputing. For example, you write "The Hamiltonian is literally built out of the metric near the boundary". The constraint equations just are not so built, and are essential to the Hamiltonian. So it is not literally—or figuratively—built out of the metric near the boundary.If you can supply a citation where anyone makes the claim you are wedded to, viz. that the constraint operators are "equivalent" to the zero operator because they both annihilate all the states in the solution space, please supply it. I cannot find anyone who agrees with this, or can even make sense of it.

In my paper, the Penrose diagram under discussion is obviously not AdS. Since the physical universe is not AdS, no realistic discussion should require that structure. The AdS boundary is timelike and the boundary under discussion is not, so if the solution depends on the timelike structure of AdS there is no reason to think it helps with actual physics.

Now: act with the same excitation at different places inside the Event Horizon. If No Hair holds, the result will be the same outside the Event Horizon and hence out to spatial infinity. You insist, if I follow, that these two different excitations must have different effects at spacelike infinity. If it is understood how the state on the CFT reflects the state in the bulk, there must be a story about how these two excitations yield different bulk states. Please cite a paper that discusses and analyzes a case like this. My reading of the literature is that the "dictionary" connecting bulk to boundary is quite incomplete and sketchy. In not, please point to the relevant literature.

All IMO only, I don't have any solid backing for these following:

1. In "Fashion, Faith and Fantasy" Roger Penrose raises the issue of the mismatch of the cardinality of the functional degrees of freedom of unconstrained physical fields in spaces of different dimensions.

If Penrose's argument is good, then IMO, the only hope for there to be a one-to-one correspondence between bulk and boundary states is that the correspondence is limited to physical states in the bulk, and that the physical state constraints effectively reduce the dimension of the space by 1.

2. Both Tim Maudlin and BHG have to be careful in their arguments about e.g., applying a creation operator behind the event horizon or e.g., annihilating the long-after-evaporation dilute gas of radiation to create a vacuum. The reason is, IMO, that the typical application of a localized operator to a bulk physical state produces a kinematic state that is not a physical state; and there is no necessary correspondence between kinematic bulk states and physical boundary states. To produce a bulk physical state from another bulk physical state, one has to have a consistent space-time history from the initial/boundary conditions for whatever did the creation behind the horizon, or for whatever annihilated the radiation of the evaporated black hole. In the latter case, I suspect that most solutions also have no black hole in the first place.

3. Thus the argument for the degeneracy of the baby universe solution has to be constructed very carefully.

I'm happy to be shown to have mistaken intuition about any/all of this (the best way to learn :) :) :) )

Arun,

1) I would like to understand the response to Penrose as well, as his observations cast doubt on the viability of the Holographic Hypothesis. (As an aside, a real holograph does not, of course, provide an example of the Holographic Hypothesis. A real holograph stores information in a flat 2-D surface about a curved 2-D surface embedded in a 3-D space. It does not store information about the entire 3-D space, e.g. the interior of the 3-D items depicted.)

2) I agree about the care here. What was in my mind was acting with a local operator and then re-applying the constraints, which (as far as I can tell) would create changes at space-like separation out to spatial infinity. In particular, it would change the ADM mass at spatial infinity. It would have to, via changing the size of the Event Horizon. As I wrote before, local changes behind the event horizon do not seem problematic because all interior events sit on connected Cauchy surfaces. Acting in the future of the Evaporation Event is a different matter, as you note, because that case requires using a disconnected Cauchy surface. I am not confident that any such solution won't contain a black hole, but it is this case that is really problematic (even thought the local excitation is connected by any Cauchy surface to spatial infinity).

3. Agreed!

A real holograph does not store information in a 2-D surface. There isn't anything like a 2-D surface in 'reality'. A holograph has a width, which depends on the resolution. This isn't nitpicking, it's actually important - there is no real-world example for the holographic principle. The other difference is that a holograph keeps information locally - you can break it in parts and still get back the whole image.

Not that it matters much to your discussion.

Tim,

"1) Like everyone else except you, the specify the Hamiltonian in terms of specific constraint equations. They could not do anything if the Hamiltonian were the zero operator, as you oddly insist it is equivalent to. Your insistence on this cannot be reconciled with any of the work I have ever seen."I have never said this. What I have said, repeatedly, is that the Hamiltonian acting on *physical states* is a pure boundary term, a fact which is made completely manifest in the paper of Fischler et. al. Let me explain again. They start with a space of arbitrary wavefunctionals of the 3-geometry. They then demand that the wavefunctions are annihilated by the constraints. These define the physical states. I don't how much more clear I can make this: the constraints are nonzero operators when acting on the full unphysical space, but they are zero (by definition) when acting on the physical states. Do you really want to argue with the statement that the bulk part of the Hamiltonian acts as the zero operator on physical states? Am I not saying the same thing that appears in every reference on this topic? I am truly baffled as to why you are having trouble with the statement that the Hamiltonian reduces to a boundary term when acting on physical states. The classical limit of this statement is just the familiar fact that you can determine the energy of an object by measuring its gravitational field at infinity.

Just to make clear that I am not saying something remotely nonstandard here, I just googled some combination of the words "General relativity, Hamitonian, boundary term" and immediately found the following statement in a talk by Gary Horowitz (an expert on gravity if ever there was one) "Diffeomorphism invariance implies that the Hamiltonian is defined by a surface integral at infinity." see http://web.physics.ucsb.edu/~gary/holography.pdf

Yes, Fischler et. al. indeed discuss topology change, which is part of why I brought this paper up. Note that I have never objected to topology change, although it remains an open issue (a statement which I know first hand the authors would agree to). They also restrict to spherical symmetry, but I don't see how that is of direct relevance here.

"The AdS boundary is timelike and the boundary under discussion is not, so if the solution depends on the timelike structure of AdS there is no reason to think it helps with actual physics."At this moment in time it turns out that we have a by far better understanding of quantum gravity in asymptotically AdS than in asymptotically flat space. It is here that we can rule out your scenario. If you want to take the point of view that this has no implications for the asymptotically flat case, then you are welcome to hold that belief. Note however, that if the AdS radius is sufficiently large the space is indistinguishable from asymptotically flat space to any local observer, so if different physics is at work in the two cases that would be a strange situation indeed. In any case, you bring up AdS in your paper.

"If No Hair holds, the result will be the same outside the Event Horizon and hence out to spatial infinity."Again, you are invoking a nonexistent quantum no-hair theorem. If you can give me an example of two quantum states that look different inside the horizon of a black hole but are indistinguishable outside then I would be happy to analyze it. Such an example would disprove AdS/CFT. If you would like to read about how different black hole microstates are distinguished you can consult papers by Mathur and other on "fuzzballs". The understanding of this is incomplete for sure, but still very impressive.

BHG,

Here is the question at hand: will differently located excitations inside the event horizon always yield different states at the boundary? If not, then the degeneracy you are on about exists just because there is an event horizon, i.e. because there is a black hole at all.

Now: how does one investigate this question?

The excitation must be represented by a local operator, otherwise the idea of having the same excitation in different places makes no sense. But operating with a local operator, if we do nothing else, takes us out of the solution space. We need to re-impose the constraints, which is why the Hamiltonian better not be the zero operator in the bulk.

Now: if we excite a mode inside the event horizon, that changes the ADM mass. This is fine: The constraints propagate the change due to the excitation out at space like separation to the boundary. But the ADM masses in the two cases will be the same, so that does not help with the degeneracy. If the no hair theorems were to hold, we would know that the boundary states in the two cases are the same. So you are forced to deny the no hair property in this gravitational theory just to keep your argument that there is no degeneracy. Standard properties of GR are being abandoned. The question is: on what basis are they being abandoned?

Is there 1) an actual calculation for a case like this which yields a differencet at the boundary that goes beyond the ADM mass so that it would differentiate the two excitations? Can you point to such a calculation and explain in what way the boundary state differs in the two cases?

Or are you appealing to the Holographic Hypothesis to argue that there must be some such difference? If the latter, then this is question-begging, since the example is being used to show that the hypothesis is false.

Once again, this issue is completely orthogonal to what happens due to evaporation. It just suggests that the Holographic hypothesis is false, and therefore there certainly will be degeneracy in the bulk states that are compatible with a given boundary state. Which is what we expect on dimensional grounds in any case.

Finally, whatever is going on, why think that the existence of disconnected Cauchy surfaces has anything to do with it? The question I just raised makes no use or mention of disconnected Cauchy surfaces. And, to repeat once more, every event inside the event horizon sits on a connected Cauchy surface that goes out to spatial infinity. Why should the solution advocated in my paper be any worse off than any other solution that is committed to the existence of an event horizon?

Is an excitation that respects the symmetry of the horizon local?

Sabine,

I'm not sure what "respects the symmetry of the horizon" means here. Suppose we form a black hole by the collapse of a real star, so the matter is not exactly spherically symmetric. The classical no hair theorem (conjecture?) implies thatthe resulting space-time is nonetheless spherically symmetric, at least asymptotically. A local excitation inside the event horizon, which might correspond to the formation of a black hole via the collapse of a slightly different star, will also lead to a spherically symmetric exterior region. So the excitation would not have to respect the symmetry.

Even if it did, there ought to be different spherically symmetric matter distributions that give the same ADM mass. So we still have the question of what features at the boundary beside the ADM mass could characterize the state.

BHG,

I am trying to work through the papers you recommend, but given my understanding of fuzzballs I think it is too much to worry about that. It is, as I understand it, very speculative and also tied to string theory. It cannot be any part of the reason that anyone had over the first 40 years to think there is a problem about information loss in black hole evaporation. I understand that many people are committed to Holography, but that also was not part of Hawking's original puzzle. At this point, the main thing I want to get straight is whether this degeneracy of bulk states that correspond to boundary states is a) really a problem and b) somehow more of a problem for the approach I am advocating than for other approaches. So far, I don't see either. I understand the sort of problem you were trying to generate from having the solution space be a tensor product state, but nothing in my approach commits one to that, as far as I can see.

Maybe some alternative approach to mine can be made to work, but my first order of business is to see if it is quickly ruled out on some grounds. I still don't see that.

Tim,

I would again like to emphasize the precise scenario I am arguing against, which is the following. Consider a late boundary time t, such that all Cauchy surfaces are disconnected. We ask: are the following statements mutually compatible at time t:

1) The CFT is in a pure state

2) The CFT has a nondegenerate energy spectrum

3) The bulk state is a pure (but entangled) state in a tensor product Hilbert space, with the two tensor factors describing the degrees of freedom inside and outside the horizon respectively. Although the full state is pure, an observer confined to the exterior region cannot determine it since he only has access to operators in the exterior region.

Answer: they are incompatible. Proof: act on a given state with an operator that acts solely on the interior factor. These two states have the same energy, since measurement of energy is done at the AdS boundary. This contradicts (2).

I believe (3) is the scenario you have in mind, but if I am misrepresenting it in some way please do clarify,

To address one of your comments, it is only when there is a disconnected Cauchy surface that there is a *chance* that the Hilbert space is a tensor product. For a connected Cauchy surface that extends to the boundary this is certainly not the case, since any excitation must be accompanied by its long range gravitational field. This is closely associated to the standard mantra that in quantum gravity there are no such things as local observables.

So far, I am not aware that you have countered the argument above except to say that maybe AdS/CFT is false.

I will address your other points later.

Tim,

I said "respects the symmetry of the horizon" and not spherically symmetric merely because black hole horizons don't have to be spherically symmetric. But fine, let's take spherical symmetry. Yes, I've heard of the no-hair theorem. But as you say, that's asymptotical, which means, realistically, never. You need to carry away all the higher moments. If you make an excitation inside the horizon that's not spherically symmetric, you must imprint the violation of symmetry outside, somehow. If it's spherically symmetric, I don't think it's local. I'm not sure what you mean by local though. Hence my question.

Besides this, I'd like to point out that if the fields aren't on compact support you'll not ever stuff them completely behind the horizon. I know I sound like a broken record about that compact support thing, but I think the relevance of this is entirely underappreciated. If you have a field that's analytic throughout space-time there's no information loss problem anywhere at any time - by definition. You only get a problem if you have a field that has a singularity (in the mathematical sense, not necessarily in the sense of a divergence) in some derivative. It's highly non-local of course.

While I'm at it, here's something that stumps me about this symmetry thing. As you pointed out, it's the non-spherical information that's carried away, while all spherical configurations (with the same mass, angular moment, charge) result in the same end-state. But for what the entropy is concerned, that's exactly the wrong way round. If you counted the entropy of this endstate, you'd have one macrostate that doesn't distinguish the radial information. Instead, the entropy seems to count the 'perpendicular' information - the part that escapes. I've pushed this back and forth in my head for a long time, but can't help thinking it feels very wrong. Do you have any thoughts about this?

BHG,

I thought we had sorted this out. To be clear: first, I was thinking of the kinematical space, not the solution space in terms of the tensor product structure and second, the claim that even the kinematical space has a tensor product structure was just off the top of my head. Nothing in the paper presupposes it. As far as I can tell, I can accept that the solution space is not a tensor product. I think that the name "baby universe solution", which is not one I came up with, is misleading here. It suggests that the "baby universe", i.e. the interior of the event horizon, is "cut off" from the rest. But it isn't, at least in terms of Cauchy surfaces: as I said, every event in the interior lies on a connected Cauchy surface. If the "disconnection" is associated with disconnected Cauchy surfaces, then it is the future of the Evaporation Event that is "disconnected" from the interior. But this does not imply a tensor product solution space. So your scenario gets no purchase on my paper. What would contradict the paper is full Holography. But as it has been presented here, as I understand it, full Holography would also violate no-cloning and is intrinsically implausible in any case.

