Thursday, May 11, 2017

A Philosopher Tries to Understand the Black Hole Information Paradox

Is the black hole information loss paradox really a paradox? Tim Maudlin, a philosopher from NYU and occasional reader of this blog, doesn’t think so. Today, he has a paper on the arXiv in which he complains that the so-called paradox isn’t and physicists don’t understand what they are talking about.
So is the paradox a paradox? If you mean whether black holes break mathematics, then the answer is clearly no. The problem with black holes is that nobody knows how to combine them with quantum field theory. It should really better be called a problem than a paradox, but nomenclature rarely follows logical argumentation.

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

First, let me point out that this is hardly news. As Maudlin himself notes, this is an old story, though I admit it’s often not spelled out very clearly in the literature. In particular the Susskind-Thorlacius paper that Maudlin picks on is wrong in more ways than I can possibly get into here. Everyone in the field who has their marbles together knows that time-evolution is unitary on “nice slices”– which are complete Cauchy-hypersurfaces – at all finite times. The non-unitarity comes from eventually cutting these slices. The slices that Maudlin uses aren’t quite as nice because they’re discontinuous, but they essentially tell the same story.

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

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

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

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

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

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

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

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

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

1,706 comments:

  1. "No one has ever demonstrated that QM computers do work in the presence of evaporating black holes."

    This could (charitably, as one should try to do) refer to TM's previous argument that BH evaporation with a remnant is something new and untested and who knows what would result? That is the argument, I believe, which he used to counter the standard-QM-model result of a tensor product. Of course this would also imply:

    1) there then is no way to predict QM experiments involving thermal radiation left by BH evaporation, contrary to his earlier claims.
    2) there then is no way of knowing if his remnant solution could work, since it disagrees with current QM methods that predict current experiments well, and we have no other theoretical or experimental way of confirming it.

    I guess, under this interpretation, his remnant hypothesis is similar to multiverse theories - untestable now, maybe, just maybe, they might be testable in some way in the future; although I think proponents of those theories would say they were grounded better theoretically.

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  2. To Anyone Still Paying Attention:

    I have said that I will no longer be responding to anonymous posters and I will not. If anyone is interested in carrying on this discussion, all it takes is posting under your real name, and if it not worth that to you then it is not worth my time and effort to respond.

    We have come to the point where foundational issues have become essential. Not because I brought them up but because BHG brought up the issue of "measurement theory" in an attempt to produce some conflict between the solution I have presented in my paper and AdS/CFT, apparently unaware that there is no "measurement theory" for AdS/CFT because, in turn there is no "measurement theory" for standard quantum theory. What standard quantum theory has, instead, is the Measurement Problem, the very problem that is a central topic in Foundations of Physics, and is either completely avoided or obscured in every standard text on quantum theory.

    To be clear: the conceptual set-up of the supposed Information Loss Paradox requires treating Quantum Theory as a no-collapse theory because collapses lose information and so no one could regard the loss of information in a collapse theory as contrary to any fundamental axioms. And the main idea that BHG and physphil have been pushing is that the set of possible *post-measurment* states of a quantum system must form a tensor-product Hilbert space, which is certainly true if every Hermitian operator corresponds to a possible measurement and the post-measurement state is given by projection onto an eigen-subspace of the Hilbert space. But taken that way, their supposed objection to my solution has no bite: it rather shows that the assumption of a non-tensor-product Hilbert space was an error to begin with. This confusion arises exactly because the description and status of "collapse of the wavefunction" in the usual presentations is obscure. In the Foundations literature, any collapse theory is regarded as changing "standard quantum mechanics".

    Con't

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  3. If you are interested in foundational questions, that is wonderful. But the beginning of wisdom in any subject is the recognition of what one does not know. In Foundation of Physics this is a particularly serious problem because so few physicists have any training in Foundations at all, and in most physics programs offer no courses in Foundations. Thus BHG and physphil are convinced of the falsehood that all quantum textbooks present the same "axioms" or "rules" or "principles". The falsity of this claim is beautifully illustrated in John Bell's essay "Against 'Measurement'", in which he takes a careful look at there different textbooks: Landau and Lifshitz, Kurt Gottfried's, and NG van Kampen's. All of these "standard" textbooks present different (vague) principles of quantum mechanics. Bell also discusses Bohm, GRW, the inadequacy of decoherence to solve the measurement problem, and much more. If you have never read this piece with great care and attention, then you have not begun a serious study of foundational questions. Indeed, Speakable and Unspeakable in Quantum Mechanics contains the best discussions of foundational questions ever produced, and anyone unfamiliar with the contents of that book does not have a thorough knowledge of the field.

    There are other places one could begin. Travis Norsen's new book Foundations of Quantum Mechanics: An Exploration of the Physical Meaning of Quantum Mechanics" is an excellent introduction. Adam Becker's What Is Real? shows how the "standard" historical account of quantum theory that is passed down from generation to generation is completely inaccurate, a piece of Copenhagen propaganda. Jean Bricmont's Making Sense of Quantum Mechanics focuses on Bohmian mechanics but goes into more mathematical detail than usual.

    For almost a century, foundational questions about quantum mechanics have been systematically obscured or denied or repressed. It should not be a surprise that this could not go on forever, and that the progress of physics would require facing those problems directly. Overcoming this situation requires becoming aware that a standard physics education provides no preparation for this task. Even a Nobel Prize in physics is no guarantee of any understanding of foundational issues.

    Au Revoir

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  4. Tim,


    I think "foundational" questions are intellectually interesting, but the idea that they are going to lead to progress on concrete questions like black hole evaporation is completely unsubstantiated and without historical precedent. I actually wish it were different, but I believe it is fair to say that the philosophy community has had no appreciable impact on physics (i.e. on what physicists actually do), and the current exchange is hardly grounds for optimism.


    Obviously, the claim that I was the one who injected "foundations" into the discussion is false, as I repeatedly emphasized that I was asking for practical rules that could be handed to an experimentalist or programmed into a computer, wording which was specifically designed to avoiding getting sidetracked in this manner. Any reader can verify that.


    As I noted above, what has been most frustrating is your continual attempt to put the most ludicrous spin on any comments that challenge your claims. To anyone who thinks I am exaggerating here, please scroll up to the messages from around Nov. 15 where we engage in a debate over Gauss' law. Here you will see that Tim faces two options: 1) he is misreading something I wrote and/or misunderstanding terminology 2) I don't understand freshman calculus. As you can read, Tim goes all in on option 2, and delights in explaining my deficiencies in this regard, until he eventually realized his error. This process was essentially repeated over and over. Physphil was the latest lucky recipient of this treatment.


    All of these antics aside, the more serious issue is that this exchange shows just how deep the black hole information puzzle really is. It does not admit any "obvious" solutions of the type Tim is pushing. Or rather, what we managed to show (and Tim did eventually concede this) is that you need to introduce structure which conflicts with that of our best understood theory of quantum gravity, namely AdS/CFT. There is actually a lot that someone interested in foundations could do with this, and I hope someone does eventually approach this in a constructive spirit.

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  5. "All of these antics aside, the more serious issue is that this exchange shows just how deep the black hole information puzzle really is. It does not admit any "obvious" solutions of the type Tim is pushing. Or rather, what we managed to show (and Tim did eventually concede this) is that you need to introduce structure which conflicts with that of our best understood theory of quantum gravity, namely AdS/CFT"

    I conceded none of this. BHG is entitled to his own opinions but not to mine.

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  6. Tim,

    Go back and read your comments from the end of March, where you made clear that you are indeed jettisoning the standard rules of QM. Namely, you propose a restricted version of the physical Hilbert space (states have to arise from connected surfaces in the past), but to compute probabilities relevant to an external observer you need to compute inner products involving states outside this Hilbert space. That is not standard QM. Note that I am not saying that this is incorrect; rather I am saying that this scenario will not arise in AdS/CFT, because the latter obeys the usual rules.

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  7. The only rule that matters now is that the state after measurement is an eigenstate of the operator.

    In the Foundations literature, any collapse theory is regarded as changing "standard quantum mechanics".

    I think the two quotes above show one of the sources of confusion in this discussion. The first quote was someone’s explanation of “standard (non-relativistic) quantum mechanics” (ala von Neumann), and it refers to a measurement resulting in a change in the state of the system. Prior to the measurement the system is (usually) not in an eigenstate of the operator, but after the measurement it is in an eigenstate. This change in the state has sometimes been referred to as “collapse of the wavefunction”. In that sense, standard quantum mechanics (as formulated and expressed by Bohr, von Neumann, et al) can be regarded as a “collapse” interpretation, because it invokes measurement as a non-unitary process distinct from Schrodinger evolution. This is in contrast to the many-worlds “interpretation”, which purports to account for all phenomena without any collapses, entirely with unitary evolution of the wave function (but then has lots of other explaining to do).

    The second quote says that any “collapse theory” is not standard quantum mechanics, but I think the term “collapse theory” is distinct from “collapse interpretation”. (Shades of Gauss’s law versus Gauss’s theorem!) I think what the quoted person has in mind are things like stochastic collapse theories, or Penrose’s gravitationally triggered collapse, etc. These are not just interpretations of quantum mechanics, they are novel theories with, in principle, discernable differences with standard quantum mechanics. But this is different from von Neumann’s collapse of the wave function, which is fairly standard quantum mechanics – and which, of course, is subject to conceptual objections over the “measurement problem”.

    For purposes of this discussion, I think the novel collapse theories are not relevant (nor is the pilot wave theory, because it can’t be made relativistic), but the standard von Neumann collapse interpretation of standard quantum mechanics is relevant, because the whole basis of the “information paradox” is a perceived conflict between the purported unitary evolution of quantum theory versus the apparent non-unitary formation and evaporation of black holes. If we allow that quantum theory involves non-unitary collapses of the wave function with every measurement (as implied by the first quote above), then the very basis of the “paradox” is undermined, unless we arbitrarily assert that the only “measurements” are at early times (before formation) and late times (after evaporation), with purely unitary evolution in between. But that is a very strange assertion – albeit one that we make all the time when using standard quantum mechanics.

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  8. "I will no longer be responding to anonymous posters"

    https://en.wikipedia.org/wiki/Ad_hominem :

    "Ad hominem, short for argumentum ad hominem, is a fallacious argumentative strategy whereby genuine discussion of the topic at hand is avoided by instead attacking the character, motive, or other attribute of the person making the argument."

    I post under a nickname, which as far as I know is an unique identifier at every site where I have posted, and have done so consistently since my first Internet comment, circa 15 years ago. I see no legal or moral reason not to continue doing so. The only hidden reasons I can imagine why someone would need to know more would be for discriminatory or harassment purposes. (Any legitimate reason could be expressed and probably accommodated, such as when people need money and I ask for a way to send it.) Others who use their full or fuller names may feel more exposed, but that was their choice and does not oblige me to make the same choice.

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  9. BHG:

    Correct me if I'm wrong, but it seems extremely important for your argument that the physical Hilbert space be exactly a tensor product space, not just approximately a tensor product space, since you need the Hamiltonian to be degenerate and not just approximately degenerate. This is an important distinction because in any no-collapse interpretation of QM, the effective collapse that we observe is always approximate. So physphill's observation for instance that you need collapse to take place so that a repeated measurement gives the same results is not quite true. If that were the case, then we would be able to determine whether the pure state of the universal wave function exists in a tensor product space by performing everyday measurements. Clearly we cannot do this.

