- (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.

## 281 comments:

«Oldest ‹Older 201 – 281 of 281Tim,

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.

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