Sabine:

I understand your concerns and share them. One question about the so-called loss of information is why the information can't be carried by the degrees of freedom of the event horizon itself, which would in turn effect the detailed form of the Hawking radiation. Classically, one appeals to the no-hair "theorems" to argue that the geometry of the external region is not affected by the details of the stress energy tensor inside the event horizon, only by the total mass, and the characteristics of the Hawking radiation are determined by the geometry of the exterior region. But the no-hair "theorem" is asymptotic, so at all finite times one can argue that the information is in subtle correlations. I also understand the issue about compact support, which is why I suggested the technique of applying a local excitation end then solving for the total solution vis the constraint equations, which would alter the solution out to spatial infinity.

There seem to be two problems with thinking this way. One is that even if there are these subtle gravitational effects out to infinity from a local excitation, the ADM mass is only defined at spatial infinity, which is where the CFT lives. The other is that there at non-gravitqtional degrees of freedom to deal with. A particle and its antiparticle have the same mass, But recovering the initial state of a black hole would require distinguishing an electron-proton pair going in from a positron anti-proton pair, even though the stress-energy distribution and total charge are the same. So instead of worrying about respecting the symmetry, one can just ask how the state of the whole system would differ at space-like infinity if there were the electron-proton pair instead of the positron-anti-proton pair. I don't see how they can differ gravitationally, so it seem that the holographic hypothesis must be wrong.

Tim,

I would like to step back for a moment to explain the general thinking of people who attack black hole issues via AdS/CFT, which I hope will put my comments in proper context. We are excited about AdS/CFT because it provides a nonpertubative formulation of a quantum theory of gravity (QG). What do we want out of a QG theory? Well, the basic demand is that it should be a theory that a) reproduces Einstein's theory in the classical limit, and b) obeys the axioms of quantum mechanics. There is strong evidence (I would say overwhelming) that large N CFTs interpreted as QG in AdS satisify these conditions. We can indeed verify that Einstein's equations emerge in the classical limit, and CFTs are manifestly quantum theories of the traditional sort. It is crucial to note that this is the ONLY example of a QG theory that we have, i.e. that satisfies these conditions. Loop quantum gravity has not been shown to reduce to Einstein gravity in the classical limit, and there are excellent reasons to believe that it won't without some massive fine tuning at best. Pre AdS/CFT string theory was just a perturbative theory, so issues of black hole evaporation were generally outside its scope. The Wheeler-DeWitt equation is entirely formal and can't be used for any computations outside the semi-classical limit. So, given that all indications are that AdS/CFT gives us a good QG theory in this sense, it behooves us to study its implications. My comments should all be read along the lines of "in the QG theory defined by AdS/CFT, such and such is impossible". It remains logically possible that there are other theories of QG where are other scenarios are possible. In particular, I not going to claim that your scenario is impossible in a global sense, although it might be, since it implies a non-unitary S-matrix and one would have to carefully study if that is compatible with all known facts. Banks, Peskin and Susskind started down this road, but I agree with others that their work is far from conclusive. To summarize, it's not so much that I am making a "holographic hypothesis", but rather that in AdS/CFT we have an explicit QG theory handed to us and we should figure out what it says. Personally, I suspect that QG is so tightly restricted that lessons learned from AdS/CFT are likely to be universal truths, but others may differ on this. I can't help adding that the argument sometimes given that "we don't live in AdS, so how is this relevant to our world?", doesn't carry much force. After all, we don't live in asymptotically flat space either. We live in an expanding cosmological spacetime in which there is a finite distance out to which we can see, or ever will see. So we will never know what the structure of our universe is at arbitrary large distance.

I have argued that AdS/CFT is incompatible with a precise claim about the final state after black hole evaporation. Now, I think you are proceeding to ask precisely the right follow up questions: when the black hole is still present, how does AdS/CFT account for the fact that in semi-classical gravity it seems like we can change the state behind the horizon in all sorts of ways without changing the state outside? This is obviously closely related to asking for the mechanism by which all this information inside the horizon eventually gets imprinted on the Hawking radiation. Some people (fuzzball and firewall enthusiasts) have come to the conclusion that the only way this can all hang together is if there is in fact a dramatic breakdown in the semi-classical description of black holes, with the region inside the horizon either dramatically altered or absent altogether. This remains speculative. The bottom line is that in AdS/CFT there exists no scenario, consistent with what we know, in which the information doesn't all come out in the Hawking radiation. But there remain huge gaps in our understanding of how this comes about in the bulk.

Arun,

"1. In "Fashion, Faith and Fantasy" Roger Penrose raises the issue of the mismatch of the cardinality of the functional degrees of freedom of unconstrained physical fields in spaces of different dimensions."If Penrose had bothered had to examine how AdS/CFT works he would have resolved this confusion for himself. In the CFT the fields indeed live in one dimension less, but the CFT fields are N x N matrices. N is essentially given by the ratio of the AdS length scale divided by the Planck scale, so we need large N for the bulk to be semiclassical. So the extra functional degree of freedom that Penrose is worried about is manifestly accounted for by the large matrix size of the CFT fields. It's as simple as that.

It seems Penrose would believe you if N → ∞ but not N very, very large but finite.

Quote:

How then are we to escape the apparent inconsistency in the functional freedoms in the two theories? Very possibly an answer may lie in one feature of the correspondence that I have, as yet, not addressed. This is the fact that the Yang-Mills field theory on the boundary is not really quite a standard field theory (even apart from its 4 supersymmetry generators), but because its gauge symmetry group has to considered in the limit where the dimension of this group has to go to infinity...the fact that the gauge group's size has been taken to

infinityfor the AdS/CFT correspondence to work could easily resolve the apparent conflict in functional freedom.End quote.

I'm curious - what is the entropy of the extremal blackholes with zero horizon area and how are they represented in the boundary CFT?

Or is this above an ill-posed question?

BHG,

Sorry for the delay: Busy week.

I think we are reaching some sort of equilibrium at last, which is nice. I guess we won't resolve the issue about the physical equivalence of operators (action on solution space vs. action on kinematic space), so let's just leave that issue. And your last post clarifies a lot of how you are thinking of AdS/CFT as the only non-perturbative theory that we can actually calculate with, so something worth taking seriously as indicative of the general character of QG. I'm sure we agree that how typical or representative AdS/CFT is, assuming the strongest claims about it pan out, is open for debate. In particular, insofar as the results depend on the space-time being asymptotically AdS it is of questionable relevance to actual physics, and certainly to the asymptotically flat situation represented in the iconic evaporating black hole Penrose diagram I am analyzing. I can't follow your objection to the observation that we don't think we live in and asymptotically AdS space-time. If the important properties of AdS/CFT are particular to asymptotic AdS spaces, then one just can't draw more general or generic conclusions from it (e.g. Holography). I have recently had a conversation that suggested that the duality really does depend critically on using AdS, which is important if true.

The issue about degeneracy of the energy eigenstates that depended on taking the solution space to be a tensor product is also moot, since I was never committed to the claim that it is. The moniker "baby universe" theory may be misleading here.

So what is left? You mention the non-unitary S-matrix objection and Susskind, Banks and Peskin. I have four comments on that:

1) Hawking introduced the idea of a super scattering matrix because he thought that the "information" just ceased to exist, so you had to trace over the interior of the black hole. My whole point is that this is wrong: on the only sense of "still exists" there is in Relativity, the interior still exists relative to any Cauchy slice that lies to the future of the Evaporation Event. My claim is that the evolution from one Cauchy state to another is always unitary and pure-to-pure, so there is no super scattering matrix needed at the fundamental level.

2) Any Cauchy-to-non-Cauchy evolution, on any view, would have to be represented by a super scattering matrix. So there just can't be anything bad or questionable about such matrices per se.

3) The Susskind, Banks and Peskin paper in particular (which has been cited to me independently, and so must have some kind of reputation) is comically bad, and completely useless. I can cite chapter an verse on this if you like, but just consider all of the work that has been done on trying to find testable consequences of the GRW collapse theory, where the fundamental dynamics is not unitary. A failure of unitarity is just not the kind of thing that will show up empirically except in very special circumstances. Similar remarks about failure of energy conservation at the level implied for GRW. There is a nice earlier paper on this by Ellis et al (cited by Banks et al) which shows what proper work on this looks like.

Con't.

4) The whole idea of trying to treat this issue using the techniques of scattering theory and S-matrices would need careful discussion. I am not making a precise claim here, but one conceptual lesson in my paper is that black holes are just not like spherical material objects. That is key to note when pointing out that the black hole event horizon (its "surface") can entirely lie in one's past light-cone while its interior does not, so the event horizon no longer exists but its interior does. Another dissimilarity recently pointed out to me by Craig Callender is that a the area of an event horizon is an invariant quantity while the surface area of a spherical object is not. These are at least reasons to suspect that interactions with and between black holes (including microscopic and "vitriual" ones, if that makes any sense) could be fundamentally different than interactions with and between particles.

FInal comment. I have suggested that we can accept AdS/CFT completely and still be consistent with my solution: the pure states on later (connected) Cauchy surfaces in the CFT would just correspond under the duality to pure state on disconnected Cauchy surfaces in the bulk. I am still curious whether anything known about AdS/CFT contradicts this possibility. Clearly, not that much about the correspondence can be known or else, e.g., firewalls could be either confirmed or disconfirmed by checking for the corresponding indicators in the CFT. And similarly, people are taking EPR = ER seriously (for reasons I can't comprehend) and that postulates a much more baroque space-time structure than my paper does. Would that structure leave calculable fingerprints in the CFT? In short, AdS/CFT has not been claimed to provide evidence for or against these other, more popular "solutions" to the "paradox". Why think it provides any evidence against the solution in my paper?

Tim Maudlin wrote:

That is key to note when pointing out that the black hole event horizon (its "surface") can entirely lie in one's past light-cone while its interior does not, so the event horizon no longer exists but its interior does.Which makes evaporation of a blackhole a truly singular event!

(pun intended)

:)

Tim,

" I guess we won't resolve the issue about the physical equivalence of operators (action on solution space vs. action on kinematic space), so let's just leave that issue."I can't quite seem to let this go, so let me just state the point one last time. The action of the Hamiltonian on physical states is that of a boundary operator. Proof: a) on the kinematical space the Hamiltonian takes the form H = [constraints] + H_bndy. b) Physical states are defined by the condition [constraints]|psi_phys> = 0. Hence c) H|psi_phys> = H_bndy |psi_phys. End of proof. This is what I and everyone else I am aware of (like in the Horowitz quote I wrote) mean when they say that in gravity the Hamiltonian is a boundary operator. Again, the only thing I want to take away from this is that the energy of a system can be determined by making measurements at the boundary, a fact which is apparent in the classical limit, for example. If nothing else, I hope we can at least agree on this.

" If the important properties of AdS/CFT are particular to asymptotic AdS spaces, then one just can't draw more general or generic conclusions from it (e.g. Holography)."As to whether asymptotic AdSness matters: of course it matters in the sense of providing a dual description in terms of CFT. However, local physics in the bulk is unlikely to care about the asymptotic structure. Imagine two situations. One in which a solar mass black hole sits in an asymptotically flat spacetime, and another in which it sits in AdS with radius of curvature equal to 10^100 Megaparsecs. It seems highly implausible that any local solar system scale measurement could distinguish between these situations. In the latter case the local bulk physics can be described in terms of a dual CFT, but one expects that the same local bulk physics will hold in the two cases. So if one uses the CFT to learn something about local physics in the bulk of AdS, we have every reason to expect that the same local physics holds in asymptotically flat spacetime. It's much more radical to suggest that local physics depends sensitively on asymptotics.

cont

cont

" I have suggested that we can accept AdS/CFT completely and still be consistent with my solution: the pure states on later (connected) Cauchy surfaces in the CFT would just correspond under the duality to pure state on disconnected Cauchy surfaces in the bulk. I am still curious whether anything known about AdS/CFT contradicts this possibility. "The trouble is that you are just making a vague suggestion here and not a concrete proposal for the structure of the bulk state. Under the most natural interpretation of what it means for the quantum state to live on a disconnected Cauchy surface I have explained that your scenario is inconsistent with AdS/CFT. Again, the most obvious interpretation of saying that the bulk state lives on a disconnected Cauchy surface is that one can make all sorts of changes to the state on the interior component without changing anything on the exterior component. I have pointed out that this is not possible because it would imply a huge energy degeneracy in the CFT, and we know there is no such degeneracy. So you must have something different in mind. I personally can't come up with any modification that is compatible with what we know for sure about AdS/CFT and which also respects the connotation of "disconnected" in the sense that information is lost to an observer confined to the exterior component. Unless you provide some precise proposal for the structure of the bulk state in this situation, there's not much more I can say. The consensus among the many people who have thought carefully about this issue over the years is that there is no scenario in which AdS/CFT (of the standard sort) is compatible with information loss to a late time observer outside the black hole. If you can present such a scenario that would be very interesting, so far I don't see any that isn't easily ruled out.

BHG,

In order:

1) " Again, the only thing I want to take away from this is that the energy of a system can be determined by making measurements at the boundary, a fact which is apparent in the classical limit, for example. If nothing else, I hope we can at least agree on this."

The Hamiltonian of the non-Relativistic theory is, of course, associated with the total energy, but its more essential role is as the generator of time evolution of the state. It is in this role that it differentiates which states in the kinematical space (in this case the space of all continuous and differentiable complex functions Psi(t,x,y,z) such that for fixed t Psi(x.y, z) is square integrable and integrates to unity (leaving aside spin). This role is much more important than the role of total energy operator. Even if one can determine the ADM mass by boundary measurements, that ignores the more essential role of H. Once one has diffeomorphism invariance, or more precisely once the t-foliation becomes a gauge degree of freedom, H cannot operate as a differential operator. But in the WdW setting it still distinguishes which states in the kinematical Hilbert space are solutions. To do that, it has to act on more than just the boundary. I hope we agree on this.

2) If the local physics in the bulk does not care much about the asymptotic structure, then why would the asymptotic structure care much about the detailed physics in the bulk? But the whole idea is that the CFT defined on the asymptotic limit *completely determines, in all detail, the physics of the bulk*. At least it does if holography holds. And if holography doesn't hold then there is no reason to think that degeneracy in the bulk has to show up as degeneracy on the boundary.