    So what is wrong with this account of Tim's scenario: not all possible states that you can think of to exist on two disconnected surfaces are actually physical states that are part of the Hilbert space. Any states that do involve disconnected surfaces are ones which formed from the evaporation of a black hole. Those states include states where there is an observer on the outside of the black hole, and when you separate the part of the state that corresponds to the observer from the part that corresponds to the rest of the world, you get a nearly tensor product state for the rest of the world after the observer becomes entangled with the Hawking radiation or whatever. Nothing about this scenario contradicts ordinary QM. And just as clearly, there is nothing about this scenario that requires a degenerate Hamiltonian because you can't just arbitrarily change things on the interior without changing things on the exterior; doing would in general cause the combination of the interior and exterior to not be a physical state which arose from a black hole.

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  10. Amos,

    Yes, I think you have it about right. If actual dynamical collapse is occurring all the time, then the BH info paradox is indeed moot. But there is no evidence for such models, and so the sensible thing to do is to ask whether unitary evolution is logically possible, and if so what it entails. What seems to be the case is that either gravity breaks down in an unexpected way, or else the rules of quantum mechanics have to be modified, and this modification affects the rules for computing probabilities independent of how a human observer actually goes about confirming these probability distributions. Tim's stance is that you don't have to give up anything: both QM and gravity can proceed as expected -- nothing to see here. It's clear that to make this work you need a huge degeneracy in the energy spectrum, since what's going on is that information is being stored in a disconnected region of space which carries no energy. I think it's logically possible that this is in fact the way nature works, and I wish Tim would be willing to defend this position. There could be an interesting discussion here. But he unfortunately seems to be allergic to the notion that a region of space can carry information but no energy, though I did my best to explain that this makes perfect mathematical sense. He's not willing to follow his position to its logical conclusion. That's how I see it, anyway.

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  11. Amos,

    I fully agree with your comment except the last paragraph, where you say the paradox is undermined by Bohr et al's interpretation and that it is "strange" to "assert" that measurements are only at early and late times. The paradox is not undermined at all, not if it is just the question of whether the formation+evaporation process is unitary on the outside or not. You would not need to assert anything if you had the ability to do experiments on evaporating black holes. Unitary versus non-unitary evolution could be tested just like we test quantum evolution in any other system. Always you have to ensure the system is isolated and not measured or otherwise interfered with in the middle.

    If you put detectors on the slits in a double slit experiment you do not see the interference pattern. That doesn't mean the particle would not have evolved unitarily or that you cannot test whether it did, it just means you interfered with it. Here if you want to test unitarity from early to late times in the black hole evolution you just have to make sure not to measure it until it is done evaporating, and then you can repeat that same late time measurement with the same initial state many times, and eventually you can conclude if it is unitary or not. As you write at the very end this is just standard QM.

    The problem for Tim is that he tried to pretend this was not true by throwing away most of the states in the Hilbert space, so he could pretend the Hamiltonian was not degenerate. But if you throw those states out the standard interpretation of QM cannot be applied because the state you should have after the measurement does not exist, and the whole thing becomes either nonsense or something very radically different than standard QM.

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  12. Travis,

    I am not sure I entirely understand your question, but here is my best shot at a response.

    In my view the term "disconnected surface" implies "tensor product Hilbert space". That is, in the quantum theory, the latter is what you really mean by the former. What would it mean for a surface to be disconnected but for the Hilbert space not to be a tensor product? If you something specific in mind here, please explain, since to me this is like talking about a sphere that is not round.

    Otherwise, I interpret your question to be "what if the surfaces were not quite disconnected, and hence the Hilbert is not quite a tensor product?". Yes, this could happen: there could a tiny bridge connecting the outer world to the would-be other disconnected region. But if you think about it, this is precisely the remnant scenario, and the bridge is the Planckian size horizon. The remnant is weakly coupled to the outside world, so the Hilbert space almost factorizes. This scenario has of course been heavily discussed over the years, and it's hard to definitively disprove it. On the other hand, it seems awfully contrived, and I don't know of anyone who currently advocates it. But it remains a logical possibility. It's not what Tim was pushing of course -- he wanted there to be no exotic physics at all, but this was what we ruled out.

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  13. What a mess. Here are some tips.

    There is no difference between a "collapse theory" and a "collapse interpretation" that is recognized in the foundations literature. If you want to make such a distinction you have to explain what that is supposed to be.

    The only quantum state that exists fundamentally is the universal quantum state and the only quantum state that has a possibility of evolving unitarily and preserving information is the universal quantum state, and the idea of verifying empirically that the universal quantum state does this by repeated measurements on multiple systems that have been prepared is absurd from multiple points of view.

    All of you pontificating about "the usual rules of quantum mechanics" have an intellectual obligation to explicitly write down what those rules are. Good luck.

    If you want a "measurement theory" that is principled physics, then you have to confront the measurement problem. If all you want in some reasonably clear but not precise way to squeeze predictions out of a mathematical formalism, then you are free to use whatever mathematics you like without concern for whether is it "physical".

    The ad hominem fallacy can only occur when one responds to an argument by attacking the person rather than the argument. Refusing to respond at all because the person chooses to hide behind a screen name is a judgment about what is even worthwhile to engage with, not an argument of any sort. To try to brand such a decision an ad hominem fallacy is only to demonstrate complete lack of comprehension of what an ad hominem fallacy is.

    Have fun.

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  14. BHG,

    As I understand it, the rules of the game are that we start with a background AdS spacetime, and then we put a quantized metric tensor on that spacetime that evolves in boundary time. The quantized metric tensor is constrained (in part) due to interaction with other quantized fields, so the full Hilbert space is the space of states of both the quantized metric tensor and the other quantized fields, with all constraints obeyed. The claim that at a particular boundary time the surface is disconnected is just the claim that there exist states in the Hilbert space in which the metric tensor describes a disconnected spatial surface. There's nothing in that statement which means that the Hilbert space has to be a tensor product space. If we had started with a disconnected background spacetime, then of course any quantized field that you place in that spacetime would have to exist in a tensor product space. But in our case, it's entirely conceivable and even plausible that you can't just arbitrarily operate on the other quantized fields without that operation causing the quantized metric tensor to no longer describe a disconnected surface. There are only certain combinations of quantized matter fields and quantized metric tensor that are allowed in the Hilbert space. An observer in the exterior would observe something slightly different if the state on the interior is changed (meaning, if we operate on the quantized matter fields in the region of space where the metric tensor is describing the interior, which is an operation we can't do without also changing the metric tensor) because that means that the black hole would have had to collapse in a slightly different manner (if it still would have collapsed at all after making that change).

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  15. …he unfortunately seems to be allergic to the notion that a region of space can carry information but no energy…

    That’s ironic, because in another discussion he was asserting that information can be transmitted without energy, and I was arguing that there can be no energyless conveyance of information.

    I fully agree with your comment except the last paragraph, where you say … that it is "strange" to "assert" that measurements are only at early and late times…. as you write at the very end this is just standard QM.

    I’d say that both statements are true: It is strange, and it is standard quantum mechanics. The "strangeness" I'm referring to is just the “measurement problem” of standard quantum mechanics.

    … you say the paradox is undermined by Bohr et al's interpretation… The paradox is not undermined at all, not if it is just the question of whether the formation+evaporation process is unitary on the outside or not.

    I agree, although the phrase “on the outside” sort of encapsulates the measurement problem. Here on earth, in the neighborhood of black holes at the center of our galaxy, we seem to be making measurements all the time, thereby collapsing the wave function and violating unitarity, like little cats, so it’s a bit, well, strange, to imagine our galaxy (or galaxy cluster, or entire universe) as an isolated quantum system evolving unitarily, with measurements made only before formation and after evaporation, since this may be at the beginning and end of the universe. I think the conservation of information under unitary evolution only works if we tally all the information in all the “branches” of the wave function that emerge from the unitary evolution with no collapse – even branches that are inaccessible to us little cats. This seems tolerable for microscopic and brief isolated experiments, but gets stranger in cosmological contexts. For large black holes, I’m thinking that the information paradox is meaningful only to the extent that we take seriously the many-worlds interpretation (as Hawking did)… which makes the empirical significance questionable.

    We might consider so-called “objective collapse” theories, designed to solve the measurement problem by positing that collapse is triggered by some mechanism we can’t control… for example, the idea ala Penrose that gravity (from which no system is completely isolated) somehow triggers an objective (and non-unitary) collapse. I think these ideas tend to founder on Lorentz invariance, but if they worked, they would undermine the information paradox. Of course, this would be a radical departure from standard quantum mechanics (not to mention special relativity). One participant here thinks the pilot wave theory solves the measurement problem, but I don’t.

    You would not need to assert anything if you had the ability to do experiments on evaporating black holes. Unitary versus non-unitary evolution could be tested just like we test quantum evolution in any other system. Always you have to ensure the system is isolated and not measured or otherwise interfered with in the middle.

    Yes, I agree completely with that. I’m not 100% confident of what the results would be, if we actually had some tiny black holes to test, and I’m also not sure the results would scale up to stellar or super-massive black holes, for which the evaporation times are so absurdly long that I wonder if it even falls in the realm of empirical science.

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  16. Wikipedia (see above for full quote) "... genuine discussion of the topic at hand is avoided by instead attacking the character, motive, or other attribute of the person making the argument ..."

    TM: "Refusing to respond at all because the person chooses to hide behind a screen name ... [is not an ad hominem]."

    Refusing to respond = avoiding discussion; because the person chooses to hide behind a screen name = attack. I think the definition in wikipedia fits. I suppose that everyone who does this would claim their motives were pure, but motives do not seem to be part of the definition.

    I have never seen TM complain about the pseudonyms of those who agree with him. Hence I feel it is a legitimate inference that it is in fact part of his argumentative technique.

    (All that added nothing to this thread since it merely rehashes previous material or makes obvious inferences. By way of apology I will donate to the site.)

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  17. There is no difference between a "collapse theory" and a "collapse interpretation" that is recognized in the foundations literature. If you want to make such a distinction you have to explain what that is supposed to be.

    I think I did, but let me be a little more explicit. Throughout the literature there are discussions of the “collapse of the wave function” that results from measurements in the standard Bohr/von Neumann formulation and interpretation of quantum mechanics (raising the measurement problem). Despite this, you declared that “any collapse theory is regarded as changing standard quantum mechanics". I was trying to help by explaining what I assumed you must have meant: There exist what are called “objective collapse theories", two examples being the Ghirardi–Rimini–Weber theory (which you have mentioned repeatedly) and the Penrose gravitational triggering theory, motivated in part by trying to avoid the measurement problem. (An ontological wave function tends to be formulable only in a non-relativistic context, so these theories don't interest me, but...) These “objective collapse” theories generally have (subtle) empirical differences from standard quantum mechanics that could (at least in principle) be tested, so they aren’t interpretations of standard quantum mechanics, they are different theories. Hence I assumed you must be referring to such objective collapse theories when you said “any collapse theory is regarded as changing standard quantum mechanics". The point is that “standard quantum mechanics” ala von Neumann already entails non-unitary “collapse of the wave function” with each measurement, so your statement can only apply to objective collapse theories. It obviously wouldn’t make sense to say that standard quantum mechanics (which entails collapses of the wave function) “is regarded as changing standard quantum mechanics”.

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  18. Travis,

    Ok, thanks for the clarification. It's important to be clear about what the goals are here. We take for granted basic properties implied by the CFT dual: unitary evolution of the full state and non-degeneracy of the Hamiltonian. We then ask whether this is compatible with info loss for an external observer in the bulk. The outcome of the discussion with Tim is that this is not possible if you follow all the usual rules in the bulk and if the standard Penrose diagram is accurate. In particular, the tensor product nature of the Hilbert space follows from the standard form the constraint equations evaluated on disconnected surfaces.

    Since you "can't have it all" in the above sense, the next thing one can ask is whether it's possible to modify things in the bulk, "just a bit", and arrive at a picture where the bulk observer sees standard physics to high precision. This is what you are asking. It's hard to make any blanket statements here since there are infinitely many ways of modifying the physical principles. I will just make a couple of general comments. First, to avoid the tensor product conclusion one apparently needs to modify the form of the gravitational constraints; roughly, one needs to add terms that couple together disconnected components. The burden here is to show that this is mathematically consistent. The constraints are associated with gauge (coordinate) transformations, and such objects are tightly constrained (their commutators must close into a well defined algebra, etc). Next, one should keep in mind that the Hilbert space of a theory is a property of the theory as a whole, not dependent on some particular solution. So when we ask questions about the Hilbert space, statements about whether certain states arose from BH evaporation are not relevant. Finally, apart from the technical issues, there is a basic physical fact that you will have to confront. Suppose the information does not come out with the Hawking radiation. We do know that the energy does come out -- if you count up all the energy in the emitted quanta you will get the mass of the original black hole. So you need to come up with a story about how all the energy can be accounted for, but not all the information about the state. The obvious way to accomplish this is for the energy spectrum to be hugely degenerate, but this is what we are trying to avoid. So even at the handwaving level, how do you plan to avoid this? Note that the remnant scenario also goes in this direction, since a remnant is an object of Planckian mass but that has arbitrarily many internal states.

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  19. BHG,
    Your reply to Travis just above prompts me to jump in again. In particular, I want to comment on two things, one minor and one that is central to the whole dispute.

    First, when you say that all the energy has gotten out of the interior but not the information, you seem to be assuming that the Hawking radiation quanta are unentangled with the interior states. But if I recall correctly, Tim does not assume this in his scenario. Rather, the radiation quanta are presumably massively entangled with the interior states, even though no standard measurements in the bulk will reveal this - the radiation looks "thermal". So, in whatever sense it makes sense to speak of information being located here vs. there, it is not the case that all the information stays trapped in the interior; it's spread over the whole Cauchy surface, i.e., bulk + interior pieces, just as one would expect.

    Second, you keep falling back on claims about "the" Hilbert space of the AdS side of the duality which are based exclusively on the fact that the late-time Cauchy surfaces are disconnected and on how one would naively describe the physically possible states on such a two-piece spacetime setup if one were starting from scratch. Tim (and perhaps Travis) object that the facts about how one would set up that kind of quantum gravity physics are not obviously relevant to the scenario in hand: we are only interested in states of the late-time bulk+interior which are such that, evolved backwards, they yield the kind of early-time states that we're assuming (black hole forms in regular AdS space then gradually evaporates). Is the set of all *these* physically possible states massively degenerate in the Hamiltonian spectrum? Maybe so, but (Tim's claim has been) you have not proven this.

    The debate seems to degenerate into one about the burden of proof. You think that the burden should be on Tim to characterize the Hilbert space (if it is one) of states to which he wants to restrict the physics, and show that the spectrum of the Hamiltonian is not degenerate in the sense that contradicts the CFT duality. Tim would say (I guess) that the burden is on you to show either that the degeneracy is inevitable, or (more directly) that the disconnected space scenario simply can't emerge if AdS/CFT is correct. But he's not accepting the mere observation, that what one would naively consider to be the space of physically possible states over such disconnected space pieces involves an overall degenerate Hamiltonian spectrum, to be such an argument. Stalemate, unless you can bring some new argument to bear that shows Tim's idea to be incoherent.

    (CONT)

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  20. What *really* does not work, as a way of breaking the stalemate, is to start talking about measurements in the bulk collapsing states, and how those states need to be "in" your Hilbert space if you're to do the physics and make predictions. Physphill made this argument most bluntly, but you offered forms of it at times as well, and it's a non-starter. If you postulate that late-time measurements truly collapse the state of the universal wave function, then you believe that quantum theory involves non-unitary evolution as well as unitary evolution, and you have blown up the key premise needed to get the information paradox going in the first place. (Pretending that the non-unitary evolution only happens at some super late time, and not in the phase of history where the black hole is forming and then evaporating, strikes me as just silly. I will assume that you do not mean to endorse this suggestion, which came up in some recent post.)

    If you don't mean this talk of "collapse" to be taken literally, but rather as what physicists can pretend happens FAPP and make good predictions, then Tim answered the objection perfectly well, I think. Use the standard "naive" Hilbert space to make your predictions all you want; there's no rule against using a mathematical framework for FAPP predictions, just because one does not take all elements of that framework to represent genuinely physically possible states. (Physicists do this all the time, in fact.) If you want to make predictions for repeated measurements, go ahead and use the "collapsed" product states to predict your subsequent results too - you won't have a problem, even though (of course!) what you're interacting with is still a state entangled with the rest of the universe, including the black hole interior, not an actually-collapsed state.

    It would not surprise me if your conclusion, that Tim's scenario is incompatible with the AdS/CFT conjecture, is correct. I hope you will try to find a different way to show that it should be ruled out. But I just don't see that all the appeals to what "The" late-time Hilbert space has to be, if we apply the "standard rules" of QM straightforwardly (or "naively"), do the job.

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  21. Carl3 said "... there's no rule against using a mathematical framework for FAPP predictions, just because one does not take all elements of that framework to represent genuinely physically possible states."

    When doing ordinary probability calculations, as I alluded to previously, you can't use a probability distribution which includes non-physical (non-obtainable) states (that is, use a Gaussian distribution to model a random variable that is restricted to the range [0,1]) to make accurate predictions. Apparently you can in making QM predictions?

    Other than that difference, which is probably lack of knowledge on my part, Carl3's exposition is similar to how I summarized things previously. BHG has not proved that a remnant solution will not work in ADS/CFT, and TM has not proved that it will.

    For an alternate argument, Dr. Hossenfelder proposed one earlier, having to do with entropy microstates.

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  22. Since both Travis and Carl3 are right on target here I will let them take up the argument, but there is one simple point that cuts through all of the rhetoric:

    What has been called AdS throughout this entire discussion is not a theory, and treating it as if it were a theory underlies the basic conceptual mistake being made.


    AdS comprises two completely separate things: A dynamical theory (some form of quantum gravity) and an additional restriction to a subset of the models of that theory, namely those that satisfy the condition of "having boundary conditions that are asymptotically AdS". In the usual way of talking, the restriction of a theory to a subset of solutions by imposition of additional conditions is not itself a theory. There are a set of models of standard GR whose boundary conditions are asymptotically AdS, but that set is not some new "theory" distinct from GR, and GR still has models that don't meet that condition. The primordial error in this debate has been acting as if "AdS" is a theory rather than a restriction of a theory. In fact, the exact mathematical nature of the restriction has never even been precisely stated. What counts as a model that belongs in the subclass of models which go in the bucket labeled "AdS"?

    Proposal 1) In order to count as belonging to the "AdS" subclass of models, every maximal spacelike hypersurface in the model must *HAVE* boundaries that are asymptotically AdS. A "maximal spacelike hypersurface" is a spacelike hypersurface that does not lie in the domain of dependence of any spacelike hypersurface that does not lie in its own domain of dependence. (In a globally hyperbolic spacetime, the maximal spacelike hypersurfaces will be Cauchy, but even if the spacetime is not globally hyperbolic there will be maximal spacelike hypersurfaces.)

    Proposal 2) In order to count as belonging to the "AdS" subclass of models, every maximal spacelike hypersurface in the model must *ONLY HAVE* boundaries that are asymptotically AdS.

    Under proposal 1, the claim that the spectrum of the Hamiltonian "in AdS" cannot be massively degenerate is plainly false. Simply take as models pairs of disconnected space-times, one of which is asymptotically AdS and the other of which isn't. Let the spectrum on the non-AdS part be massively degenerate. Done.

    Under proposal 2, the solution I have been defending does not count as an instance of AdS. Take a maximal spacelike hypersurface that is disconnected (and in every evaporating black hole spacetime there will be such hypersurfaces): the external disconnected piece is asymptotically AdS but the interior piece is not, so according to this proposal the model is not part of the "AdS" subclass of models of the gravitational theory.

    So either one defines what counts as "being AdS" is a way that plainly violates the claim that the Hamiltonian in AdS is non-degenerate, or one defines it so that the spacetimes that model blackhole evaporation in an initially AdS spacetime don't count as being part of "AdS". Either way, the claim that the solution I have been defending is touched by the criticism collapses.

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  23. BHG,

    You say:

    "Next, one should keep in mind that the Hilbert space of a theory is a property of the theory as a whole, not dependent on some particular solution."

    This is of course true. But we're clearly talking here about a superselection sector of the full Hilbert space of all states that satisfy the constraints, since it is trivially true that completely disconnected spacetimes satisfy the constraints. So how do we know, once we've restricted ourselves to the superselection sector in which the solutions are asymptotically AdS, that we don't have some disconnected surfaces in the sector which don't form a tensor product space? This is why it seems like you need to actually look at the particular solution to see if it should be included in the superselection sector. What am I misunderstanding here?

    As far as confronting the basic physical fact that you need to account for all of the energy: it seems like you are using the term "energy" in multiple ways. The energy that is measured by the boundary Hamiltonian can't just be the energy that is associated with the mass of the black hole, since if that were the case then nondegeneracy of the spectrum would imply that all spacetimes which contain the same amount of mass are exactly the same, which is clearly absurd.

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  24. Carl3,

    About info getting out, when people say that the "info has not gotten out" they mean that the state is entangled between interior and exterior, so is not fully accessible to the external observer. No disagreement there, I believe.

    You say we are only interested in the specific states arising from certain black hole evaporation events. Not me: I am interested in the structure of the theory as a whole, because the entire point of thinking about the BH info paradox is to figure out what it tells us about the laws and structure of quantum gravity. So I would like to know whether the full spectrum of the Hamiltonian is degenerate; after all that is what we would compare with the spectrum of the CFT Hamiltonian. Aside from that, the idea that you can just toss away states you don't like is highly dubious. Generically, if I study dynamical evolution starting with some class of states and then declare that the final states that arise are the only ones that are allowed in theory, everything will fall apart. I gave an example of this earlier: if you scatter electrons and protons toward each other from infinity you will never form a hydrogen atom, but the hydrogen atom states must be in the Hilbert space for consistency.

    We know that the CFT obeys the ordinary laws of QM, hence so too should the theory in the bulk (since they are the same theory in different variables). The idea that to compute probabilities of certain outcomes you are *required* to evaluate inner products" of states that are not in the Hilbert space certainly doesn't sound like standard QM to me, and certainly is not something that physicists do all the time, as you suggest.

    Finally, I've said this before, but getting into any philosophical issues about measurement and collapse is a complete red herring in my view, since there is no indication that BHs bring anything new to the table. We should try to use the usual cookbook rules which we know "work", and if it turns out that they somehow fail for BHs, well, that would be interesting indeed.

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  25. Carl3,

    "First, when you say that all the energy has gotten out of the interior but not the information, you seem to be assuming that the Hawking radiation quanta are unentangled with the interior states. But if I recall correctly, Tim does not assume this in his scenario. Rather, the radiation quanta are presumably massively entangled with the interior states, even though no standard measurements in the bulk will reveal this - the radiation looks "thermal"."

    This is a basic misunderstanding. You are backwards. It is because of the entanglement that information is lost (to the exterior). If the exterior quanta were in an unentangled state, they could encode all the information about the initial state and there would be no paradox.

    "...we are only interested in states of the late-time bulk+interior which are such that, evolved backwards, they yield the kind of early-time states that we're assuming (black hole forms in regular AdS space then gradually evaporates)."

    The problem is, you cannot only consider such states and still follow the basic rules of QM. You cannot calculate probabilities for the results of (external) measurements, and the states after those measurements do not make sense. Bhg made that point many times and Tim never responded, like with a formula that says how to calculate those things.