3) I don't at all see why "the most obvious interpretation of saying that the bulk state lives on a disconnected Cauchy surface is that one can make all sorts of changes to the state on the interior component without changing anything on the exterior component." I could see the argument if the *space-time* were disconnected, but no argument at all just because the Cauchy surface is. Once more: take a disconnected Cauchy surface, and act on the interior part with a creation operator. Then in the corresponding solution, there is a bit more mass inside the event horizon, and the event horizon has a slightly bigger area, and that changes the geometry in the external region out to infinity. It has to because the ADM mass changes in the solution. If the part of the space-time where the interior piece lives were disconnect (in the 4-dimensional topology) from the part where the exterior piece lives, I could see an argument. But it isn't.

I should also add that this talk of information " lost to an observer confined to the exterior component" is very odd. Unless the spatial structure of the space-time is compact, no external observer will ever have access to the full information on any Cauchy slice: no Cauchy slice will sit in anyone's back light-cone, ever. So this idea of information being "lost" to an observer is strange. The relevant information, the information needed for the evolution to be deterministic and unitary, is information on a complete Cauchy surface. You can't lose what you never had, and this is information nobody ever has in any case.

About your consensus: I fail to see why the scenario you ask about is not exactly the scenario that one reads off the Penrose diagram in the way I have. You have said that no one takes the Penrose diagram seriously. Be that as it may, one *can* take it seriously and you get the result I describe in the paper. What I don't see is any incompatibility of that scenario with AdS/CFT. The only thing you mention is the degeneracy argument, which I have answered above.

Tim,

On the remote chance that anyone ever reads this, I will give the argument one more time in an even simpler fashion, since my basic point has still not been addressed as far as I can tell.

All I will assume about AdS/CFT duality is that there is a correspondence between states on the two sides, with the energy of a bulk state being equal to the energy of its corresponding CFT state. Now, consider the region connected to the boundary, long after black hole evaporation. By making measurements solely in this region we can determine the energy of the bulk state. According to you, there is information loss in the sense that measurement made purely in this exterior region cannot determine the full (pure) quantum state of the entire system. If so, it must be that knowing the energy is insufficient to determine the full quantum state, which is to say that there must be more than one state of the same energy. But if the CFT has a nondegenerate energy spectrum, then there is a clear contradiction.

That's it in a nutshell. It would be great if this argument could be addressed without getting sidetracked into issues of kinematical Hilbert spaces etc. The logic is so simple that it's hard to imagine a loophole here, except to imagine that the very notion of AdS/CFT is somehow inconsistent, many concrete computations notwithstanding.

Black Hole Guy,

This argument is indeed clear. It is also clearly incorrect, or at least committed to a thesis so strong that it cannot possibly be right. So let's take on this argument.

The claim you are committed to is this: "If so, it must be that knowing the energy is insufficient to determine the full quantum state, which is to say that there must be more than one state of the same energy. But if the CFT has a nondegenerate energy spectrum, then there is a clear contradiction."

To which the answer is: of course knowing the energy is insufficient to determine the full quantum state! Or, to be even more precise, knowing the ADM mass (or the analog in AdS/CFT that is defined at spatial and/or null infinity) is insufficient to determine the full quantum state. If whatever AdS/CFT asserts implies that it is sufficient to know the energy to determine the full quantum state, then it is clearly wrong.

None of this has anything at all to do with disconnected Cauchy surfaces or evaporating black holes. Take a bulk state with a solid sphere of mass M and otherwise vacuum. Now take another bulk state of hollow spherical mass M but otherwise vacuum. These are different states with the same energy and ADM mass. If you assert that the existence of these two distinct bulk states is inconsistent with AdS/CFT, then AdS/CFT is inconsistent with plain gravitational possibilities. So quite apart from the issues we have been discussing, AdS/CFT as you understand it is refuted. QED.

So I don't think this can be the argument you have in mind. But it seems to be what you wrote. Please explain how this does not immediately refute your principle.

BHG: "On the remote chance that anyone ever reads this..."

I check this thread regularly! But the high point was the exposition regarding the canonical quantization of gauge theories, and how it works for general relativity and for AdS specifically.

BHG,

I thought I responded to this already, but the response never seems to have shown up. Here it is.

This can't be the objection you have in mind, because if it is it shows that the AdS part can't be a theory of gravity at all, irrespective of any issues about black hole evaporation. You state: "If so, it must be that knowing the energy is insufficient to determine the full quantum state, which is to say that there must be more than one state of the same energy. But if the CFT has a nondegenerate energy spectrum, then there is a clear contradiction." But knowing the energy of the bulk state (i.e. the ADM mass, or whatever the analog is for AdS) cannot possibly be enough to pin down the state in the bulk. Consider a solid sphere of (ADM) mass M. And a larger hollow sphere made of the same material of ADM mass M. They have the same energy but are different states.The asymptotics of the states will be the same. So if this contradicts AdS/CFT, AdS/CFT can't be right. QED

If the CFT state is non-degenerate in its energy spectrum, then it cannot possibly be in 1-to-1 correlation with the energy state of the bulk. So if this is the argument that you have been worried about, it simply refutes AdS/CFT as you conceive it.

I also have read this thread with much interest. In my opinion, it could serve as a classical example of the difference between scientific and philosophic reasoning.

JImV: I'm not sure what you mean. Can you give an example of the difference you see? All we have been doing is trying to get clear about what significance AdS/CFT has for the solution I am advocating. BHG thinks it rules out the solution and I can't see that it does any such thing. What would a "scientific" vs. "philosophical" approach to such question even look like?

(I have a response to BHG's latest post, but for some reason it has not appeared yet.)

TM, I wanted to give details in my previous comment, but decided that might be invidious; also I did not want to spark further debate in a thread which has already strained our host's hospitality.

Perhaps, however, a better way to state my opinion of this thread is that it provides a good example of the differing ways that a scientist and a philosopher might approach the problem of understanding the nature of reality (e.g., is quantum reality unitary or not).

Tim,

Your comment greatly clarifies why you are having trouble seeing the argument. I am indeed asserting that the CFT has a nondegenerate energy spectrum*, and so does the bulk gravity theory, and that the latter fact is not in contradiction with any aspect of known physics in the bulk.

Essentially, you are missing a basic fact about quantum mechanics, which is that the spectrum of energies of bound state collections of interacting particles or fields is generically non-degenerate, except for those degeneracies due to symmetries. In your example of a solid sphere and a hollow shell, it seems that you are thinking about classical configurations; classically there can indeed be multiple configurations of the same energy. The situation is totally different in QM, in particular the energy is quantized for bound states. So, if you try to write down a concrete quantum mechanical theory realizing your example -- say a collection of particles interacting via Newtonian gravity and repelling due to some other force -- you will find that the energies of your solid and hollow configurations will never match. In QM of interacting particles, there is simply no reason for the energy levels of distinct states to match exactly unless there is some symmetry that relates the states. I doubt there is any known example where such a thing occurs. So to be very precise, if you specify for me your QM system, and only tell me that the system is in a bound state of such and such an energy, then I can indeed in principle tell you precisely which state the system is in, up to the action of symmetries. To forestall any confusion, it is important that the particles be interacting here, and that we're looking at bound states so that the energy spectrum is discrete. Both facts are true of bulk physics in AdS.

Having cleared this up, perhaps you now see why your scenario is impossible?

* footnote: this is a slight lie, as the CFT typically has a finite dimensional symmetry group which can lead to small degeneracies, but this is not relevant to the basic issue at hand, and I can address it if asked.

black hole guy,

Any unitary operator will do. Besides, the Hawking radiation isn't in a bound state with the black hole.

JimV: This is not idle curiosity on my part. From the very start, long ago, I said that this is not a matter of philosophers vs. physicists, it is a question in the foundations of physics. As it turns out, a standard physics education just does not cover foundations of physics, basic conceptual questions about quantum theory, etc. Even the fundamental conceptual understanding of an intrinsically clear theory, such as GR, is can be absent in the most accomplished physicists. So I really wonder where in this discussion you see philosophy vs. physics. As far as I can tell, it has all been about getting clear on the physics (in some cases, e.g. the exact content of AdS/CFT, without success) and its bearing on the "information loss paradox". Getting clear on the physics should be something both physicist and philosophers do, and I can't see that there are different ways to go about it. As I have said before, the greatest philosopher of physics in the first half of the 20th century was Einstein, and in the second half was Bell. But maybe I am taking the import of your comment incorrectly.

BHG,

Your comment certainly clears up why you think the scenario is impossible, but only by committing yourself to something that seems plainly impossible. Taken at face value, you are claiming that just giving you a single real number (the ADM mass or its analog), the entire precise quantum-gravitational state of the entire universe—well, really, the entire quantum state of the universe—up to symmetries can be recovered uniquely. I am frankly quite astonished that you can believe what you have written down, as it is a much more radical and incredible claim than anything to do with black hole evaporation or information loss.

The only possible way this claim could even begin to be defended is if you are considering only energy eigenstates, which will, of course, be stationary. But the scenario we are considering—a black hole forming and then evaporating—is evidently not a stationary state. We are supposed to be investigating a gravitational situation that admits of a classical macroscopic description at least in the early and late times (leaving out the whole formation-and-evaporation era). We can, for example, consider the black hole to have been formed by a collapsing solid sphere of dust, or a hollow shell of dust, or a huge collection of elephants, or whatever. And in the case of dust, the matter density is freely variable, so there are initial solid spheres of whatever radius and density you like, and hollow shells of whatever radius and density, and solid cubes and hollow cubes, and elephant-shaped collections of dust, etc. Indeed, one of the main pieces of information that is supposed to be lost in the information-loss paradox is which of these various possible initial conditions led to the black hole in the first place. And what you appear to be claiming is that some single real number will in principle contain the information about which of these scenarios obtained. But since the scenarios can be continuously rescaled for total energy (by continuously increasing the density of the dust, or the radius of the sphere, or the size of the elephant) than is just plainly impossible. Any amount of energy (i.e. expectation value of energy) can be instantiated in any of these scenarios.

Put it yet another way. In QFT we build up states by starting with a vacuum and operating with creation operators on it. But the creation operators are themselves parameterized by an index that is continuous: the location of the state created. So there are continuum many distinct one-particle states, all with the same energy. You can say these are related by a symmetry, but that argument goes away already with the two-particle states. There we have the relation between the two locations, which can also vary continuously and no symmetry connects all of the two-particle states to each other. And it clearly gets worse for symmetry considerations as the number of particles goes up. You simply can't believe that any single real number characterizing the energy can tell me which of these continuum-many states has been created, can you? The energy will be massively degenerate in this folium of states. If the CFT energy spectrum can't correspond to this, then all the worse for AdS/CFT, at least understood as a Holographic Hypothesis.

Needless to say, I do not see at all why my scenario is impossible. What you seem to have argued is the AdS/CFT, as you interpret it, is impossible.

Sabine,

Can you amplify on your comment "Any unitary operator will do"? Do what? I'm not following, and I do hope this point can be sorted out definitively. We seem to reaching bedrock here.

Tim,

The simplest example that comes to mind is an entanglement swap. Changes the state, doesn't change the total energy. Basically that's what I did to show that the firewall argument is wrong, but that wasn't a bound state, so not entirely sure that applies here.

The example I used in an earlier post was two modes of energy E vs one mode of energy 2E. Clearly for that you'd need bosons though. My point was that the *expectation value* of the stress-energy tensor doesn't distinguish both - particle number is a real quantum effect. Now of course if you claim you couple quantum gravity to the actual operator, then things look different. I've tried for a bit to see whether one can prove they're mapped one-on-one but no success (didn't try very hard though). Clearly one needs additional assumptions. It's possible AdS delivers these, I don't know. So maybe bhg is right.

Tim,

I am happy that you are astonished by what I write, because this means that I am close to helping you to see the light

According to you, we can disprove AdS/CFT as follows. First, (up to the action of a finite dimensional symmetry group whose existence I will often suppress) we know that the canonical example of N=4 super Yang-Mills theory on S^3 x R has a discrete energy spectrum. This is simply because the spatial geometry is a 3-sphere which is compact. But quantum gravity on AdS_5 with boundary conformal to S^3 x R can't possibly have a discrete energy, spectrum, right? Right....?

At the risk of sounding snarky, for which I apologize, I think this interchange does illustrate a difference between physicists and philosophers. In particular, you are making some claims based on your "intuition", without having actually tried to do a computation to check these claims. This may also address some of Bee's statements, which I did not actually follow.

Let us do the following exercise. Let's first quantize a free scalar field in AdS_{d+1} with the standard boundary conditions. There is a vacuum state of course, and then there is a space of 1-particle states. The 1-particle states have a discrete energy spectrum, and the states are labelled by discrete quantum numbers (namely the angular momentum quantum numbers and the energy). This is different than the spectrum of 1-particle states in Minkowski space, which is indeed continuous. The difference is due to the fact that AdS acts as a gravitational well, essentially a gravitational version of a box. In any case, the existence of a discrete spectrum is a mathematical fact which anyone can verify for themselves. Now we build up the full Hilbert space by considering 2-particle states, 3-particle states, and so on. At the level of non-interacting particles we can indeed have energy degeneracies. E.g. it is possible for some 2-particle state to have the same energy as some 1-particle state. But now turn on interactions, say from gravitational attraction, which are inevitably present. One can check completely explicitly that this lifts the degeneracies -- the 2-particle state gets its energy lowered due to gravitational attraction. So even in lowest order perturbation theory, one can see that the spectrum becomes non-degenerate. So the statement that the energy spectrum of quantum gravity in AdS is discrete and non-degenerate (up to the action of symmetries) is certainly true at low energies, and is perfectly consistent with all known facts.

cont

Note that I in my statements I was very careful to say that we are considering interacting particles with a discrete spectrum, since this is the case for the actual theory in AdS. These carefully worded provisos seem to have gotten lost in the ensuing discussion.