    "If you postulate that late-time measurements truly collapse the state of the universal wave function, then you believe that quantum theory involves non-unitary evolution as well as unitary evolution, and you have blown up the key premise needed to get the information paradox going in the first place. (Pretending that the non-unitary evolution only happens at some super late time, and not in the phase of history where the black hole is forming and then evaporating, strikes me as just silly. I will assume that you do not mean to endorse this suggestion, which came up in some recent post.)"

    Every physicist that deals with QM has been using those rules for the last 100 years. They have calculated the results of experiments using those rules, they have done experiments and the prediction agreed. 100 years of results support the idea that QM evolution is unitary when the system under study is isolated, and that measurements are well modeled by collapse. Now all of a sudden we cannot use them? Strange.

    You refer to "the phase of history where the black hole is forming and evaporating", but for the third time, small black holes can form and evaporate in very short times. Those are the only black holes anyone will ever have any chance of studying if they ever can address this experimentally. Any large black hole will be long lived and interact with lots of things and unitarity will be impossible to test. But small black holes could be studied (given high technology to make them) and this question could be settled in the lab, no different from any other QM experiment.

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  26. JimV wrote: “When doing ordinary probability calculations, as I alluded to previously, you can't use a probability distribution which includes non-physical (non-obtainable) states (that is, use a Gaussian distribution to model a random variable that is restricted to the range [0,1]) to make accurate predictions. Apparently you can in making QM predictions?” My answer is: Yes you can, and yes you can. If you start with a Gaussan distribution over both physically possible and non-possible states, you can use that to calculate any physical outcome you like; you just have to conditionalize on the outcome being found in [0,1] using the standard formula for conditional probability. A simpler example: suppose I tell you that in rolling a pair of dice, even numbered outcomes are physically impossible. You wonder how I intend to calculate the probability of obtaining a result of 7, if I have ruled out half of the states in my (standard dice-throw) probability distribution as unphysical. Simple: it is the standard probability of rolling a die and getting an odd result AND a 7, divided by the standard probability of getting an odd result. (Since 7 implies odd, the probability is just Pr(7)/Pr(odd).)

    Similarly, calculating where A is an observable that acts as identity in the hole interior, using the standard QM prescription, is essentially just computing the probability of getting the various results a_i outside, given that the state in the interior piece is whatever it actually is acc. to Psi. I am implicitly conditionalizing on the non-occurrence of any of the myriad physically impossible states that would (if considered physical) imply degeneracy of the energy spectrum of the full Hamiltonian.

    I don’t see how any of this leads to trouble, though I admit I may be missing something. In any case, your Gaussian analogy strikes me as handled by simple conditionalization.

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  27. Good, no disagreement about information.

    You say, “I am interested in the structure of the theory as a whole, because the entire point of thinking about the BH info paradox is to figure out what it tells us about the laws and structure of quantum gravity. So I would like to know whether the full spectrum of the Hamiltonian is degenerate; after all that is what we would compare with the spectrum of the CFT Hamiltonian.” Good, let’s start here. In particular, we already know that the spectrum of the CFT Hamiltonian is non-degenerate (up to the usual symmetries), so the spectrum of our AdS Hamiltonian will similarly be non-degenerate, assuming AdS/CFT. And we know (I will take your word on this) that the quantum gravity theories studied under AdS/CFT are able to model black hole formation. What we don’t know yet is what they say about evaporation.
    Now, along comes a person suggesting that perhaps what happens in BH evaporation is just what one sees in the Penrose diagram, with Cauchy surfaces becoming disconnected for late times. What we then would like to know is: can this situation arise if our AdS/CFT theory is correct? In order to answer this question, we need to know whether our theory permits states to arise that involve two disconnected bits of 3-space metric, *starting from ordinary physical states in an ordinary (no spatially disconnected Cauchy surfaces) asymptotically AdS space.* This is an interesting question I suppose, and I have no idea how one might attempt to answer it. But my point is that one cannot answer *this * question by simply looking at the answer to an entirely different question, namely: “What’s the natural Hilbert space of states to postulate for a QG theory set in a world with two disconnected spatial hypersurfaces?” The answer to this question may indeed be: “the full tensor product space of states on each piece – hence, a Hilbert space with massive degeneracy in the spectrum of the full Hamiltonian. And hence, this QG theory will not be our AdS/CFT theory.” But this fact about what is “natural” to postulate if one decides to start by assuming a disconnected space does not, it seems to me, prove anything one way or another about whether in our AdS/CFT theory such disconnected surfaces can *arise* in late times, post-BH-evaporation.

    CONT.

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  28. CONT. [The first part of this was a reply to BHG, forgot to label it as such at the top!]

    In connection with the distinction I’m trying to draw here, let me echo a point Travis made a few days ago (April 6th), which I don’t believe you addressed. “The claim that at a particular boundary time the surface is disconnected is just the claim that there exist states in the Hilbert space in which the metric tensor describes a disconnected spatial surface. There's nothing in that statement which means that the Hilbert space has to be a tensor product space.” You replied to Travis on April 7th and basically re-asserted your claim about what is not possible under AdS/CFT in the first paragraph, without addressing the point Travis was (I think) trying to make, and then went on to act as though he was proposing some new physics in the bulk. But that was not the thrust of his post, which continued: “If we had started with a disconnected background spacetime, then of course any quantized field that you place in that spacetime would have to exist in a tensor product space. But in our case, it's entirely conceivable and even plausible that you can't just arbitrarily operate on the other quantized fields without that operation causing the quantized metric tensor to no longer describe a disconnected surface. There are only certain combinations of quantized matter fields and quantized metric tensor that are allowed in the Hilbert space.” Travis is speculating not about some new physical theory, but rather about whether the Hilbert space for the AdS part of our AdS/CFT theory might not (a) allow states in which, at some times, the metric tensor describes a disconnected spatial surface, but (b) not allow degeneracies in the spectrum of the full Hamiltonian. Telling us how one would set up a QG theory in a spacetime that one takes, ab initio, to be disconnected does not answer the question we are raising about our AdS/CFT theory that starts ab initio with ordinary physical states on a connected, asymptotically AdS spacetime.

    Regarding how we can use the standard QM rules for generating predictions, see my response to JimV.

    Finally, I agree that the measurement problem should be kept off the table in this discussion. Whether one likes to admit it or not, the context of the information loss paradox is a context of assuming universal unitary evolution, which means Everett/Many-Worlds. This means that, when we make measurements, we do not collapse the universal state; rather, the universal state simply evolves in such a way that in the various decoherent branches that emerge, observers “see” the various possible measurement outcomes. That these observers in different branches can pretend that the quantum state “collapsed” due to their measurements FAPP, shows us nothing about the actual universal wave function or its Hilbert space; it only shows that this whole “evolution is always unitary” story is empirically possible in the first place.
    The cookbook rules of QM you refer to work for one of two reasons. Either wavefunctions DO collapse on measurement, really, in which case we can forget about the information loss paradox. (And then I would ask you to tell me when, and why, these collapses occur.) Or (and this is what I presume you believe) wavefunctions DON’T collapse, ever, but because of decoherence we can get along very well pretending that they do in many experimental contexts.

    [About micro black holes, see my reply to Physphill.]

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  29. Physphill:

    About entanglement and information loss, you are right that the full quantum state at earlier times cannot be reconstructed just from observations (ideal complete ones) made in the bulk in late times. As I tried to say, there is information in the interior *and * in the exterior. But you are wrong to say “If the exterior quanta were in an unentangled state, they could encode all the information about the initial state and there would be no paradox.” At least, this is not true if the exterior quanta are in an unentangled state because one of your collapse-inducing measurements has been performed on them; in that case the information would be lost forever.

    Regarding other points you make, please see my replies to JimV and BHG.

    About the standard rules of QM that have been used for the last (nearly) 100 years, which BHG rightly called the “cookbook” rules, of course they work well enough, and if that makes you want to believe that collapses really DO occur in some circumstances, it’s fine by me! I would ask you to tell me if you know when and why this occurs, i.e. what physical situations provoke collapse; but if you don’t know, that’s fine too. Just be aware that many physicists today are convinced that real collapses DON’T occur, ever, and it just looks to us as though they do because of decoherence.

    Finally about micro black holes, it’s an interesting argument you offer here, because if your claims about them were right, we could get the information loss paradox going even if there are collapses and the universal wave function does not always evolve unitarily. The paradox would be kind of a micro-paradox: we want to say that our micro BH evaporates by unitary evolution, but the state of the Hawking quanta post-evaporation does not appear to have the information necessary to encode the initial states pre-evaporation. What gives?
    I call this a micro-paradox because it seems to me that this puzzle would be vastly less interesting than the question I put to you above: exactly when do wave function collapses occur, and why? But I grant you that it would still be a puzzle.

    But I have a further response to your micro black holes objection, which is this: we have no very good reason to believe that such things are in fact physically possible. They have not been demonstrated to be a consequence of “our” AdS/CFT theory of QG, as far as I know. There are just some calculations designed to show that if one treats quantum particles as basically classical in some regime, and also assumes that GR applies without modification at these energy regimes and distance scales, then a tiny black hole should arise. (And immediately evaporate, if Hawking’s calculations or other evaporation calculations also apply.) Interesting, but far from convincing proof. So I would say that the information loss paradox is best understood as a puzzle in the context of ordinary astrophysical-scale black holes.

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  30. Carl3, thanks for the response on conditional probabilities. I agree they can handle certain cases, although I am not sure it makes sense to conclude that a random variable which has the range [0,1] is a subset of a Gaussian distribution - it seems likely to me that it would not be Gaussian, but if it happens to be, okay. For another example, for times to failure, which can not be negative, we tend to use the Poisson distribution, rather than conditioning on a Gaussian distribution.

    I like the use of conditional probability to weed out unphysical states in a QM model of evaporated BH thermal radiation, but that seems, perhaps naively on my part, to rule out the use of the standard QM prediction calculation, unless you are saying that will automatically happen with the standard calculation, whereas I would have thought you would have to first calculate all possible physical states resulting from BH evaporation (which we don't know how to do, absent a theory of quantum gravity) in order to condition on them.

    (My education is not the purpose of this thread, so there is no need to respond.)

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  31. Travis,

    "This is why it seems like you need to actually look at the particular solution to see if it should be included in the superselection sector. What am I misunderstanding here?"


    I am sorry, but I just don't follow what you are saying. Let me say this, inthe hope that it addresses your point. In every other QM theory I am aware of, to decide whether or nor some candidate state is in the Hilbert space or not you don't have to evolve forward or backward in time, you just examine it at the given time in question. You ask that it obey any constraint equations and has finite norm, finite energy, etc. . As far as I can tell you seem to be defending a novel restriction on allowed states that has no counterpart outside of black hole evaporation, and appears to be invented purely to avoid an uncomfortable conclusion. On top of that, it seems internally inconsistent to me, for the reasons that have already been given.


    "The energy that is measured by the boundary Hamiltonian can't just be the energy that is associated with the mass of the black hole, since if that were the case then nondegeneracy of the spectrum would imply that all spacetimes which contain the same amount of mass are exactly the same, which is clearly absurd."


    You may wish to rethink your statement. The boundary Hamiltonian measures the total mass of the system, and it is indeed the case that that as far as one can do any explicit computations its spectrum is discrete and non-degenerate (up to symmetries). I suspect you are confusing classical and quantum notions of energy, but I'm not sure. I will note that this seems related to one of the elementary errors that Tim repeatedly made (makes): he thinks that I am making some fantastical claim by saying that if you take a bunch of particles in AdS with some mutual interactions that the spectrum will be discrete and non-degenerate (up to sym). Of course, he would realize his error if he tried to do the actual computation.