I was also careful to say that to determine the state of the system you have to give me the energy eigenvalue *and tell me what the Hamiltonian of the system is*. So yes, if you tell me what the Hamiltonian of your sphere and shell system is, and tell me what the energy eigenvalue is, then I in principle tell you what the state is, by putting the Hamiltonian on an arbitrarily powerful computer.

Again, the main point seems to have gotten lost, so here it is again. In standard examples of AdS/CFT the CFT has a discrete and nondegenerate spectrum, hence so too must the AdS theory. This latter statement is consistent with explicit low energy computations of interacting fields in AdS, and is further consistent with all known facts. This is the only result I need: the quantum gravity theory defined by the CFT necessarily has a discrete and non-degenerate spectrum, and this is compatible with expectations, and with all explicit computations that have been done. On the other hand, your scenario requires a highly degenerate spectrum, so cannot be realized in any standard AdS/CFT example.

Bee,

"The simplest example that comes to mind is an entanglement swap. Changes the state, doesn't change the total energy.Could you clarify this? if your quantum theory has a non-degenerate spectrum, (which is a pretty generic statement) how are proposing to change the state without changing the energy? I am not following..

BHG,

You are being rather betrayed by your own argument. There are several points to make.

First, we do not live in AdS. The original Penrose diagram I have been commenting on is not in AdS. I have been letting that go on the assumption that you would argue that whatever important features AdS has for our question will transfer over to a physically realistic case, i.e. the case we actually care about. But by insisting on a closed universe for deriving your results, you guarantee that they are, after all, completely beside the point.

Second, as a point of logic your argument does not go through. You start with gravity off and have degeneracy. Then you turn gravity on and say that that breaks the original degeneracy. But if it does then it can just as well create new degeneracies that were not there originally. So your argument that there is no degeneracy not associated with a symmetry is simply invalid.

Third, your appeal to the Hamiltonian here is, shall we say, ironic? I have been arguing ad nauseam that the Hamiltonian of the bulk interior is *not* the same as the zero operator even though it has the same action on the folium of solutions. And you have been arguing the opposite. But if the Hamiltonian is the zero operator in the bulk, then your supercomputer will not have much to work with! You can correct this by admitting that you meant all along what everyone else means by the Hamiltonian.

Fourth, your closed AdS geometry obviously has no spatial or null infinity to it. So what was all that talk about the ADM mass about? That is only defined on an asymptotic boundary, and now you don't have one.

Fifth, you seem to be confusing the energy eigenstates with the set of states with a given expectation value for the energy. I already said that if you are only looking at energy eigenstates then you can't possibly model a collapsing and evaporating black hole. But even if the eigenstates are discrete, since the set of solutions is a Hilbert space (as you insist) there are also superpositions of the eigenstates. For these, the "energy" can only mean the expectation value for the energy. States with the same expectation value will be massively degenerate ( and the "spectrum" of expectation values will be continuous).

Is that enough for the moment?

bhg,

It's non-degenerate for a basis I assume? What prevents you from taking superpositions?

I assume our host can verify that this is the same BHG as before?

Bee,

It's non-degenerate for a basis I assume? What prevents you from taking superpositions?I guess I would need some more detail to understand what the context of your argument is. I was trying to explain to Tim the fact that the energy spectrum of a bound state of interacting particles is expected to be non-degenerate (I can't tell from his subsequent comments whether I succeeded in this or not). Was your comment meant to address this point? We are talking here about the spectrum of the Hamiltonian, which is a basis independent notion. Perhaps you are talking about operations that change the state while holding fixed the expectation value of the energy? -- I can't tell.

BHG,

There were so many issues to raise about your post that I forgot the main one: so what?

Even though I found and still find your claims incredible, let me grant every single one of them. So the gravity theory in the bulk has a discrete and non-degenerate energy spectrum. So what?

You keep saying that my solution to the information-loss puzzle requires massive degeneracy in the energy spectrum of the bulk. But it doesn't. We have already been through this. It does not imply that one can do whatever one like inside the event horizon and not have it any impact outside. If you change the state inside then, I guess, by your lights the energy inside changes so the event horizon changes, and everything out to infinity changes. And if you change the state outside the event horizon, there will also be changes in the energy (and mass) out to infinity. So where is the problem?

"...holographic CFTs must have an enormous degeneracy of states at high energy..."

page 132 here:

http://www.hartmanhep.net/topics2015/14-adscft.pdf

Arun,

No, I have no way to check who submits comment under a pseudonym.

bhg,

Yes, that's what I mean. You can change the state without changing the energy.

Arun,

The sentence in the pdf that you link to I think is a sloppy use of the word 'degeneracy'. This is a coarse-grained degeneracy, not what bhg is referring to.

BHG,

I retract point 4. Can have null infinity.

Tim,

I commented earlier on your points 1-5, but this doesn't seem to have appeared. In the meantime:

I retract point 4. Can have null infinity.This comment illustrates what I find so frustrating about our exchange. As background, let me note that I am probably more sympathetic to the role of philosophy in physics than 99% of my colleagues, and I am always interested in broadening my horizons. I am of course less familiar with the style of public discourse in philosophy than in physics, so some of our failure to connect might be due to different expectations about how a dispute should proceed. But with all that said, I still find it remarkable that after all of our back and forth about black hole evaporation in AdS you apparently do not understand the most basic feature of AdS, in particular its causal structure. I am very surprised that you did not ask for clarification about this or look it up or work it out for yourself. First you heaped scorn on my arguments by saying that

"your closed AdS geometry obviously has no spatial or null infinity". Now you follow it up by claiming that AdS has null infinity. Both claims are stated with apparent confidence, not prefaced with something "please correct me if I am wrong.." or "it is my understanding that...". Maybe this is how arguments proceed in philosophy -- I don't know.In any case, a very short computation (or google search) reveals that null geodesics hit the boundary of AdS at finite time, so there is no null infinity. Further, the boundary is at infinite spatial distance (as computed along an outward directed spacelike geodesic), so AdS has a spatial infinity. The Penrose diagram for a cross section of AdS is a rectangle. The analog of the ADM mass in AdS is defined at spatial infinity. I already explained way back in this thread that for this reason AdS has no notion of Bondi mass, but apparently that comment made no impact. The totally standard statement I am making is that the conformal boundary metric of AdS_{d+1} is that of S^{d-1} x R. Energy is defined as an integral over this S^{d-1}, just like ADM mass in asymptotically flat space is defined as an integral over a sphere at spatial infinity.

I am willing to further clarify my arguments but I ask in exchange that you respect that fact that I am not just blowing smoke. I have been very careful to explicitly flag any comments I make that are speculative or which I have some uncertainty about. When I write something like that the Hamiltonian in GR is a boundary operator you treat me (and Gary Horowitz) like I am an idiot and don't appear willing to listen. It is very frustrating...

BHG,

The point Sabine is making is the same one I was making, and is very straightforward. You claim that there is some single number that (up to symmetry) characterizes every possible physical state in the bulk uniquely. That is, each state (up to symmetry) is assigned a unique such number, so from knowing just that number I can in principle recover the state. And you thing that number is something like "the total energy of the state", and is associated with the Hamiltonian.

But all you have actually argued for is that the spectrum of the Hamiltonian is discrete and non-degenerate (up to symmetry). And all that tells you is that there is unique number (up to symmetry) assigned to every eigenstate of the Hamiltonian. But the eigenstates of the Hamiltonian are a set of measure zero in the space of solutions. And not only are they a set of measure zero, they are all stationary, so one of them cannot possibly be the relevant one for our problem.

The rest of the physically possible states—the set of measure one—are superpositions of energy eigenstates. So what is the number that is supposed to uniquely characterize these (up to symmetry)? The only thing in the neighborhood is the expectation value for the energy. But the expectation value is massively degenerate, even apart from energy considerations. For every pair of eigenstates with different energies, there is a superposition with every intermediate value for its expectation value. So the set of expectation values for the energy is both continuous and massively degenerate, so there is no single number that characterizes the state (up to symmetry) uniquely.

It was the latter claim that I found incredible, not that the spectrum is discrete. To repeat: since the states we are talking about can't be energy eigenstates, it is kind of irrelevant to our issue.

BHG,

Sorry about having rushed to judgment on the infinities. I was not careful about separating the claim that states on a compact space must have a discrete spectrum (obviously true) from the claim that bound states even in a non-compact infinite space must have a discrete spectrum, so I thought the claim was that AdS is compact. I'm not sure why we can be certain that the relevant state is a bound state though. The reference would have to be to the complete state in AdS which, in our case, would have to include the Hawking and anti-Hawking radiation. The photons are not bound. Indeed, they are supposed to have a thermal spectrum, so should have all possible energies. But this is all not worth arguing about, since we do not agree that the discreetness of the spectrum bears on the issue at hand. As I said a ways above, the idea that the Hilbert space of solutions is a tensor product state is not anything I am committed to, so if that is what you are appealing to to argue that my solution is committed to a continuous spectrum, I can just accept that the Hilbert space does not have a tensor product structure. I can't see anything in the solution committed to it.

Since you seem convinced that the discreteness of the energy spectrum somehow settles the issue about the solution discussed in my paper, we need to understand what that argument is. I can't see it.

Tim,

OK good, I think we are back on track. I don't think I ever claimed that every state is uniquely characterized by its energy expectation value, since that statement is clearly false. The only claim I was making was about discreteness and non-degeneracy (up to symmetry) of the spectrum of generic systems analogous to physics in AdS. Anyway, I think we are in agreement about this now.

You are of course correct that when we form a black hole and let it evaporate we are dealing with a time dependent process, and so necessarily dealing with a nontrivial superposition of energy eigenstates. In that situation, even if I prepare many identical copies of the system and measure the energy of each, I cannot determine the state of the system. I can determine the amplitude in front of each energy eigenstate by measuring the frequency of obtaining each possible energy, but I can't determine the relative phases. Similarly, just by making measurements of the energy of many identical copies of a system, I cannot determine whether the state is pure or mixed, again because I cannot measure relative phases this way.

While these are all relevant issues, I would like to set them aside momentarily so that I can clarify one point which will also help me to understand the scenario you have in mind. In general, please interpret my remarks not as a claim for a complete proof that no scenario along the lines you are proposing is possible, but rather that I can't come up with any interpretation that fits with known facts about AdS/CFT.

Once again, let us consider the evaporated black hole at some very late time as measure by a clock on the boundary. My understanding is that you want to claim that quantum mechanics holds as usual, so that the full system is in a pure quantum state at this time, but that information is lost as far as measurements confined to the region connected to the AdS boundary are concerned. We need to translate this into a precise mathematical statement. A concrete proposal is that the full Hilbert space is a tensor product, with one factor representing the region connected to the boundary, and that the result of black hole evaporation is an entangled state. This particular proposal can be ruled out, since if the Hilbert space has this structure then I can always think of considering states (not that they would be produced from black hole evaporation) which take the form of, say, the external vacuum state tensored with various choices for the state in the other factor. This is ruled out since it implies a degeneracy of zero energy states, which contradicts the CFT. Now, I realize that you say that you are not committed to such a tensor product structure. But I wonder if you have any other specific proposal to make? Or maybe not, and perhaps you are just raising the possibility that another proposal might exist that accords with AdS/CFT. Either way, I would like to know.

BHG,

Excellent! I think that there are a couple of things to say here, which may help us. It is not that I have a definite answer, but I think that even without that this particular question can be seen not to be problematic. We see this by analogy.

Suppose we are in a normal, non-evaporating-black-hole scenario, where the Cauchy surfaces are always connected. Then I take it we agree that the state on each Cauchy surface, together with the exterior curvature of the surface, contains full information about the entire space-time.If the initial state is pure, then the state on every Cauchy surface is pure. From any one of the Cauchy surfaces and the Hamiltonian we can in principle recover the entire global state at all times. But now suppose one were given not that full information but only partial information: the state on just part of the Cauchy surface, a part that connects out to spatial infinity. The state would, I take it, be a density matrix. It would specify the expectation value of every local observable on that part of the Cauchy surface. And it would not contain full information about the quantum state on the whole Cauchy surface: different complete states would be compatible with the density matrix. In some sense, the density matrix would be derived by "tracing out" the degrees of freedom associated with the remainder of the Cauchy surface. (I put "tracing out" in scare quotes since I'm not positive what the exact mathematical procedure is. But it is clear that the pure state on the whole Cauchy surface does determine expectation values for all the local observables on just the part, and so determines a density matrix at least for those observables.)

If we call just the part of the Cauchy surface Part, then the state on Part is a density matrix that does not contain full information about the global state. Different global states are compatible with Part. And if we call the remainder of the Cauchy surface Rest, there will (generically) be a density matrix associated with Rest, and furthermore generically the state of Part will be entangled with the state of Rest.

con't

Now I just want to raise the same questions here that you are raising. Does that mean that the full Hilbert space is a tensor product of a Hilbert space for Part and a Hilbert space for Rest? Or not? That is just not apparent to me. But whatever the answer is, I would say that the very same answer holds for, say, Sigma In and Sigma Out. They are also parts of a complete Cauchy surface. Indeed, the surface I called Sigma critical, which contains the Evaporation Event, has just the form of Part and Rest: a Cauchy surface divided into 2 parts by the Evaporation Event. The disconnected Cauchy surfaces that follow Sigma Critical would presumably have nearly the same physical structure as Sigma Critical.

It is also important that we get clear on what we mean by "the full Hilbert space". Do we mean the kinematical space or the solution space? I would not be very surprised if the kinematical space were a product space. And I would not be very surprised if the solution space were not. (Indeed, I would not be surprised if the solution space were not a Hilbert space at all!)

Now maybe your argument equally shows that the full solution space cannot be a product space of a Hilbert space associated with Part and a Hilbert space associated with Rest. If so, then the same should hold for Sigma In and Sigma Out. On the other hand, it is possible (as far as I can see) that the full kinematic space is a tensor product of a Hilbert space for Part and a Hilbert space for Rest. That does not mean that we are free to put whatever state we like on each and expect there to be a solution for that state. This is because there are constraint equations that have to be satisfied, even apart from the Hamiltonian constraint.

So if I were to guess, I would guess: kinematic space is a tensor product space, solution space is not. But my point is that I would make exactly the same guess for Part and Rest as for Sigma In and Sigma Out. I can't see that the disconnectedness of Sigma In and Sigma Out marks an essential difference. If we agree about that, then we agree to let the chips fall where they may.

bhg,

Since we agree that the energy doesn't uniquely specify the state, this means that the vacuum isn't uniquely defined. Indeed, there are infinitely many vacuum states that generically differ in their entanglement structure. This is still so if you use the spectral energy density instead of the total energy. Do we still agree on that?

BHG

(This is the first part of the post whose end is above.)

BHG,

Excellent! I think that there are a couple of things to say here, which may help us. It is not that I have a definite answer, but I think that even without that this particular question can be seen not to be problematic. We see this by analogy.

Suppose we are in a normal, non-evaporating-black-hole scenario, where the Cauchy surfaces are always connected. Then I take it we agree that the state on each Cauchy surface, together with the exterior curvature of the surface, contains full information about the entire space-time.If the initial state is pure, then the state on every Cauchy surface is pure. From any one of the Cauchy surfaces and the Hamiltonian we can in principle recover the entire global state at all times. But now suppose one were given not that full information but only partial information: the state on just part of the Cauchy surface, a part that connects out to spatial infinity. The state would, I take it, be a density matrix. It would specify the expectation value of every local observable on that part of the Cauchy surface. And it would not contain full information about the quantum state on the whole Cauchy surface: different complete states would be compatible with the density matrix. In some sense, the density matrix would be derived by "tracing out" the degrees of freedom associated with the remainder of the Cauchy surface. (I put "tracing out" in scare quotes since I'm not positive what the exact mathematical procedure is. But it is clear that the pure state on the whole Cauchy surface does determine expectation values for all the local observables on just the part, and so determines a density matrix at least for those observables.)

If we call just the part of the Cauchy surface Part, then the state on Part is a density matrix that does not contain full information about the global state. Different global states are compatible with Part. And if we call the remainder of the Cauchy surface Rest, there will (generically) be a density matrix associated with Rest, and furthermore generically the state of Part will be entangled with the state of Rest.

Bee,

We must be talking about different things here, since I doubt either of us is confused about something this basic. Perhaps you are referring to energy density, rather than the total energy of the system? I am always referring to the total energy. The statement I am making is that knowing with certainty the total energy is enough to uniquely fix the state (modulo symmetries). As a concrete example, if you hand me a scalar field theory in a box, with some specified cubic and quartic terms in the potential (to break all symmetries) and tell me that if I were to measure the energy I would get some value E with certainty, then with a powerful enough computer I can uniquely reconstruct the full quantum state. Again, I am not talking about expectation values here, but rather the energy I would get from a single measurement.

Agreed?

Tim,

" Indeed, I would not be surprised if the solution space were not a Hilbert space at all!"Let me focus on this statement, both because it is important, and because I am confident I can change your mind here. If the space of solutions of all the constraints cannot be made into a Hilbert space then you have a breakdown of quantum mechanics. Specifically:

1) By definition, the only states that can ever be physically realized are the physical states, which are those annihilated by all the constraints. Let |\psi> be such a physical state.

2) Now suppose I am in the state |\psi> and measure some physical observable X associated with the physical operator O_X. (O_X commutes with the constraints so takes physical states to physical states). How do I compute the probabilities for finding the possible values x, where x denotes an eigenvalue of O_X? The normal rule in quantum mechanics is the following. I first identify the space of states |\psi_x> which are eigenstates of O_X with eigenvalue x. The states |\psi_x> are themselves physical, since they are obtained upon measurement of the physical observable O_X. Then the the probability to find x is given by |<\psi_x|\psi>|^2. It had better be the case that these probabilities are all positive and sum up to 1. Now, the statement that the physical space is not a Hilbert space is the statement that there is no inner product defined on the space of physical states. But then you have no way of carrying out the above procedure, since <\psi_x|\psi> is not defined. So please tell me, how do you then propose to make any predictions in this theory?

I think what might be confusing you here is the issue of the technical challenge of finding such an inner product in quantum gravity, versus the requirement that one is necessary to do quantum mechanics. You might find some of the discussion in the review arxiv 0501114 useful, and where you will read:

"At any rate, the challenge here — as with any other constrained quantum mechanical system — is to find a subspace that is annihilated by all the constraints, and to define a physical inner product which yields the ‘final’ physical Hilbert space."

I have never heard anyone else make the claim that you can have a well defined theory without having a Hilbert space structure on the physical space. If you know of any such claims in the literature please pass them on.

BHG,

I think this is issue about the solution space is a bit of a red herring. The main reason that I mention the idea that the solution space may not be a Hilbert space is not because I think that it could fail to be an inner product space, but because there are more conditions to being a Hilbert space than just being an inner product space. In particular, we need that there is a limit in the space of every Cauchy sequence: the space has to be complete. It is not obvious (to me anyway) that the solution space has to have this property.

The only positive reason I have suspect that the solution space may not be a Hilbert space, as I said a while ago, is just that Thiemann discusses a case exactly like this, in which he makes a point of saying that the solution space is not a Hilbert space. And the only reason that I kept coming back to it was to repeat that while Thiemann has a Hilbert space, H, that is the kinematical space, and the solution space, D, is not. It just reinforced the idea that talk of "the Hilbert space" of a theory is typically talk of the kinematical space.

So you don't have to work to convince me that the solution space is an inner product space. If it is really important that the solution space also be complete, then there is some interest in seeing why that is, but even in that case I don't think anything important hinges on it. Nothing I am aware of, anyway.

Tim,

I am happy to not get into technical issues of completeness. This would be especially pointless in the case of Wheeler DeWitt quantum gravity, which is anyway highly formal and ultimately ill-defined. However, I do want to insist that whatever we mean by quantum gravity in the bulk we are assuming that it is a quantum mechanical theory, so it has a space of physical states that obey the requisite properties of a QM theory. Also, it is only these physical states that have any relevance to AdS/CFT. Whether some approach to constructing the bulk theory involves constructing some larger unphysical space of states as an intermediate step is immaterial, since these states never occur (by definition) and have no counterpart in the CFT.

Now to get to the point of your earlier comment. In a theory without gravity, it is definitely possible to construct a pair of distinct quantum states that cannot be distinguished by any measurement outside some spherical (say) volume. E.g, one state could be the vacuum state, while the other state could be the vacuum state with a local unitary operator applied near the origin. By causality, no measurement performed outside a sphere of radius R can detect the operator until a light signal has had time to travel from the origin to the boundary of the sphere.

However, the situation changes in a theory of quantum gravity, specifically one that can be dual to a CFT via AdS/CFT. I want to argue that any change in the state nominally near the origin can always be detected by some measurement far away. This measurement will generically involve measuring correlations of fields at large separations. There is no violation of causality, because no single observer can make these measurements in a time less than the time it takes for light to have traveled from the origin to the exterior region. This issue, phrased slightly differently, was studied in the relatively early days of AdS/CFT in papers of Polchinski, Susskind, and others. I will attempt to defend this position as soon as I get a chance to think of a way to phrase the argument effectively.

Tim,

Here's the followup to my last message. First, the proper context here is to think of the quantum gravity theory in the bulk as being *defined* via the dual CFT. What I mean by this is that the CFT is a well defined quantum theory, and by now it is pretty firmly established that it correctly yields Einstein's theory coupled to matter as a low energy effective field theory in AdS. Any theory that accomplishes this is properly called a "Quantum Theory of Gravity". Yes, it is in AdS rather than "our world", but note that we have no idea what the asymptotic structure of our spacetime is -- we of course live in an expanding cosmological spacetime, and asymptotically flat space is as far removed from that as AdS is.

Next we need to know that any two states in the CFT can be distinguished by correlation functions of local operators. Further, such correlation functions are equal to the boundary values of the associated bulk correlation functions -- that's part of the AdS/CFT dictionary. It follows that the bulk theory defined via AdS/CFT has the property that any two states can in principle be distinguished by measurements at the boundary. There simply do not exist two states that look the same near the boundary, and only differ deep inside the bulk. If we were talking about a nongravitational theory this would indicate some pathological feature of the bulk theory so defined, but in a theory with gravity there is no contradiction with any expected property of the theory. I emphasize that the measurement one needs to do to distinguish states might be quite subtle, involve correlations of fields at large distances, and something as simple as expectation values of local operators are not fine enough probes.

The extension of this to the evaporating black hole case should be clear. I prefer not to refer to any of this as a "holographic hypothesis". It's better to say that this is how things work in the theory defined via AdS/CFT. It's possible that there could exist other theories of quantum gravity where things work differently. It just happens to be the case that there is no other theory that is on remotely as firm a footing as AdS/CFT.

BHG,

OK, so you are cutting the Gordian knot and just defining the bulk physics via the boundary physics. That takes some questions off the table. Let's see where it goes. (I'm not sure why you do not like the term "holographic hypothesis". Certainly your intent is to form—by construction—a holographic theory.)

So here are, to begin with, just some technical questions. You seem to be assuming that every operator on the boundary corresponds to some operator in the bulk. You appeal to the AdS/CFT dictionary here. But as I understand it, the dictionary is very, very incomplete. It exists for only a few operators, and maybe even then only in certain circumstances (low energy, for example) is the correspondence even conjectured. If so, then we really have no idea how to understand the physics of the bulk. It would not be clear which observables on the boundary to use to answer particular questions about what is going in in the bulk. To that extent, this isn't really a theory of gravity in an N + 1 dimensional space-time in the way we normally think of it. But let's leave that aside for a moment.

Here is another question, which I need to get clear on. You seem to be assuming, and have for a long time, that any operator on the boundary either *is* or in some sense *directly corresponds to* a local operator in the bulk. The bulk operator, indeed, has to be localized enough to count as "far away from the middle" of the bulk. Because there are references to what an observer "far from the black hole" can determine just by making local measurements in her region. Indeed, the talk is rather as if the CFT physics literally lives *on the boundary of the bulk*. Now this is problematic for several reasons. One is that the "boundary of the bulk" (e.g. null infinity and spatial infinity" are not really physical locations at all. Mathematically, having brought infinity in to a finite distance, as it were, by a conformal transformation, one can then add on fictitious "boundary points" to the space-time. But these are fictitious points and the boundary a fictitious space. It is not at all clear how any operator that "lives on the boundary" of the bulk could also live in the space-time of the conformal theory. There is, for example, the simple problem of dimensionality: the bulk operator on the fictitious boundary lives on an N + 1 dimensional manifold with a boundary. The boundary operator line in an N-dimensional manifold. The one can't be literally the same as the other. What sort of "identification" is being posited here?

Can the quantum state of either the CFT or the bulk theory be determined by measurements? Well, not in the normal way to think about it: measurements on a single unknown system cannot be guaranteed to yield accurate information about the system. One can appeal to quantum tomography here, but that requires an indefinite number of freshly created systems, all created in the same state. How to appeal to such a condition when discussing the entire universe is beyond me.

Con't.

The duplication of operators from a formal point of view is also puzzling. Apparently, for every CFT operator there is a bulk operator that is somehow guaranteed to give the same outcome as the CFT operator. But the CFT operator also corresponds to—or even *is*—an operator on the "edge" of the bulk in the bulk theory. But it sounds like that violates the no-cloning theorem. It is certainly quite bizarre if one brings in collapses or some other stochastic mechanism to solve the measurement problem. If a measurement of some kind is made on the edge of the bulk, what is to enforce the condition that that particular result must be forthcoming in the correlated state in the middle of the bulk? So in the bulk theory there will have to be two quite different operators that always give the same answer: one in the bulk and the other out at the edge. Does this pair of perfectly correlated operators in the bulk theory then correspond to two perfectly correlated operators on in the CFT? And if so, does this pair of operators in the CFT then give rise to yet another pair in the bulk? So then there must be four perfectly correlated operators in the bulk, which correspond to four in the CFT, which means eight in the bulk, rinse and repeat. It seems to follow that there must be infinitely many operators both in the CFT and in the bulk that are guaranteed to both have the same expectation values on all states and to give the same particular outcomes on all states. This seems frankly incredible.

Anyway, these are some questions whose answers would help me understand the theory.

But let's suppose all of these questions can be answered adequately. I have to say that I still don't see what bearing this has on our main question. Insofar as we are able to understand any of the physics in the bulk is a way relevant to our main question, it must be possible to to interpret it in temporal terms as an initial state (with matter in a state that will form a black hole) and ends with the black hole having evaporated, Hawking radiation, etc. These states have to be defined relative to some space-like hyper surface in the bulk. My claim has been that 1) these space-like surfaces have to the Cauchy and 2) due to the spatio-temporal peculiarity of black hole evaporation the initial surface is connected and the final one disconnected. Can you give me some idea of how this particular model of quantum gravity is going to illuminate that claim? How can appeal to the CFT settle it one way or the other?

Tim,

I didn't follow all of your comments, particularly those regarding the "duplication" issue. But I hope the following will address some of these concerns.

First of all, you are vastly underestimating the precision with which the AdS/CFT dictionary is understood. For example, one of the very early tests of AdS/CFT was comparing the spectrum of operators of N=4 super Yang-Mills to the spectrum of supergravity fields on AdS_5 x S^5. The precise matching that was found involves several infinite classes of CFT operators/ bulk fields. The entire spectrum of the supergravity theory at energies below that of black holes is accounted for in the CFT. This has since been extended to include stringy excitations as well (yes, you can see string theory emerging out of the CFT ). We also known with great precision the asymptotic spectrum of high dimensions operators in certain CFTs, and there is a precise match with the entropy of black holes using the Bekenstein-Hawking area theorem. This has over time been developed to produce a matching between these quantities that includes quantum gravity corrections to the area theorem. All of this is to say that we know a huge amount about AdS/CFT, and there is no doubt that it yields a quantum theory of gravity that in the classical limit looks just like what we expect a theory of gravity to look like. No other approach to quantum gravity comes remotely close to achieving this.