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  32. Carl3,

    the problem I refer to is just as bad if you want to use many worlds/decoherence as a model for measurements. The problem is that a unitary transformation applied to the exterior (which would be the effect of an external measurement) does not preserve the subset of states that Tim wants to keep. It is the same problem as for a projective measurement. That also takes you into the "forbidden" part of the Hilbert space, which just goes to show that the whole idea cannot work. I think this is best treated in the standard, projective interpretation, where the problem is most obvious. My point all along (which I think is really bhg point) is that this is a very radical modification of QM.

    About micro black holes, I do not understand you. That has nothing to do with AdS/CFT. Small black holes evaporating quickly are a prediction of the same theory as large black holes evaporating slowly, in AdS or in flat spacetime. All the calculations apply equally well in both cases (unless you make the black holes so small they are near the Planck size, but there is no reason at all to do that). My black holes could evaporate in a millisecond, that is 40 orders of magnitude separated from the Planck time and the calculation is just as controlled as for a large black hole.

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  33. BHG,

    Ok, let me ask you: how are you determining which states are allowed in the Hilbert space? It is clearly not the case that all states which satisfy the constraints, and have finite norm, and have finite energy are allowed, since a disconnected spatial surface satisfies these conditions. The Hilbert space of all possible spatial surfaces and matter distributions is being restricted somehow in order to accommodate AdS, and the idea is that that restriction might still contain some disconnected surfaces which don't form a tensor product space because not all of the states that would be necessary to form a tensor product space survive the restriction.

    And ok you're right about the energy; I take back what I said before. I'll have to think some more on that.

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  34. Travis,

    The situation with the spectrum of the Hamiltonian is actually much more counter-intuitive even than it sounds. Question: are there *any* eigenstates of the Hamiltonian in the space of "physical states" apart from the vacuum? Since an eigenstate of the Hamiltonian would have to be a stationary state, and since vacuum AdS appears to be unstable against perturbations, the answer would seem to be "no". So apart from the vacuum, every physical state must be a superposition of energy eigenstates. But the energy eigenstates themselves are not in the physical Hilbert space. That is, for every physical state but the vacuum, there is only an expectation value for the energy, or the ADM mass. And there is nothing preventing many different states having the same expectation value. This is the thing that gets confusing: in classical GR, every state has an ADM mass and many have exactly the same ADM mass. But you cannot equate "having a classical ADM mass M*" with "being an eigenstate of the Hamiltonian with eigenvalue M*". In fact, as I have just argued, it seems that the spectrum of the Hamiltonian in AdS is not just discrete and (nearly) non-degenerate, but with respect to the "physical states" there is only one value in the spectrum! There may formally exist mathematical solutions with other eigenvalues, but how can they be physical?

    Of course, that poses the question whether there are any eigenstates of the Hamiltonian in the CFT other than the vacuum state. It is hard to see why there wouldn't be, but I don't know the answer.

    Basically, just as the "static solution" to the field equations that Einstein sought by adding the cosmological constant turned out to be an unstable equilibrium, as far as I can tell AdS is an unstable static state. That raises in an even more stark way the question of what sort of "theory" AdS is supposed to be. As I have already noted, it is not even clear what it means to be an "AdS state" in this context.

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  35. Travis,

    We start from the assumption that all surfaces are allowed, provided there is precisely one noncompact surface, which asymptotes to AdS. The question is then whether there are superselection sectors or not. If we start with a wavefunction with support on only connected surfaces, can it evolve in time to the disconnected sector? If you trust the standard Penrose diagram then you are led to believe that such connected -> disconnected time evolution can occur. But note that this conclusion is not on solid ground, because to change the topology of the surfaces you necessarily need to pass through a singularity, and we don't know the rules of quantum gravity for this. On the other hand, if we start from the CFT and work backwards we know that disconnected surfaces are ruled out for the reasons I have given. So the CFT predicts that there *are* such superselection sectors and if you start in the connected sector you stay in the connected sector. For this to happen the standard Penrose diagram must be wrong, and this is what the fuzzball/firewall scenarios are proposing. I.e semiclassical QG breaks down in a dramatic fashion inside the horizon.

    Two important points which sometimes get lost. First, there may not be a unique solution to this problem. I.e. there may not be a unique "theory of quantum gravity in AdS". My point has always been that AdS/CFT does supply us with one (and so far the only) such theory, and I am arguing about what happens in that theory. Second, the claim that the Penrose diagram breaks down is of course a very strong claim, and I want to emphasize that for this reason I regard the whole situation as paradoxical. My stance is opposite to that of Tim: he thinks that the standard Penrose diagram scenario is not in tension with AdS/CFT (although he admits he knows essentially nothing about the latter...) so there is no paradox, while I think there is no scenario on the market that is fully satisfactory.

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  36. BHG:

    "We start from the assumption that all surfaces are allowed, provided there is precisely one noncompact surface, which asymptotes to AdS. "

    An answer! So you start from an assumption that is in direct conflict with my solution and the derive—via a rather tortured and tortuous chain of reasoning—that your theory is in conflict with my solution. Sigma_In is not compact.

    Why did you waste so much of our time?

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  37. BHG:

    Something doesn't seem right about that explanation. The only thing that AdS/CFT tells you is that whatever states are part of the superselection sector which contains a wavefunction with support on only connected surfaces, the Hamiltonian must be nondegenerate with respect to those states. If you include disconnected surfaces such that every interior corresponds to an exterior with different energy, then you maintain the nondegeneracy. What's the problem?

    And Tim does bring up a good point. The disconnected surfaces don't need to be energy eigenstates, and in fact since evolving them backwards in time does by hypothesis change the state, they actually can't be energy eigenstates under Tim's model. So all that is needed is that some superpositions of the energy eigenstates of the nondegenerate Hamiltonian result in disconnected surfaces. And obviously there is no restriction that the superpositions all have to have different expectation values of the energy, so it's not even a necessary restriction that each interior must correspond to an exterior with a different energy.

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  38. Tim,

    Please justify your claim: "Sigma_In is not compact. "

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  39. Travis,

    "If you include disconnected surfaces such that every interior corresponds to an exterior with different energy, then you maintain the nondegeneracy. What's the problem?"


    The problems are legion. I already explained some of them and here's one more for you to chew on. So, you are proposing that the CFT Hilbert space is dual to states corresponding to a single connected component together with certain states corresponding to two components. Why stop there? After the black hole evaporates, we can collect the radiation and collapse it to form another black hole that evaporates, hence causing the surface to split once again. So now you will have to say that the CFT Hilbert space also includes states corresponding to three components. We can continue this as many times as you like, and you will conclude that the CFT Hilbert space includes states with an unbounded number of components. Further, we can do this while restricting the total energy to be below some specified value. These states must all be present in the theory at any given time, since the Hilbert space is a time independent property of the theory. Hence we conclude that there are an infinite number of states below a finite energy. This contradicts the CFT.

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  40. Section 3.2 of this https://arxiv.org/pdf/1609.00026.pdf may be worth a quick look.
    Lectures on Gravity and Entanglement
    Mark Van Raamsdonk
    https://arxiv.org/pdf/1609.00026.pdf
    -Arun

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  41. BHG,

    It's not exactly clear that you can accomplish collapsing a black hole an infinite number of times with a finite amount of energy; you need some energy in order to collect the radiation back together into a black hole. In any case, as Tim pointed out before, no black hole state exists as an energy eigenstate, so the total energy can only refer to the expectation value of the energy, and there's no restriction on how many states there can be with a given expectation value.

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  42. Travis,

    I don't see either of those objections as carrying any force. Any energy you "use" to create the black hole stays in AdS since it is a closed system, so can be used again. Actually, you don't have to use any energy at all: due to thermal fluctuations the radiation will clump up from time to time and collapse to a black hole; this is well known. And of course I was not talking about energy eigenstates, I was saying that we form wavepackets built out of states with energy below some (arbitrarily high) cutoff. One can certainly create black holes under those conditions.

    To help you see how absurd the proposal is, let me ask you what you are claiming the Hilbert space to be? Surely it can't just consist of at most 2 disconnected components, since that would not allow more than one black hole to ever form. But 3 seems just as arbitrary, and then where do you stop? I believe if you ponder this you will see that this scenario cannot get off the ground.

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  43. BHG—

    Justify it? That is what you get by taking the Penrose diagram seriously, and is completely clear in my paper. If you put in the Evaporation Event, it can belong to exactly one Cauchy surface, and in order to be Cauchy all of the other In parts have to be surfaces that are open at that boundary.

    Pp. 10, 11 of the manuscript:

    "Consider a spacelike line that goes from a point on r = 0 to the Evaporation Event. Let it be open at the Evaporation Event. In effect, the Evaporation Event functions for all the slices inside the event horizon in exactly the same way as Spacelike Infinity functions for the spacelike slices outside the horizon, with the difference that the Evaporation Event represents a real, physical event in the space-time."

    Could not be clearer.

    If you don't like having the EE as a physical event (Wald doesn't) then it is all the more obvious that the In parts are open.

    But let's let the In part be closed, as apparently you want. The content of the In part evidently contributes nothing to the ADM mass of the Out part. Since you have been equating the energy with the ADM mass, and that with the spectrum of the Hamiltonian, then the spectrum of the Hamiltonian in AdS is massively degenerate *according to your own definition of what AdS is*. So by your own definition, once you provided it, we have refuted the AdS/CFT conjecture. That has exactly nothing to do with my solution to the information loss paradox. Essentially, you yourself have proven that AdS/CFT is incoherent, and now you are try to attack my solution to the paradox by noting that incoherence!

    As I said, AdS is not a theory. It is, if anything, the restriction of a theory (string theory or some other theory of quantum gravity) to some subset of its models. All of this talk about the spectrum of the Hamiltonian (by which you have always meant the boundary Hamiltonian) being non-degenerate had to be relativized to that restriction. The tacit restriction—which is fine—has been to single connected space-times. But what you have been assuming throughout this discussion is that if the space-time is connected then the space-like surface on which the wavefunction is defined is always connected. The entire point of my paper is that contradicts the very Penrose diagram that everyone uses. At this point, as in your response to Travis, you are pointing out the untenability of your own claims and trying to tie that untenability to my solution to criticize it.

    Take the very definition of AdS you yourself provided—"We start from the assumption that all surfaces are allowed, provided there is precisely one noncompact surface, which asymptotes to AdS. "—and try to justify your own claim that the spectrum of the Hamiltonian in AdS is non-degenerate (up to symmetry). You can't. The disconnected compact surfaces contribute nothing to the ADM mass of the one open surface (obviously), and so with even one such closed surface the Hamiltonian is massive degenerate. If you insist that the Hamiltonian is not massively degenerate in the CFT, and that that would destroy the duality, then the AdS/CFT conjecture itself is wrong, as you are interpreting it. All of this out of your own mouth.

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  44. BHG,

    If there is an energy cutoff for the energy eigenstates that are included in the wavepacket, then I agree that the scenario has to fail because the system will be in a superposition of a finite number of states and hence will have a finite recurrence time, so any black hole collapse must eventually become uncollapsed. The only way for it to work is if the black hole states are superpositions of an infinite number of energy eigenstates; and it's possible to do this while keeping the expectation value finite. So then there would actually be nothing contradictory about allowing an unbounded number of collapses.

    In any case, ok: I'm beginning to doubt that it can work. On the other hand, it definitely seems that the nondegeneracy of the Hamiltonian is not enough by itself to rule out Tim's scenario; much more complex arguments are needed. Would you agree with that at this point?