The basic AdS/CFT dictionary at low energies (meaning states that can be well described by local quantum fields in the bulk) goes as follows. Each bulk field is a function of d+1 bulk coordinates, subject to the field equations. If we go out to the boundary, the bulk field behaves as an overall fixed function of the radial coordinate times some function of the d remaining coordinates. After stripping off the canonical radial factor we get an operator that is just a function of the d "boundary coordinates", and the statement is that this bulk operator is isomorphic to a corresponding CFT operator. The test of this statement is that one can independently compute correlation functions of these objects using either the bulk or CFT equations of motion, and the results have to match. And they do in the many cases where this has been checked.

cont

So the CFT naturally gives us a new way to compute the results of any bulk experiment that involves making measurements at the boundary, where by "at the boundary" I mean in the precise limiting sense described above. For example, you can imagine an observer near the boundary who sends in collapsing matter and then makes measurements at a later time to see what comes out. The CFT can be thought of as a machine that spits out answers to such questions. And one answer it spits out is that in a situation where we expect a black hole to be formed, the state at late times that is accessible to boundary measurements is a pure state. What the CFT machine does not provide is a "bulk story" for what is going on. That is up to us -- we need to infer a bulk story that is compatible with all the boundary measurements that are governed by the CFT.

The essential black hole paradox is that on the one hand when we do low energy experiments the CFT predictions are fully compatible with a bulk story involving familiar quantum fields governed by local physics (general relativity coupled to matter). On the other hand, the CFT predicts that all the information in black hole evaporation comes out in the Hawking radiation. These two statements seem incompatible at face value -- indeed because of what you have been saying: that in the normal Penrose diagram the late time slice that is connected to the boundary is not a Cauchy slice.

So something has to give, and that brings us to the frontier of current research. One school of thought is that low energy physics breaks down in a dramatic fashion in the presence of black holes, e.g. maybe black holes don't form at all, but are replaced some exotic configuration. Or maybe physics somehow becomes nonlocal in the presence of black holes. Or the rules of quantum mechanics need to be modified. Of course, the challenge here is how to introduce such radical modifications without their effects spilling over into non-black hole physics. The point is that this is not just wild speculation (as is the case for most of quantum gravity research) since any scenario is subject to the tight logical straightjacket of being compatible with the CFT. The CFT defines a theory of quantum gravity in the bulk, and the challenge for us is essentially to rewrite the CFT equations of motion in term of the bulk variables, so that we can translate the CFT story of a given process into a bulk story.

About "duplication of operators"... Tim, whether or not it is ultimately the ontologically correct attitude, for now I would suggest that you think of the CFT space-time and the AdS space-time as entirely distinct (so that in particular, the CFT space-time is only *isomorphic* to the AdS conformal boundary, rather than being "numerically identical" with it, as I believe you would say in philosophical jargon). Then, when you think about combinations of CFT operators, you can maintain a sharp distinction between their interpretation in terms of the CFT space-time, and their interpretation in terms of the AdS space-time.

Here is how it will work. Everything is ultimately defined in terms of operators in the CFT space-time. Operators at a single point in CFT space-time, correspond to asymptotic states entering or exiting AdS space-time, from a single point on the boundary (it's the AdS analog of the asymptotic states familiar from perturbative QFT in flat space). Averages of operators over *regions* of CFT space-time, correspond to local operators *within* the AdS space-time.

The details of this mapping really are a lot like Plato's cave. An object in the cave casts a shadow on the wall, and as you get closer to the wall, the shadow gets smaller. In a similar way, an object in the AdS bulk has a spacelike "shadow" on the AdS boundary - a region of boundary points that are spacelike separated from it. Move the object closer to the boundary, and this region can shrink. In the asymptotic limit, the object and its shadow are the same size.

Using the CFT as the fundamental framework, we can reverse this dependence. With a small region of CFT space-time, we can define events within a small wedge of AdS space-time. Larger and larger regions of CFT space-time allow us to describe more and more of AdS. Maybe the quickest way to see how this works, is to read up on MERA, and how a stack of coarse-grainings of an n-dimensional field theory, defines an (n+1)-dimensional hyperbolic space. MERA seems to be profoundly similar to how AdS/CFT works. (Figure 1 in hep-th/0606141 and figure 1 in 1403.3426 might also help a little.)

Anyway, there is no duplication of operators. There are CFT operators at a point in CFT space-time, corresponding to asymptotic states at the AdS boundary; and there are averages of CFT operators over regions of CFT space-time, corresponding to operators defined within a wedge of the emergent AdS space-time.

Mitchell,

Thanks a lot. That is actually the way I had originally assumed it worked, especially since the "boundary" points of a space-time are typically just mathematical fictions, not real parts of the space-time. But Black Hole Guy was expositing things so that it appeared that the space-time of the CFT was literally a submanifold of the AdS space-time.

I do have a bit of a worry about this, and maybe you can help me out. I was rather surprised when I first found out how restrictive the conditions are for a space-time to be "asymptotically flat". I naively thought that is just meant that the curvature went to zero as you went to infinity in all directions, and then I started puzzling about space-times with conical singularities...well, the details don't matter. But I came to realize that the condition of asymptotic flatness was much more restrictive than I had thought. I know that this condition—for the "boundary" of the AdS to be isomorphic to the space-time of the CFT—is a different condition, but is it also highly restrictive? Is there reason to think that every solution, or a generic solution, or a solution subject to certain conditions (e.g. island universe') will meet it? That is, that every possible state of the AdS theory can be modeled by the CFT? If not, then there would be an issue about whether an evaporating black-hole solution is one of the special class.

Tim - I am an amateur spectator of AdS/CFT research. A few names are coming to mind here (Berenstein, Horowitz, Maloney) but I hesitate to say exactly what their work means for your question. I will think about it further, and if Black Hole Guy doesn't step in, I'll say something a week from now.

I wouldn't say that the conditions for being asymptotically flat or AdS are very restrictive from a physical point of view. Start with reasonable initial data, say corresponding to various planets orbiting each other, and then let it evolve in time. Certainly, black holes and their evaporation are pretty irrelevant for this issue, since an evaporating black hole, viewed from outside, doesn't look appreciably different from a radiating star. All the exotic properties of black holes have to do with the features inside the horizon, and this is decoupled from what goes on out at infinity.

Sorry to be so long. I got caught up in a bunch of things.

I suppose that what we have here is really a technical question that may be unknown. I recently read, by the way, that AdS is inherently unstable: the asymptotically AdS space will collapse into something else one way or the other. I don't know how significant that is, but it was interesting.

If you agree with the intervention by Mitchell—which makes sense to me—then the AdS spacetime does not literally contain the CFT space as its boundary. That is what I always thought. Now let's suppose there is a holographic relation between the two: the instantaneous state of one is compatible with exactly one instantaneous state of the other. But this mapping is very convoluted and complex, if only due to the difference in dimensionality between the two spaces. So I am still stuck with my original question: granting all this, how do you know that a state on a connected Cauchy surface on the boundary does not map to a state on a disconnected Cauchy surface in the bulk? I'm sorry, but I just don't see the reasoning.

Tim,

Regarding the question at the end of your message, I believe this has been addressed in my previous remarks so I will be brief. To make your question precise you need to tell me exactly what you mean by saying that the bulk state lives on a "disconnected Cauchy surface". That is, I am asking for a definition at the quantum level, in terms of states and operator algebras. My interpretation of "disconnectedness" is that each component has observables that are realized as operators, and that the operators associated with one component commute with the operators associated with the other component. The problem is that there is no way to realize a corresponding construction in the CFT. To say that the CFT can describe a bulk with a disconnected Cauchy surface in the above sense, we would need to find these mutually commuting operators. As soon as one realizes that the CFT Hamiltonian is associated with a boundary operator in the dual spacetime it becomes clear that this construction cannot be realized, because the only CFT operators that commute with the Hamiltonian are symmetry generators, and these clearly not sufficient to describe the interior component of the bulk Cauchy surface.

Once more: the Hamiltonian in quantum (or classical) gravity is a boundary operator. Operators living in a component of a Cauchy surface disconnected from the boundary commute with the Hamiltonian, assuming the above meaning of "disconnected". But in the CFT essentially no operators commute with the CFT Hamiltonian. Thus the CFT state cannot be dual to such a bulk state. The CFT evolution is unitary. However we are eventually able to translate this unitary evolution into a coherent story in the bulk, this story will not involve disconnected Cauchy surfaces in the above sense.

BHG

I take it that we all agree that the bulk theory satisfies the ETCRs? So all the operators on one part of a Cauchy surface commute with all the operators on any other part? That is true no matter whether the Cauchy surface is connected or not. If you were correct (which I do not believe, as you know) that the Hamiltonian operator for the bulk were a boundary operator for the bulk, then it would immediately follow that every bulk operator that is not associated with the boundary would commute with the Hamiltonian. And then, by your own argument, the correspondence fails completely. AdS/CFT is dead.

Time to rethink.

Tim,

Sorry, but anyone who has studied quantum gauge theories or gravity knows that physical operators do not all commute with each other at spacelike separation. You are taking a fact you've learned (or heard) about theories without gauge or diffeomorphism invariance and assuming this holds in general, and this is also why you fail to understand how the Hamiltonian in gravity can be a boundary operator. I can help you understand these issues if you are willing to listen. For example, I suggest you try to understand why it is that in QED the time component of the vector potential A_t has a nonzero commutator with the electron field at spacelike separation. This is a standard textbook fact, and is tied to the fact that the charge operator in QED is a boundary operator.

BHG,

I am indeed quite surprised by this claim. The demand for the ETCR for all observables is common and widely presented as an absolute requirement in QFT, both Aberlian and non-Abelian. Indeed, this is the reason more take that theory to be Relativistic: if the observables commute, then you can't send superluminal signals. Of course, the vector potential is usually not regarded as observable, too. So I would love the hear about this, especially if your account of black hole evaporation hand on it.

"Abelian"; "many take" rather than "more take"; "hangs on it" rather than "hand on it". Sorry. Too tired.

Tim,

It is indeed true that local measurements performed at spacelike separation had better commute or causality would be violated. So to the extent that a local measurement is implemented by a local operator, such local operators must commute at spacelike separation. I certainly have no quarrel with any of this, but I what I am saying is something different. What I am saying is false is the statement that in gravity or gauge theory the Hilbert space can be built up by acting with a collection of local operators that commute at spacelike separation.

Let's consider QED in flat space, and imagine a large region of space with a spherical boundary of radius R. There is a vacuum state, and the Hilbert space of the theory can be built up by acting with operators on the vacuum state. One such state corresponds to an electron localized near some point p. There is an operator, call it O(p), that acts on the vacuum to create this state. Another operator is E_i(R), representing the electric field on the boundary sphere. We ask: does O(p) commute with E_i(R) at equal times? The answer is no, even though the two operators are spacelike separated.

QED is a gauge theory, and in such theories we have to be careful to define physical operators; i.e. those that act on physical states and produce other physical states. One way to do so is to demand that physical operators be gauge invariant. The electron field operator by itself is not gauge invariant, and hence not a physical operator. To understand this point, let's consider how an experimentalist would create a 1-electron state in a manner consistent with charge conservation. The person could zap the vacuum with some energy sufficient to produce an electron-positron pair, and then move the positron off to infinity. Crucially, upon doing so one will be left with the electric field lines running between the electron and positron, or rather running from the electron to the boundary sphere. The moral is that the physical gauge invariant operator corresponds to creating the electron along with its Coulomb field. Clearly this operator is nonlocal despite it nominally being associated to the point p, and so it's no surprise that this operator fails to commute with other operators that are spacelike to p. There simply doesn't exist any local, physical, operator that creates the 1-electron state. The construction of these physical operators that create electrons was, I believe, first explained by Dirac in 1955.

cont

The other approach to constructing physical operators is to "fix the gauge". In most QFT books you will find QED quantized in "Coulomb gauge". In this approach one imposes conditions on the gauge fields and then solves associated constraint equations to remove all unphysical degrees of freedom. But the action of solving the constraints involves integration, so is nonlocal. This eventually yields commutation relation that are nonzero at spacelike separation. See for example eq. 15.11 of Bjorken and Drell where the commutation relation between A_t and the electron field is written down, and is most definitely nonzero at spacelike separation.

This structure is consistent with the statement that in QED the charge operator is a boundary operator (it measures the electric flux passing through the sphere at R). This charge operator has nonzero commutation relations with the physical operators that create charged particles inside the sphere.

In gravity, the analog of charge is energy-momentum, and all of the above carries over accordingly. Any physical operator that creates a particle of nonzero energy-momentum is necessarily nonlocal. The Hamiltonian is a boundary operator that measures the gravitational flux through a large sphere, and it has nonzero commutation relations with physical operators that creates states in the interior.

On the other hand, if there is a "baby universe" the situation is different. The "gravitational field lines" of objects in the baby universe cannot extend out to the asymptotic region disconnected from the baby universe, and so the Hamiltonian defined on the boundary of the asymptotic region will commute with the baby universe operators. That is, baby universe states have zero energy. Exactly the same would be true for QED defined on a disconnected baby universe spacetime: the charge operator would commute with all physical operator that create states in the baby universe. The baby universe states have zero charge (in general, one cannot have a nonzero charge on a compact space, as is clear from thinking about the field lines).

The relevance to AdS/CFT is that in a CFT essentially all operators have nonzero commutators with the Hamiltonian, so there is no room for baby universes.

BHG,

I think I understand all of these comments—they are not really unfamiliar—but I don't see what relevance they have.

First point: of course any physical state will satisfy Stokes' Theorem, or its quantum analog. So measurements "at infinity" can yield information about the total charge in the interior. Assuming, of course (which is essential!) that there are no other boundaries in the interior! Which may be the very question under dispute.

But however that comes out, just the total charge is not even close to what you want, which would include the exact charge distribution in the interior. So this observation gets us almost no distance toward what you wanted to prove.