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  45. Tim,

    \Sigma_in certainly looks compact by any reasonable definition. It describes a 1-parameter family of 2-spheres interpolating between r=0 and r= r_min, where r_min is a PLanckian distance near the evaporation event at which classical geometry breaks down. So if you consider the region away from evap event where geometry makes sense, then you see that \Sigma_in a finite volume space that has the topology of a 3-sphere with a tiny 2-sphere cut out. That's a compact space and how you are getting a noncompact space out of this is anyone's guess. Truly bizarre.

    As for the rest of your message, I can see that you are not bothering to read what I write, since your objection has already been answered. As I explained, the CFT says that surfaces cannot dynamically split, so even if the full bulk Hilbert space contains disconnected surfaces, one has superselection sectors consisting of a definite number of disconnected components. As usual, it is completely consistent to restrict the theory to one superselection sector, and the statement is that the CFT is dual to the superselection consisting of a single connected component. You are proposing that there are not superselection sectors since the number of components can change dynamically. In this situation you are correct that the Hamiltonian is massively degenerate, and hence you are correct that your scenario is incompatible with AdS/CFT.

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  46. Travis,

    OK, good. In case there's any doubt, I can keep coming up with arguments showing why this can't work. Here's yet one more. On general grounds, we can consider the energy eigenstates of the theory (i.e. the Hamiltonian is certainly diagonalizable; we know that from the CFT). In this case, what is the criterion supposed to be that tells us whether an energy eigenstate is allowed or not? It certainly can't involve evolving the state back in time to see if it was connected in the past, since energy eigenstates are time independent. Good luck coming up with any criterion beyond saying that all states that satisfy the constraints are allowed, in which case you get the energy degeneracy.

    In my view, the statement about the non-degeneracy of the Hamiltonian gives the general reason why Tim's proposal fails, and at the very least one has to come up with some argument about how the black hole interior is going to retain information yet not give rise to any degeneracy in the spectrum. But to actually disprove it one indeed needs to examine specific proposals, all of which have so far been dead on arrival. If you can come up with a proposal that does pass basic consistency checks I would be most interested to hear it, but I am highly skeptical that this exists.

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  47. To quote from the above:

    "It is interesting in general to understand which CFTs are holographic. There are specific examples (e.g. maximally symmetric large N Yang-Mills theories) for which lots of evidence for a dual gravitational description exists. However, we don’t have a set of necessary and sufficient conditions to tell us whether any particular CFT is holographic.

    It is believed that having a gravity dual that looks like Einstein gravity coupled to matter requires a CFT with a large number of degrees of freedom (“large N”) and strong coupling. There are also more detailed conditions on the spectrum of states/operators of the theory; roughly, these conditions say that the CFT should have only as many low-energy states as we would expect for a theory of gravity on asymptotically AdS spacetime. On the other hand, it is plausible that any UV-complete theory of quantum gravity on AdS can be associated with a CFT, because the gravitational observables could be used to define a conformal field theory."

    What I found intriguing waa the statement: "...the CFT should have only as many low-energy states as we would expect for a theory of gravity on asymptotically AdS spacetime." Here the argument rages that the theory of blackhole evaporation that Tim Maudlin provides has too many states for the CFT; and on the other side, it seems if you don't pick a CFT with the right number of low-energy states, you won't get an AdS/CFT. That is - if gravity in asymptotic AdS has a degenerate ground state, then the CFT better have it too, or you don't have AdS/CFT.

    It seems to me that the CFT is like Goldilock's porridge, it had better be just right. But without being able to trace to those detailed conditions, all this is just conjecture.

    -Arun


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  48. BHG

    "So if you consider the region away from evap event where geometry makes sense, then you see that \Sigma_in a finite volume space that has the topology of a 3-sphere with a tiny 2-sphere cut out. That's a compact space and how you are getting a noncompact space out of this is anyone's guess. Truly bizarre."

    How about explaining this reasoning. By analogy: take a 2 sphere and cut out a point (i.e. a 1-sphere). Are you disputing that the result is non-compact? That would truly be bizarre. So by what principle are you claiming that a 3-sphere with a 2-sphere cut out is compact? I would say that is an obvious example of a non-compact manifold. So 1) are you disputing that a 2-sphere with a point deleted is non-compact? Or do you grant that and hold that a 3-sphere with a 2-sphere deleted is somehow different? I don't know how to even begin responding because I can't even imagine what principle you are appealing to.

    Anyway, the question is moot. It is actually quite irrelevant whether Sigma_In is compact or non-compact: either way it does not contribute to the ADM mass of Sigma_out, so the spectrum of that energy operator will be massively degenerate *by your own definition of the physical states of AdS. Therefore, by your own criteria, the CFT cannot be dual to AdS: given the space of physical states for each, the Hamiltonian of the CFT is non-degenerate and the Hamiltonian of AdS is massively degenerate. BY the precise criterion you have been using to try to show that my solution is incompatible with AdS/CFT, what you have shown is that AdS cannot be dual to the CFT in the way you are demanding.

    As for "the CFT says that surfaces cannot dynamically split", that is of course the Cauchy surface *in the CFT*. But the reaction of the geometry of a surface in the CFT to the geometry of its dual in the bulk depends on the dictionary relating the CFT state to the bulk AdS state, a dictionary neither you nor anyone else has.

    Your appeal to superselection sectors now a new gambit. Why don't you explain in detail how you are determining what these superselection sectors are?

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  49. BHG,

    What is wrong with the proposal I gave in my previous comment? That states which form black holes are superpositions of an infinite number of energy eigenstates and so are capable of collapsing an unbounded number of times?

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  50. Travis,

    To make sense of that you first need to define your rule for which energy eigenstates appear in the theory (that's the hard part). You should first define the full Hilbert space, and after that talk about which states yield black holes.

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  51. Tim,

    If you take a compact manifold and remove a region you typically get back another compact manifold, just one with a boundary. Note that I made a slight typo in what I wrote: I meant to say that you have 3-sphere with a 3-ball removed, giving a compact manifold whose boundary is a 2-sphere. If you remove a single point things are indeed slightly different. But this is indeed all rather moot since all one can really say is that geometry breaks down near the singularity. I think it should have been obvious from context that when I said that "only a single noncompact surface is allowed" I meant only a single surface with an asymptotic region in which classical geometry makes sense.

    I think we are actually making progress in that you now seem to grasp my basic objection: as soon as you allow the dynamical splitting of surfaces then you are led to degeneracy, in conflict with the CFT.


    The trouble is that you say that my appeal to a superselection sector is a "new gambit". Oh dear. I have been making this point right from my earliest comments in this thread. E.g. here is one comment of mine from the first week (there are many others):

    "So if we start in the AdS vacuum and prepare a collapsing shell of matter by acting with boundary operators these bag of gold states (or the closely related baby universe states) are totally irrelevant, since by definition of being in a different superselection sector they will never be accessed in the course of time evolution."


    As I have explained many times, the superselection sectors are labelled by the number n of connected components, and the CFT is dual to the sector n=1. Once again, from the bulk point of view the existence of such superselection sectors looks like a radical claim, and no one should simply accept it. My point is that it is the only scenario on the market that is not immediately ruled out.

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  52. BHG—

    Can you please—if you know them—try to use standard definitions of terms. What I meant by "compact" is what "compact" means in topology. Your comment: "That's a compact space and how you are getting a noncompact space out of this is anyone's guess. Truly bizarre." was itself truly bizarre. What is "anyone's guess" is what you meant by "compact".

    I was under the misimpression that you knew what "compact" means in topology and were using the term properly. My initial response was written under that misimpression, and is now moot. But your assertion that "it should have been obvious from context that when I said that "only a single noncompact surface is allowed" I meant only a single surface with an asymptotic region in which classical geometry makes sense." is really beyond the pale. It should have been obvious by context that you were misusing the term "compact" in a way that has no relation at all to its actual topological meaning? And yet it was not obvious to you from my response that I was taking the term "compact" to have its actual topological meaning, and that you should have cleared this up by explaining the novel use you were putting that term to? The most charitable construction I can put on this whole episode is that you just didn't know what "compact" means, made a kind of blind guess, ridiculed me because my response assumed you actually knew the meaning of the term, and are now trying to put the responsibility of all this on me for not telepathically knowing what private definition of "compact" you were using.

    Your comment "If you remove a single point things are indeed slightly different." illustrates the situation. Not by your idiosyncratic definition of "compact" it isn't. So my best guess is that you were aware that a compact space with a point removed is non-compact but never actually worked out why. That's fine. No reason for you to have studied topology. Just be aware that you don't know what you don't know, and that you are likely to make mistakes throwing around technical terms of a discipline you have not carefully studied.

    Now that we have sorted out that confusion, let's move on to the next. What do you mean by a "superselection sector"? I will not assume that your are using the standard meaning of the term, so please provide a) a definition of what you mean and 2) the method by which you are determining a superselection sector of a theory. Please be as precise as possible, because unlike "compact" there is not a really clear standard usage here.

    Let's review the bidding. You have been going on for some time about having some notion of the "physical Hilbert space" of AdS which is not the same as the space of solutions of the fundamental theory (in this case some sort of quantum gravity) restricted to solutions that meet the condition of "being AdS", whatever that means. Then, in response to Travis, you wrote "We start from the assumption that all surfaces are allowed, provided there is precisely one noncompact surface, which asymptotes to AdS. " Good. if "by allowed" you mean "are allowed in the physical Hilbert space" and if by "non-degenerate" you mean "non-degenerate with respect to the states in the physical Hilbert space" then we are done: The Hamiltonian of AdS is massively degenerate, and by your own criterion you have refuted AdS/CFT. This has nothing to do with my proposal at all. Now you are trying to somehow block this obvious conclusion by appealing to superselection sectors. I don't see how any such appeal is even relevant, and I also suspect that you are not using the term "superselection sector" in a coherent way. So either define "superselection sector" yourself or cite the definition you accept. No point wasting more time on the semantics.

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  53. https://arxiv.org/abs/hep-th/9912119
    "We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other."

    Does the converse hold - black hole horizons require disconnected-boundary-at-infinity?

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  54. Tim,

    bhg probably meant it's bounded, not compact. something else, I assume you read the Manchak, & Weatherall comment. Do you have something to say about it?

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  55. Arun,

    I am guessing the answer to your question is no because of the discussion we were having like 500 or so comments earlier. The horizon might have formed (indeed: should have formed) at finite time and it's enough if the information was able to leak to the boundary before it formed. If, on the other hand, you already know you have disconnected boundary-at-infinities, the only way to get this (without disconnecting the manifold I am assuming) is to causally disconnect pieces. But of course that just returns us to the question whether any physical state can have information that remains caught in a region that is never in causal contact with the boundary. Now the thing is that once you assume AdS/CFT works, you basically preclude this possibility by assumption.

    While I am at it, let me remind everyone that the purpose of my blogpost was to say you can't solve this problem with mathematical logic.

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  56. Sabine,

    Sure, Manchak & Weatherall initially made some really odd claims in virtue of using a completely erroneous conformal diagram that they mistakenly asserted to be equivalent to the standard Penrose diagram I used. In fact, in their original draft, their diagram was of a space-time that was not even temporally orientable, and so could not be equivalent to the Penrose diagram. In the updated version they posted to the ArXiv they have half-fixed the problem: the space-time no longer is non-orientable, but at the price of the Evaporation Event having multiple future light cones, contrary to my diagram. In short, their comments make no contact with my proposal.

    How they ended up in this position is instructive: the started from a fine conformal diagram of Wald's that was just vague about the Evaporation Event and then filled in more details (drew more light cones) in a way that Wald certainly did not have in mind. The relevant figure is Figure 7. In their original draft of the paper there was a single horizontal light-come at the top, looking rather like a bow tie.

    The moral is that one should always use strict conformal diagrams with fixed rules about the light cones or else one is in danger of getting confused.