Second point, and I have said this several times, in my Penrose diagram there is no point in the interior of the black hole that cannot be connected to the boundary by a continuous Cauchy surface. The situation is just the opposite: there are points in the future of the evaporation event that can't be connected to the interior of the black hole by a single continuous Cauchy surface. So the situation does not at all suggest that there is something in the interior of the black hole that is "screened" from the outside. It is that there are facts about the exterior world that are shielded from the inside of the hole. Not paradoxical at all.

Tim,

If these points are familiar then how come you are having trouble accepting the fact the Hamiltonian in quantum gravity is a boundary operator? It is not fundamentally different from the statement that the charge operator in QED is a boundary operator. I have provided pedagogical explanations, pointed you to quotes from renowned experts affirming this fact ("Diffeomorphism invariance implies that the Hamiltonian is defined by a surface integral at infinity." -G. Horowitz), etc. Not sure what else I can do.

Perhaps it will be more productive if I pose the following question to you. Consider some late time at the asymptotic boundary, long after the black hole has evaporated. Let H be the Hamiltonian defined on the asymptotic boundary at this time. Question: in your conception of the Hilbert space of this theory, is it or is it not the case that all physical operators (save symmetry generators) have a nonzero commutator with H? If "yes", then no degrees of freedom are invisible to an asymptotic observer. If "no" then there is a corresponding degeneracy in the energy spectrum of the theory.

BHG,

Of course the answer to your question is "no". I can't understand how anyone could believe otherwise. As you say, you build up a folium of states by starting with a vacuum state (let's leave aside the complication that this could already be degenerate!) and acting on it with creation operators. So I have creation operators for 1-particle states than can be associated with any given place on the interior. Why should't operating with the same operator at different locations on the vacuum produce different states with exactly the same asymptotic character (same net charge, mass and spin, for example). Are you suggesting that every single application of that creation operator yields a state with a different energy? How is that even possible? There are non-denumerably many different locations to create a particle, so there should be correspondingly many different values for the energy, with each state having its own energy? Why would that be: it seems incredible.

The answer you have suggested, as I understand it, is that the AdS/CFT correspondence demands that the Hamiltonian on the boundary match the Hamiltonian in the bulk. And since the CFT Hamiltonian is non-degenerate so must the AdS one be. But that is, at the moment, a completely question-begging argument. You are just presupposing a holographic theory, not proving that this is one.It seems completely implausible that a single number could characterize all of the details of the bulk state, for the reasons just given. And I never did concede that the AdS Hamiltonian is a boundary operator rather than a bulk operator, for reasons we've been through many times.

The charge operator gives the net charge of the system which is, of course, just a number and by Stokes' Theorem equal to both a surface integral and a volume integral. Sure, Net charge. But that is an entirely different claim than the claim that a boundary operator could reveal the exact charge distribution inside rather than the net charge. Do you maintain this about charge, that there are no two distinct states with the same net charge? If not, what's the difference? I would expect boundary operators in AdS to provide the sort of information we expect:total charge, mass(energy) and angular momentum, exactly the usual "hair" on a black hole.

We keep coming around, in some fashion or other, to your claim that the Hamiltonian of the bulk theory is a boundary operator, and the zero operator in the interior, which I have never accepted. And I can't find anyone who I ask that the claim makes sense to. So maybe we are just stuck here. It's a pity after all this time, but I don't know how to proceed.

The implications of diffeomorphism invariance for any theory are notoriously tricky. It is easy to fall into the trap of concluding, e.g., that there can be no local observables at all, and the only observable features are global quantities, but that is plainly false. Maybe if you can give me the citation for Horowitz I can study that. But I know that there are fairly highly regarded experts who seem to have become confused over this point, and used it to reject all local observables.

So if you give me the citation, I'll try to work through the paper. But for now we seem to have hit a wall.

Thanks for all your help. You have been very patient.

Cheers,

Tim

BHG

Since you put in the exception for symmetries, and you might argue that all my one-particle states are related by a symmetry, let me change the example to remove that issue. Start with the vacuum. Create a six-particle state with six creation operations associated with points that form an equilateral octahedron (in some frame). Consider all of the states formed by scaling the sides of the octahedron up and down. Now consider the state formed by six such operations that form a pair of scalene triangles that are many times further apart from each other than the length of any of their sides. Scale that configuration up and down. These two types of configuration are not related by any symmetry. Each contains six identical particles with the same rest mass. As one scales them up and down the gravitational potential energy in each configuration changes continuously.

Now: are you claiming that no pair of states, one from the first set of states and one from the second, has the same expectation value for the boundary Hamiltonian? That can't possibly be correct. But if not, how can that Hamiltonian fail to be degenerate? Or more directly, how can any set of measurements taken at infinity possibly distinguish all of these states from one another?

If you are not claiming that, then can you at least, in this simple context not involving black holes, explain precisely what you are claiming? Because if that is not what you are claiming then I have not understood at all the argument you are trying to bring against my solution. If we can clear up this point then maybe we can make further progress.

Thanks.

Tim,

"There are non-denumerably many different locations to create a particle, so there should be correspondingly many different values for the energy, with each state having its own energy? Why would that be: it seems incredible. "If you actually go and work out the energy spectrum of a particle in AdS you will instantly see why this argument is nonsense. AdS acts like a gravitational well, so changing the location in the way you are imagining costs energy.

"Do you maintain this about charge, that there are no two distinct states with the same net charge? If not, what's the difference? "Of course I do not maintain that. The obvious difference is that energy is universal while charge is not. There are lots of operators that commute with the charge operator, but many fewer that commute with the Hamiltonian (only symmetry generators like rotations etc).

" We keep coming around, in some fashion or other, to your claim that the Hamiltonian of the bulk theory is a boundary operator, and the zero operator in the interior, which I have never accepted. And I can't find anyone who I ask that the claim makes sense to. "Pick any reference you like that discusses the Hamiltonian in the canonical formulation of gravity. I guarantee that they will explain that the Hamiltonian consists of a sum of two terms: a spatial volume integral and surface integral. The former vanishes whenever the equation of motion are satisfied, so the Hamiltonian on-shell reduces to the surface integral. Either you are garbling the issue when you discuss this with your correspondents or they are seriously confused. In electrodynamics the operator that generates gauge transformation has exactly the same structure: a volume integral that vanishes by Gauss' law plus a surface integral that yields the charge. I take it you disagree with this as well?

The Horowitz quote is taken from a talk (http://web.physics.ucsb.edu/~gary/holography.pdf). I only included it to quickly refute your claim that I am making a controversial statement about the Hamiltonian in gravity.

It's a pity that we can't get to discussing the real issues, but I do think this was illuminating in terms of distinguishing how physicists versus philosophers think about problems. I think this thread illustrates the distinction well.

BHG,

I doubt we are going to be able to work through this, but just some clarifications.

1) Your comment about AdS being a gravitational well once again suggests that however these arguments are supposed to go, they are restricted to AdS. Since a standard black hole is not in AdS I can't see that these results can be diagnostic about the general case. Occasionally you say that we can't tell whether actual space-time is or isn't AdS, but by that same token we can't be sure it is AdS, And in any case, if this is a problem of principle it better have a solution in all physically possible circumstances. The closer your remarks are tied to AdS the less relevant they seem to the supposed paradox.

2) I can't follow your claim about Gauss' Law. That law does not state that anything is equal to the sum of a volume term and a surface term: the usual explication is that it proves that the volume integral of the divergence of a field equals the surface integral of the flux over the boundary.

3) However it is with a single particle in AdS, my example of six particles seems decisive against one understanding of your claim. Can you either explain why the example is not decisive or else explain exactly what the claim is?

4) I really have no idea what the whole"physicists vs. philosophers" thing is supposed to be about. We seem to be hung up on various technical claims that we straightforwardly disagree about, as well as some simple points (e.g. whether you think of the space of the CFT as literally a subspace of the bulk or rather just isomorphic in some way to an asymptotic boundary of the bulk) that just can't get cleared up. Your insistence that the Hamiltonian is the zero operator just because of its action on the solution space is perhaps the main disagreement. The action of any operator on its eigenstates with eigenvalue zero is obviously the same, confined to them, as the action of the zero operator, but that just does not make it the zero operator. If it were the zero operator, every state would be a state with eigenvalue zero! But the whole role of the operator is to distinguish states in the kinematic space into the solutions and the non-solutions. I don't know any clearer way to put this, and the point is so manifest as to seem beyond dispute. It certainly is to everyone I talk to. But this has nothing to do with physics vs. philosophy as far as I can see.

5) My diagnosis of the errors underlying the supposed information loss paradox is what it is, but you appear to be reacting more to the name "baby universe theory" than to any of the detailed argumentation. In particular, you keep saying that what is going on the the interior of the black hole will be inaccessible in some way from the boundary, and that distinguishes the interior from other places in the bulk. I have pointed out repeatedly that this is just not so: every event in the interior lies on many connected Cauchy surfaces that reach to the asymptotic boundary. So I can't see any place at all that your objection—which I still can't get—could tell against my diagnosis.

If you want to address any of these questions that would be great. Or if they are unclear, I can try to sharpen them up.

Tim,

My patience is running out, mainly because all these points have already been explained in agonizing detail. Very quickly

1) AdS is the one case where we understand quantum gravity in significant detail, and surely it's relevant that your proposed scenario is ruled out here.

2) Please reread. I clearly stated and used precisely what Gauss' law is, which the statement that the divergence of the electric field equals the charge density.

3) How does six particles help? In general, a system of interacting quantum particles confined to a box (or by a gravitational well) will have a nondegenerate spectrum, modulo those degeneracies associated with symmetries (like rotations). Again, this becomes obvious if you actually take pen to paper and try to work it out.

4) For the nth time, all that matters for anything being discussed here is how operators act on the actual, physical states of the theory. Whatever some operators do when acting on some unphysical states is totally irrelevant to any physical question (by definition). It is simply a theorem that the Hamiltonian is a boundary operator when acting on physical states, as the quote from Horowitz makes clear. I sincerely doubt that you have encountered someone versed in canonical gravity who would disagree.

5) you already agreed in a previous post that your scenario contradicts AdS/CFT. When I asked whether there are lots of operators that commute with the Hamiltonian at late times you said yes. This implies a large deneracy, but the CFT has no such degeneracy. This was my point all along, and now you have agreed, even if you apparently don't realize.

BHG

I don't think you are taking the time now to even read what you write. For example, you wrote, and I quote,

"In electrodynamics the operator that generates gauge transformation has exactly the same structure: a volume integral that vanishes by Gauss' law plus a surface integral that yields the charge. I take it you disagree with this as well?"

And just now:

"I clearly stated and used precisely what Gauss' law is, which the statement that the divergence of the electric field equals the charge density."

Do you really not want to correct something in one or both of these claims? Trying to make sense of what you even have in mind here is beyond my capacity, and your additional claim that everything is clearly stated and precise just makes the situation worse. I am going to be very explicit and brutally honest about what I have just quoted, since that seems to be what it might take to get you to attend to what you are claiming.

First: yes, I certainly disagree with what you asked about, because it is plainly wrong.

Gauss's Law is one instance of the generalized Stokes theorem. That theorem does not state that any volume integral vanishes. It just doesn't. That is just a plain mathematical error. Yet that is what you have written. Gauss's law simply does not state what you have said it does. One more time, and this is not hard, Gauss's law states that the volume integral of the divergence of a vector field equals the surface integral over the boundary of that volume of the flux. Period. That's the theorem. So the only way that there can be "a volume integral that vanishes by Gauss's law" is if the corresponding surface integral also vanishes. In that case, the integral of the divergence would vanish in the volume and since—by what you also somehow manage to call Gauss's Law, viz. that the charge density is the divergence of the electric field—the total change would have to be zero. Of course, if the net charge is zero, then the volume integral of the divergence of E is zero, and by Gauss's Law or Gauss's Theorem the surface integral of the flux is *also* zero. But nowhere does Gauss's law state, or imply, that anything is the *sum* of any volume integral and any surface integral. It states that a certain volume integral is *equal to* a certain surface integral, and hence their sum is twice what each is individually.

Now either you understand what I just wrote, and you now want to correct your claims, or you don't, in which case the problem is not my understanding of AdS/CFT but your understanding of basic vector calculus. Either you are being sloppy—and refusing to acknowledge that when it is pointed out—or you are very seriously confused. This is not physics vs. philosophy, this is basic first year mathematics. And the additional tone of frustration in which these plain mathematical errors have been conveyed—as if I am too dense to get the point—is just making the situation impossible.

On top of that, I haven't a clue what "the operator that generates gauge transformation"—as if there is only one possible gauge transformation—even means.

Con't

Con't

To back up your claims, you cite the "renowned expert" Gary Horowitz. When I ask for the source of this quote, I get a set of slides in which their is no attempt at all—none—to explain and justify it. Horowitz just throws out some more names (Balasubramanian, Marolf, Rozali) of more renowned experts, I suppose. I stated clearly above that there is a tremendous amount of confusion, even among the "renowned experts" about the implications of diffeomorphism invariance. I will state that I have studied and thought about this particular question—because it lies at the center of the so-called "hole problem"—more intensely and extensively than you have. If you think you understand what diffeomorphism invariance implies here, fine. Then we can have a discussion. But if all you are going to do is throw out some quotes with no argumentation or explication and expect me to accept them because they are from "renowned experts", then there is no point at all in continuing. Either you actually think you understand the arguments given by the experts or you don't. If you do, we can discuss, but leave the appeals to authority out. If you don't, then just admit it: you don't really understand how they justify their claims, you are just repeating what you have found written somewhere and can't explicate it or defend it.

This discussion has gone on so long because I believed you might have some understanding of something that was relevant to my diagnosis of the information paradox, and that I could learn something useful. But after all this time what you are providing are erroneous claims about basic theorems of vector calculus and assertions without a shred of argument that are just attached to names that you think I must bow down before.

Do now want to correct either or both of the quoted passages above?

Do you think that you, yourself, understand the implications of diffeomorphism invariance well enough to defend the quotations you cite?