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  57. Sabine,

    Can you explain what you mean by "mathematical logic" in this post? I would not use the term that way. What do you have in mind?

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  58. Tim,

    All I am saying is that the requirement that a solution to the problem be mathematically consistent will not allow us to conclude which solution is the correct one. Eg, I believe AdS/CFT to be mathematically consistent. I don't think it's correct in the sense of that it doesn't describe the situation in the real universe. I also don't think your solution is inconsistent. I also don't think it's correct in describing what's actually going on in actual black holes. There are various other proposals for solutions that I think are mathematically consistent but not well motivated on physical grounds. Of course the people who propose them think differently. Fine by me. But then the only thing I can conclude is that math won't solve the problem.

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  59. Sabine,

    I know that you don't have a dog in this fight, but can you say what you take the content of "AdS/CFT" to be? BHG could never answer that precisely, so I don't know even what it is to say it is mathematically consistent.

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  60. BHG,

    Why do I need to define a rule for which energy eigenstates appear in the theory? The point is that we don't actually know enough about the correspondence between the states in the CFT and the states in the bulk, so one possibility is that whatever the energy eigenstates in our AdS/CFT superselection sector turn out to be, the states which evolve into black holes will be states that are superpositions of an infinite number of them, and the claim is that this possibility is not so far ruled out by any argument.

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  61. Tim,

    I merely mean that the CFT can alternatively be described as a gravitational theory in an asymptotic AdS space-time with one dimension higher (as defined through the generating functional in the usual way). I am not sure how essential supersymmetry is to make that correspondence work; I think no one really knows, just that the known examples that are reasonably well understood are all supersymmetric.

    There are probably asymptotic AdS spaces which cannot be described this way. When I say I think it's consistent I merely mean that I see nothing wrong with just not considering those that aren't in the correspondence. Of course this kind of means you have solved the problem by defining it to be solved. But at least mathematically I see no contradiction with doing that.

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  62. Travis,

    So you are saying that black holes can only form from wavepackets built out of an infinite sum of energy eigenstates. If you were to cut off the sum at some maximal energy, no matter how large, then apparently the black hole would not form. That might be the case, but if so it's not clear that's it's much different from what I am saying, since it amounts to a claim that black holes do not form under circumstances in which you would expect them to. Beyond that I can't say much, since you're not providing a general rule for which states are allowed in the Hilbert space. The proposal is not specific enough for me to critique.

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  63. Tim,

    Compact vs. noncompact: the request for standard terminology here is misplaced. The surfaces corresponding to \Sigma_in necessarily approach the singularity where geometry breaks down. One has to be flexible and use physical reasoning, which is what I was doing. Anyway, I don't think this point is worth arguing about.

    As for superselection sectors, the statement is that the CFT Hilbert space describes the bulk Hilbert space built on connected surfaces that are asymptotically AdS. By superselection sector, all we need to say here is that time evolution preserves this Hilbert space, i.e. surfaces never split. I want to emphasize that I have been saying this consistently from the beginning. You will not be able to come up with any trustworthy argument that surfaces must split under time evolution, since such a change of topology necessarily involves a singularity where known equations break down.

    I would also like to note that there are (literally) many hundreds of papers performing detailed computations verifying, for a large class of states, that the energy spectrum of the CFT matches that of AdS gravity on a single connected component. My wording may suggest to you that I am invoking vague conjectures, but there is a large and detailed literature that supports this.

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  64. "You can't solve this problem with mathematical logic" - true; but you can rule out suggested solutions, perhaps?

    ===
    Anyway:

    The full extended spacetime of an eternal blackhole in AdS requires two asymptotically AdS boundaries and two decoupled CFTs. With this doubled state, from the first of the papers I cited above "The remarkable conclusion (emphasized in [13, 14]) is that by taking a specific quantum superposition of disconnected spacetimes, we obtain a connected spacetime, as depicted in figure 4."

    ( https://arxiv.org/pdf/1609.00026.pdf page 11).

    So, if quantum superposition of disconnected spacetimes can produce a connected spacetime in this case, what forbids disconnected classical boundaries at infinity? If this phenomenon (quantum superpositions of disconnected spacetimes can produce a connected spacetime) is a valid one that has to be included in quantum gravity, then whither this picture of AdS/CFT, and its alleged mathematical consistency?

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  65. Arun,

    I don't think there is anything that forbids disconnected boundaries, unless you forbid them. Regarding my comment about mathematical consistency. You can make literally anything consistent by dropping sufficiently many assumptions (or refusing to write them down). The "firewall problem" is a good example for this. Taken at face value, it is a proof of contradiction that rules out AdS/CFT. Unless of course you go and discard one of the assumptions you wanted your solution to fulfill.

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  66. BHG,

    Ok, fair enough. It's hard to gauge though just how weird it would be that no superpositions with an energy cutoff can form a black hole; the energy eigenstates might be pretty delicate, as Tim pointed out, and so any non-delicate state could plausibly require a superposition of an infinite number of them.

    In any case, it does seem strange that this was considered a paradox before AdS/CFT; none of these arguments about degeneracy would have been available before. Seems like it should really be called the AdS/CFT paradox.

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  67. bhg, travis,

    Regarding the issue of which superpositions are allowed, just let me add that you shouldn't create cases which are no longer described by classical geometries because then it's not clear what is even meant by a gravitational theory. You can create all kinds of "paradoxa" with that already in GR, for example by asking what's the radiation created by a superposition of two black holes with opposing spin. The total spin of the so-created quantum state is zero, which would result in a different spectrum of Hawking radiation than the superposition of the spectra belonging to each hole. So it looks like black hole evaporation isn't even linear. The problem with this argument is if you take such a superposition you are no longer in the semi-classical limit, so it's not paradoxical after all. I suspect (but do not know) that you run into similar problems with taking arbitrary superpositions in AdS. Best,

    B.

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  68. BHG—

    I really don't care whether you use terminology in a standard way so long as you explain clearly how you are using it. You introduced the term "compact", and if you had no intention to use it with its standard topological meaning then you ought to signal that and explain how you intend to use it.

    Similarly with "superselection sector". You are now using that term, and, forewarned by your way with "compact", I requested that you define that term as you intend it. You have not attempted to do that. All you have done is assert that the words "superselection sector" are somehow supposed to imply that "surfaces never split". How those words are supposed to imply that is not explained. What "superselection sector" means such that this supposed implication follows is never explained. At present, in essence, this is what you have written: "Surfaces in AdS cannot split because of the superselction sectors, and what I mean by "superselection sectors" is that surfaces do not split." The only part of that argument that is missing is the argument part.

    As for needing a "trustworthy argument that surfaces must split", I never purported to have an argument that they must. All I pointed out what that if you takes the standard Penrose diagram seriously (with the EE), then some Cauchy surfaces are connected and some are not. That is just a fact. And that if you want information to be preserved in the globally hyperbolic space-time, then you better consider only evolution from one Cauchy surface to another. Only such evolution has a chance to preserve information in both time directions. These are just plain facts.

    For a long time you have been trying to argue that states on disconnected surfaces *cannot* be int he physical Hilbert space, based on some supposed considerations from AdS/CFT. The idea is that if you allow *all* sates on disconnected surfaces into the physical space then the spectrum of the Hamiltonian in the AdS become massively degenerate, while the spectrum of the Hamiltonian in the CFT is not. Fine. But it hardly follows that if you allow *any* states on disconnected surfaces into the physical Hilbert space that the same conclusion follows. It doesn't. If the physical Hilbert space is the space of all states on connected asymptotically AdS slices that satisfy the constraints *and all other states that arise from these by time evolution* then we could get some states on disconnected surfaces in the physical Hilbert space without allowing all states on disconnected surfaces in.

    If you insist on forbidding this scenario, and label this refusal "superselction sectors" then there is not much else to say. You are simply stipulating that you will not allow the scenario, and not grounding that refusal in any physical principle.If you think that "superselction sectors" means anything more than that, state exactly what that extra meaning is.

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  69. Tim,

    As I said, my issue with your proposal in its current form is that to make it be self-consistent you have to depart from the standard rules (and here I just mean the practical rules) about how one would set up QM in the spacetime represented by the Penrose diagram. You may as well take your proposal to its logical extreme: the Hilbert space of the theory consists of precisely one state, which is the state that our world happens to be in. If you are willing to bend the rules in the way you have, then this setup will by construction reproduce all possible observations. What you are proposing is essentially the same. As I have said repeatedly, in all known quantum systems you determine the Hilbert space at time t by examining candidate wavefunctions at that time without any need to evolve them (arbitrarily far) into the past. In the case at hand, the criteria that someone educated in the standard formalism would apply to a candidate wavefunction is that it obey the constraints, has finite energy, and finite norm. None of these conditions involve a time evolution. Similarly, rules for probabilities would be computed by inner products of states in the physical Hilbert space, not outside of it. So if your claim is that you can trust the standard Penrose diagram and then apply ordinary QM to it, I just don't see that you have accomplished this. As a practical matter, if your scenario were correct it would make it essentially impossible to do physics. For example, suppose I want to know if some operator O is a physical operator, i.e. is the state O \psi in the physical Hilbert space if \psi is? According to you, I need to evolve O\psi back in time, uncollapsing all of the (arbitrarily many) black holes that may collapsed over the course of history, and see if this state emanates from some connected surface at some point in the past. Obviously, this is not feasible.

    I am certainly not just randomly stipulating rules that forbid your scenario. I am saying that in the only theory of quantum gravity that is sufficiently well defined that one can hope to answer this question, namely AdS/CFT, your scenario is not realized. We know a huge amount about the connection between states in the CFT and those in AdS in the regime where energy densities are not too large. The CFT beautifully reproduces, without any assumptions, the spectrum of quantum gravity in single connected asymptotically AdS space. That is, the CFT reproduces standard Einstein General relativity coupled to ordinary matter in this regime, and does this in a framework which obeys all of the *standard* rules of QM. That's what gets people excited. If one were to introduce disconnected surfaces in the bulk and apply the standard rules, then the spectrum would disagree with the CFT. Hence the CFT Hilbert space does not include anything like such disconnected surface states, and so there is apparently a superselection rule saying that even if such states "exist" in some meta-sense, one can self-consistently restrict to the single connected component sector, and this is what the CFT describes. So I am not stipulating this rule, I am simply listening to what the CFT tells me by the explicit computations contained in the hundreds/thousands of papers on this topic.

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  70. Travis,

    The thing is, black hole formation is supposed to be anything but delicate. GR tells us that if you take an arbitrary state with sufficient energy density then it will collapse to form a black hole. So appealing to some delicate rules about whether an (arbitrarily high ) cutoff is allowed or not really cuts against the grain.

    Whether pre AdS/CFT this should have been called a "paradox" or not, I think there is some truth to what you say. This term arose once the problem entered the particle physics community, which is deeply wedded to the idea of a unitary S-matrix. It is surprising that there weren't more efforts to see if Hawking's non-S-matrix scenario could be made compatible with the rest of physics. It's not easy, since once you give up a unitary S-matrix there is a vast range of possibilities, and its hard to know what rules to impose. But "challenge" would have been a better term than "paradox".

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  71. Tim,

    P.S.

    And there's also the energy eigenstate problem: you have no rule for determining if an energy eigenstate is in the physical Hilbert space or not, since your time evolution does not apply. The CFT of course has a complete set of energy eigenstates, so this is perhaps the most concise way of stating why your proposal is incompatible with AdS/CFT. Any response to that?

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  72. Sabine,

    That's a good point; and I can add that, in general, energy eigenstates in quantum mechanics tend to be the least like classical states, which would make it all the more plausible that any state which actually describes a classical geometry would consist of a superposition of an infinite number of energy eigenstates.