There are other passages in your recent post that I could go through in this detail, but that would take too long. I think this suffices to make the point. The issue is not physics vs. philosophy, it is the failure to either understand or accurately explain the claims that physicists—and you yourself— are making.

If this is the level of precision and rigor you think sufficient to justify your claims then there is no reason to go on with this. You are wasting both of our time.

None of my business, but,

Statement:

"In electrodynamics the operator that generates gauge transformation has exactly the same structure: a volume integral that vanishes by Gauss' law plus a surface integral that yields the charge."

Rebuttal:

"Gauss's Law is one instance of the generalized Stokes theorem. That theorem does not state that any volume integral vanishes. It just doesn't."

The meaning I took from the first statement is not that it defines Gauss's Law, but rather that Gauss's Law can be used to show that a particular volume integral is zero, in a particular case.

For example, see https://arxiv.org/abs/0809.1764 which contains this quote:

"By Gauss' law the zero modes of the Faddeev-Popov kernel constrain the physical wave functionals to zero colour charge states."

(I found this by googling "Hamiltonian Gauss's Law" which gave a number of hits, mostly pdf's on how to derive Hamiltonians for quantum behavior.)

Guys,

At this point I have no clue what you are talking about, but of course what Tim says is technically correct, Gauss' law merely states that certain volume integrals can be converted into surface integrals. Having said that, this might have been what bhg meant.

I think it's fairly easy though: If there is a theorem that states what bhg claims, then certainly there is a proof for that theorem, so please provide a reference.

JimV,

Of course I don't know the exact context of this quote, but since the net color of any system is zero, one would expect the volume integral (and hence, by Gauss's law, the surface integral) for color charge to be zero. This does not claim that any integral is a sum of a volume part and a surface part. I have a suspicion that some of this is a garbled report of what is supposed to follow not from Gauss's law but from diffeomorphism invariance, which is a topic about which there is endless confusion. but that's just a guess.

I'm probably misunderstanding this, but isn't bhg just referring to the Regge & Teitelboim paper that the Hamiltonian is a volume integral plus a surface contribution, where the volume integral vanishes on-shell? Or are you still debating which states are and aren't physical and should be allowed in qg?

Sabine,

That sounds pretty plausible, so maybe that's it. I know of the paper but have not studied it, so I'll see if I can get anything out of it. May go over my head.

Tim,

Your latest messages have reached a new height of absurdity. Where to start...

1) Please open either of the two most standard books on electromagnetism (Griffiths or Jackson). There you will find that Gauss' law is the name attached to the equation div(E) - \rho = 0. What *you* are calling Gauss' law, which is a basic theorem in vector calculus, is what these books call the "divergence theorem". My usage is totally standard: just try typing "Gauss law electromagnetism" into google. Please report your findings. I find your rant quite amusing and illuminating

2) The Horowitz quote: this is also hilarious. Early on in this exchange I pointed you to the paper that establishes what I have been saying about the Hamiltonian in GR. The paper is "Role of surface integrals..." by Regge and Teitelboim. Frankly, I doubt that you will be capable of understanding this paper, as it requires more than a superficial understanding of GR, but anyway this is a classic paper in the field (844 citations) that everybody in GR knows about. You can't blame me that you won't or can't absorb this. As I said, I brought up the Horowitz quote only to quickly refute your claim that I am the only one who says that the Hamiltonian in GR is a boundary operator. The reason Horowitz doesn't include references along with this claim is because it (the RG paper) is standard knowledge (.e.g see Bee's message).

cont

cont

3) Now let's move to my statement "In electrodynamics the operator that generates gauge transformation has exactly the same structure: a volume integral that vanishes by Gauss' law plus a surface integral that yields the charge" which seems to have sent you into a downward spiral of confusion. As I will now explain in full pedagogical detail, it is precisely stated and precisely correct. Here we go...

Let's take EM coupled to a charged scalar field \phi of charge q. A gauge transformation associated with gauge parameter \lambda acts on the vector potential A_i and scalar field \phi as: \delta A_i = d_i \lambda, \delta \phi = q \lambda \phi (I will ignore signs and factors of i). By definition, the generator of this transformation, call it Q_\lambda, should give the commutation relations [Q_\lambda,A_i] = \delta A_i, [Q_\lambda, \phi] = \delta \phi. Let us now construct this operator. THe basic equal time commutation relations at our disposal are [E_i(x), A_j(y)]= \delta_{ij}\delta^3(x-y), [J_t(x), \phi(y)] = q \phi(x)\delta^3(x-y). Here J_t is the time component of the conserved charge current for the scalar field; i.e it is the matter charge density J_t = \rho, and E_i is the electric field. We then immediately see that the correct generator is Q_\lambda = \int d^3x ( E_i(x) d_i \lambda(x) + J_t(x)\lambda(x) ). We now integrate the first term by parts, taking care to keep the surface term since \lambda is allowed to be nonzero at infinity. This gives Q_\lambda = \int d^2x \lambda n_i E_i + \int d^3 x (-div(E)+J_t(x))\lambda(x). So we have two terms: a surface term and a volume term. The volume term vanishes by Gauss' law: div(E) - J_t = 0. We are left with a surface term at infinity. Let's take \lambda to go to a constant at infinity. Then, we see that Q_\lambda is simply the asymptotic value of \lambda times the flux of the electric field out of surface. But this flux is just the total electric charge of the system. So we have Q_\lambda = \lambda(\infty) Q_{tot}. As you see, what I said is completely correct. Was that so hard?

Note that in the canonical formulation of EM Gauss' law is a constraint, not a dynamical equation: the phase space is constrained to the hypersurface where the Gauss law is obeyed. So Q_\lambda is a pure surface term on the actual phase space of the theory. In the quantum theory the statement is that Q_\lambda is a pure surface term when acting on any physical state.

bhg,

It is merely a matter of terminology and only tangentially relevant, but I do side with Tim on the matter of what "Gauss' law" refers to: It's an integral identity. The equation you refer to div(E) - \rho = 0 (and similar relations in other theories) is clearly not Gauss' law. You use Gauss' law to rewrite the (integral over) this relation which gives an equation that can be interpreted as saying the flux through some surface is sourced by the charges inside.

I don't have the books you mention here (or any other textbook for that matter), but I am very sure that when Weinberg in Grav & Cosm discusses the GR version of Gauss' law that's what he's referring to, an integral identity (his point being to show that the derivative of the determinant doesn't make trouble but just converts the partial into a covariant derivative). Now, I'm from a somewhat different community than you, but I have never heard Gauss' law being referred to in the way that you do.

perhaps this is why Purcell and Morin thought to alert the undergraduates about it :)

Electricity and Magnetism (Edward Purcell, David Morin), 3rd ed

Sec 2.13 "Distinguishing the physics from the mathematics"

this is indeed an amusing discussion at an elementary level (i can imagine how it might be frustrating for the bhg!)

Bee,

Really? What name do you give to the Maxwell equation div(E)-\rho =0, or the equivalent integral version \int E.dS = Q? This Maxwell equation is called Gauss law in every text or reference that I know of, and this also seems to be the case on Wikipedia and all of the hits on google that I see. The theorem from vector calculus is usually called the divergence theorem or sometimes Gauss' theorem. The distinction between a "law" and a "theorem" is meant to clearly distinguish a physical principle from a purely mathematical relation. I also just checked Weinberg and he calls the vector calculus result a theorem and not a law, so I don't understand your claim here. So what is the basis of your strong statement that "The equation you refer to div(E) - \rho = 0 (and similar relations in other theories) is clearly not Gauss' law." when it seems that every reference I can find does call this equation Gauss' law. Perhaps there a counterexample out there somewhere, but I am truly perplexed by your claim that I am adopting some idiosyncratic nomenclature.

BHG,

Actually, this is very useful. Once we have sorted out the terminological issues, your derivation makes exactly the point I have been making all along, and exactly at the point where you invoke what you call Gauss's Law. (And there is just a difference of terminology, as Sabine notes.) So this is actually quite perfect. From the critical part of your derivation:

"So we have two terms: a surface term and a volume term. The volume term vanishes by Gauss' law: div(E) - J_t = 0."

Now note: what exactly did you refer to as the reason that the volume integral goes to zero? You referred to Gauss's Law: a specific relation between div(E) and J_t. A specific, physical relation, and a rather important one, between those two quantities. There are states in the kinematic space where that particular relation does not hold. It is a requirement for a state to be in the folium of physical states that the relation does hold. If it did not hold, then the volume integral would not go to zero. There is important physics in that relation. Even, as you say, a physical law.

Now: consider div(E) - J_t as an operator. That operator *vanishes on the physical states*. And that operator *is not the zero operator*. It is importantly not the zero operator. It is essentially not the zero operator. It is an operator that codes essential physical structure, indeed essential physical law—the very law you call Gauss's law. The zero operator does no such thing. The zero operator contains no physics. The zero operator does not encode any physical law. If you were under the impression that the volume integral vanishes for the trivial reason that div(E) - J_t is the zero operator, that is for the trivial reason that the value of that operator on *every* state, no matter what it is, is zero, then you would be making both a physical and a mathematical mistake. You would not understand the physics.

So your derivation make absolutely clear precisely the point I have been insisting on, namely the incorrectness of the assertion that the Hamiltonian *is just* a boundary operator. It isn't, and can't be and never was. It makes no sense to claim that. The Hamiltonian is not just a boundary operator, which means that even if one were to think of the space of the CFT as literally the boundary of the space of the bulk, it would be an error to say that the two theories have the same Hamiltonian. And it is not clear in any case whether the space of the CFT is literally the boundary of the bulk.

Con't

Con't

The equation div(E) - J_t = 0 is enlightening about our problem in another way. That is a slightly irregular way to write the relevant physical relation. The more usual way is div(E) = J_t. These are evidently mathematically equivalent, so what is the point? The point is that every physical equation, every equation that specifies the actual physics being postulated, can be written (trivially) as Something = 0. So written, the condition for being a solution to the equations, the condition for being a physical state, will be being a solution to Something = 0, where the Something is an operator. And that operator will, by definition, act the same on all the physical states, all of the solutions, exactly as the zero operator does. But that does not, in the least, *make* Something the zero operator! If it did, then any physics would reduce to the claim that Zero(psi) = 0. But that is true for all psi and has no physical content. If you add: the physical theory is Zero(psi) = 0, but of course only on the physical states, then that is not a whit better, since we evidently have no specification of what the physical states are.

So although a misunderstanding about what constitutes Gauss's law got us here, it got us to exactly the right place. When you say "X is a boundary operator", or, more exactly, "The Hamiltonian of the bulk is a boundary operator".is it the case that you have in mind this sort of thing as an example? If so, then it is a clear example where it is not correct to say that the Hamiltonian of the bulk is a boundary operator, for the reason I just gave. Now lets push this even further. You want to say that in this case the "charge operator is a boundary operator, and therefore can be measured at the boundary". Leaving aside the point just made: yes, the net charge of a system can be determined just by measurements made on the boundary of the system. There was certainly no reason to make a big deal about that since it is a familiar consequence of what you call Gauss's Law and what I call Gauss's Law. That is, by your Gauss's law, rho = div(E), and by my Gauss's Law the volume integral of Div(E) = the surface integral of (E), i.e. the flux. So everybody knows that that net charge can be determined by measurements at the boundary. But the net charge is just one number. It does rather little to specify the exact distribution of charge in the bulk. Many different bulk states have the same net charge.

Now if I am understanding your claim correctly, you are claiming not merely that the total energy in the bulk can be determined by measurements at the boundary, which is trivial, but that *everything* about the bulk, in all detail, can be determined by *just thee measurement of the Hamiltonian at the boundary*. If that is your assertion, I think it cannot possibly be correct, and my six-particle example proves it. It is essentially the claim that the entire, detailed physical state of the bulk can be boiled down to one, single number, measurable at the boundary. And if that's the claim, I don't believe it. If that is not the claim, then please state what the claim it.

Further to J's comment:

Electricity and Magnetism (Edward Purcell, David Morin), 3rd ed Sec 2.13 "Distinguishing the physics from the mathematics"

...

Turning to mathematics, we introduce the divergence, which

gives a measure of the flux of a vector field out of a small volume.

We prove Gauss’s theorem (or the divergence theorem) and then

use it to write Gauss’s law in differential form. The result is the first

of the four equations known as Maxwell’s equations (the subject

of Chapter 9). ...

bhg,

I just know this equation as "Maxwell's first equation." Besides, I probably was conflating "law" with "theorem". My textbooks were all in German where it's called "Satz" (lit: sentence). In any case, I think this was one of the easier points to clarify ;)

To the vast readership of this thread: I exhort everyone to respect the proper usage of the words "law" and "theorem". Referring to a purely mathematical relation as a "law" can cause serious confusion for students, not to mention more advanced researchers, as evidenced by the recent kerfuffle.

bhg,

agreed. misunderstanding/confusing basics at undergrad level can lead to more serious confusion [reason enough for Purcell and Morin to add the section i cite, see also JimV's comment].

the following comments (on you) seem particularly ironic in this context!

"But after all this time what you are providing are erroneous claims about basic theorems of vector calculus"

"the problem is not my understanding of AdS/CFT but your understanding of basic vector calculus. Either you are being sloppy—and refusing to acknowledge that when it is pointed out—or you are very seriously confused. This is not physics vs. philosophy, this is basic first year mathematics"

all the best for further discussion(s) with Tim on this :)

J,

You are absolutely right: my bad. I know the theorem as Gauss's Law rather than Gauss's theorem, and unfortunately both what we may call Gauss's Theorem and Gauss's Law were at play in the derivations. Without all the extra steps, it is not possible to see how you get a situation where there really is a sum of a volume and a surface integral, the former of which uses Gauss's Law to do. So it was my error, and I went off too quickly. The solution was to be more explicit and clear, and that solved everything.

Now: let's apply the same technique toward solving the confusion of the zero operator with an operator that happens to yield zero on solutions and we can make more progress.

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