    Tim,

    I believe that what is meant by the superselection sector is just what you said: "the space of all states on connected asymptotically AdS slices that satisfy the constraints *and all other states that arise from these by time evolution*." BHG isn't saying that this immediately implies no splitting, but if the splitting can supposedly happen with states that contain an energy cutoff then there is a contradiction with your model.

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  73. Bee,

    I think I'm getting closer to understanding your point about mathematical consistency.

    Consider though:

    1. There is a General Relativity we know and love that applies in our world, from Mercury perihelion precession, gravitational lensing, GPS, gravitational waves, etc. This is a mathematical structure that we can map into our world. This theory may not be entirely consistent mathematically, it breaks down, e.g., when quantum effects become relevant, or at a singularity.

    2. There is a "General Relativity", probably better termed "a metric theory of gravity" in AdS/CFT that may even be entirely mathematical consistent and may be capable of being fully quantum, resolves singularities, etc., etc. However, it may not apply at all to our world. Because I don't want to say "General Relativity sometimes applies to our world and sometimes does not", I'm disinclined to call the gravity in AdS/CFT as "General Relativity". It is the same bunch of mathematical symbols maybe, with the same set of legal mathematical operations, theorems, etc. that apply, but we don't know if it is physically the same thing.

    Does this capture what you were saying?



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  74. Arun,

    I wouldn't say that GR (all by itself) is inconsistent. Singularities are not mathematical inconsistencies, it's just that we think of them as unphysical. GR is inconsistent with QFT, that being the issue that prevents us from just calculating what happens with stuff that falls into black holes.

    But, yes, assuming that AdS/CFT is consistent (which it almost certainly is in some formulation) this doesn't mean it has anything to do with the world we live in. Also, the GR in AdS/CFT is strictly speaking a limiting case that in reality would never be achieved anyway. But thist is (interestingly enough) not how the bh infloss problem is supposedly solved.

    Best,

    B.

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  75. BHG—

    Again, you are now attempting to operate on my turf now—foundations—and again I assert that what your are claiming about the "standard rules" is complete fiction that you are just making up. There are no such "standard rules". Period. The way to respond to this is either 1) write them down or 2) cite some text where they are written down. Until you do this you are not even trying to have a serious discussion.

    You appear to believe—possibly sincerely—that you have somehow distilled these "rules" in your own mind in the course of your physics education. I can tell you with not a speck of uncertainty that you have not. There simply are no such rules, and certainly not any that satisfy the conditions that you keep pulling out of thin air. If you have not bothered to read Bell's "Against 'Measurement'" you can start there is get a feeling for how different texts are both internally inconsistent and incompatible with each other on foundational issues. But since you seem uninterested in foundational issues, let's at least agree that you not propound any more "rules" that my solution violates without citing a text in which such a "rule" is explicitly articulated.

    Take the very first "theory" that you learn: the non-Relativistic quantum mechanics of N-particle systems (omitting spin). The standard presentation (and this really is standard) includes these "principles". 1) The Hilbert space is the space of square-integrable complex functions on the corresponding classical N-particle configuration space. 2) the Hamiltonian takes the form p^2/2m + V(x) for some function V(x) (including the zero function for the free theory). 3) the position operator and the momentum operator are x and -ih-bar x/dx in position representation. 4) The spectrum of an observable is the set of eigenvalues of the Hermitian operator that represents that observable. If you challenge that any of these are presented as "principles" I will provide citations.

    So: in the free case, the Hamiltonian is just p^2/2m. What is the spectrum of the Hamiltonian? The standard response is that it has a continuous spectrum, but that standard response violates your "rules". Following your rules, the Hamiltonian has no spectrum at all. Nor does the position operator. Nor does the momentum operator.

    I conclude that your "rules" are a bunch of hooey. And if you think the situation gets better when you bring in Relativity and have to worry about renormalizability, good luck with that.

    Answer in this simplest possible setting, or just stop bluffing about these "rules" that you go on about.

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  76. BHG,

    The fact that black hole formation is not delicate is exactly my point. What I'm saying is that it seems like any *energy eigenstate* must be delicate because it must be stationary. Thus, it seems plausible, since such states must be very exotic and almost certainly not even interpretable as classical geometry at all, that almost all normal states will be superpositions of an infinite number of them, similar to how all physical states of an ordinary free particle in an infinite space are superpositions of an infinite number of energy eigenstates.

    Also, you're accusing Tim of not having a way of determining which energy eigenstates are included in the Hilbert space; but how is it that you are determining that?

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  77. Tim,

    It would be so refreshing if you could communicate like a normal person and cut out all the snide comments and childish provocations.

    I have said, and will say it again, that I don't see how your attempt to turn this into a philosophy of physics discussion is going to be productive. Your single particle example seems pretty pointless - just put the system in a box and all is well. Anyway, I happen to have Weinberg's QFT laying next to me, so I will refer to it for the standard rules of QM. They are (you can read his discussion for the fine print):

    1) Physical states are represented by rays in Hilbert space

    2) Observables are represented by Hermitian operators

    3) If a state is in state \Psi, and a measurement is done to test whether it is in one of the mutually orthogonal state \psi_n, the probabilitiy is |<\psi_n,\Psi>|^2

    These are the rules that those of us who actually "do" QM adhere to, and they apparently work like a charm in describing everything from QM effects in the early universe, to particle collisions at the TeV scale, to the macroscopic world of superfluids and superconductors.

    Now, your proposal clearly does not adhere to these (or so far any) principles. I will ask again: what is your rule for determining whether an energy eigenstate is in your Hilbert space? As I said, this seems to be the most concise way of seeing why your proposal is incompatible with AdS/CFT. If you are hoping your free particle example is going to save you, you are sorely mistaken, since the energy eigenstates of a single particle in AdS are strictly normalizable.

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  78. Travis,

    You are making this seem way more complicated and mysterious than it is. A basic exercise that everyone does when they first learn AdS/CFT is to work out the low energy eigenstates on the gravity and CFT side, and verify how they beautifully agree. You can consider a collect of n particles in AdS, interacting via gravity and possibly other fields depending on the precise construction. In the large N limit the particles are free, and one explicitly compute the normalizable wavefunctions of all the particles. The result agrees with the spectrum of single trace operators in the CFT. Then interactions are included in the 1/N expansion. Again one computes the energy eigenstates as normalizable states, and compares the energy shifts to the anomalous dimensions of the CFT operators. Everything here just follows the textbook rules. Of course the energy eigenstates are not described in terms of classical geometry, since they are quantum states. As usual, one can form coherent states that do reduce to classical geometry in the hbar-> 0 limit. My point is that there is absolutely no mystery here about how to compute and characterize the energy eigenstates on both sides of the AdS/CFT correspondence. The precise agreement one finds was one of the highlights of the very first papers on the subject.

    So far Tim's proposal fails to incorporate the above, and so is incompatible with AdS/CFT. This has nothing to do with the philosophy of QM as far as I can tell.

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  79. BHG—

    So: exactly which of Weinberg's three rules are you claiming that I have violated? I have no idea.

    When I asked you what counted as a state in AdS (which—once again—is not a theory but a restriction of a theory) you responded "We start from the assumption that all surfaces are allowed, provided there is precisely one noncompact surface, which asymptotes to AdS. " Do you affirm or deny that this is your understanding of a condition for being a state of AdS?

    If we take you at your word, then the spectrum of the Hamiltonian in AdS—as *you* have just defined it—is massively degenerate. Even if the spectrum of the Hamiltonian operating just on the connected AdS piece is non-degenerate, that does not take account of the other disconnected pieces. As I have repeatedly said, you can't criticize my proposal for being inconsistent with AdS/CFT if by your own definition AdS/CFT is inconsistent for the exact same reason all by itself. You have never responded to this observation. I see no reason to take your supposed criticisms seriously until you do. I made precisely this observation on April 12. You never responded.

    As far as tone goes, just check the sequence above. You made snarky comment about turning compact manifolds into non-compact ones by deletion—an absolutely standard thing to do—and when I respond that it is standard you act like it is my problem for expecting terminology to be used in the proper way. You never acknowledged your error. Then you start in talking about superselection sectors. When I ask you to define what you mean by that, you never answer.

    Then you have all the rhetoric about "all of us who actually *do* QM". Well, all of us who actually *study the foundations of QM* know a lot more about whether there are is a set of clear "rules" than you who have never so much a read a book on the subject.

    Your technique is simple. I answer every single one of your questions, so then you complain that I had not provided any "measurement theory". You have never explained what that is even supposed to mean. If all you want is some practical rules of thumb for making predictions, they are what they always were, there is no problem at all. So then you demand something more, but what the something more is is never clear.

    How about this: you claim that you can compute a spectrum of energy eigenstates for a collection of n particles in AdS interacting under gravity. Good. Do you agree that every energy eigenstate is a static solution? And do you agree that n particles interacting under gravity will not be static? If you agree to both of these, then what is it you are calculating? If you don't, which do you reject? These are absolutely straightforward questions.

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  80. BHG,

    Coherent states *do* contain an infinite number of energy eigenstates, so if that is the accepted way of describing states in AdS which reduce to classical geometry, then we can't find any contradiction with Tim's scenario.

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  81. BHG,

    I applaud you for taking the time to write down the rules of quantum mechanics from a standard text, as Tim challenged you to do. The rules you cite are indeed typical, in being perfectly *un*clear, at precisely the point where a vague notion is introduced: measurement. In particular, notice that the rule #3 you cite from Weinberg is not even grammatical, and hence is even more ambiguous than it inevitably must be by the simple fact of introducing the undefined notion of 'measurement'. I copy it here:

    3) If a state is in state \Psi, and a measurement is done to test whether it is in one of the mutually orthogonal state \psi_n, the probabilitiy is |<\psi_n,\Psi>|^2

    This is ungrammatical because it does not tell us what sort of event has the probability |<\psi_n,\Psi>|^2. (I am here using an extended notion of 'grammatical' on which nonsense phrases may be called ungrammatical even if they obey the rules of syntax. For example, 'Colorless blue ideas sleep furiously' is ungrammatical in the sense I use here.) If this rule #3 were grammatical, then one could just detach the consequent and use it as a stand-alone sentence. But (say) 'The probability is 1/3' is meaningless - unless, of course, we can use context to fill in the missing '... of ___ ...' in the middle.

    The problem is, there are at least two ways of filling in the missing bit to make Weinberg's rule #3 grammatical. One would be:

    3a) If a state is in state \Psi, and a measurement is done to test whether it is in one of the mutually orthogonal state \psi_n, the probability that the measuring apparatus indicates outcome n after the measurement interaction is |<\psi_n,\Psi>|^2

    This is clear enough, and I have no problem with it. It makes clear, though, that quantum mechanics is simply a recipe or instrument for making probabilistic predictions.

    A second version of 3) would be

    3b) If a state is in state \Psi, and a measurement is done to test whether it is in one of the mutually orthogonal state \psi_n, the probabilitiy is |<\psi_n,\Psi>|^2 that the state of the system collapses onto the ray \psi_n, and hence(?) that the measurement apparatus indicates outcome n.

    This is the way Physphill likes to understand QM, and I also have no problem with it. BUT if we take it seriously, then there is no question of a star-sized system evolving unitarily for billions of years (unless the conditions required for a 'measurement' to take place are recondite in the extreme), and hence no information loss paradox.

    Under either 3a) or 3b), we still need to be told what counts as a measurement, and why such physical interactions have a special place in the foundations of our physical theory.

    In case you want to take a look at the article by Bell that Tim keeps mentioning, you can find it at: http://www.tau.ac.il/~quantum/Vaidman/IQM/BellAM.pdf

    It is not too long a read.
    Best,
    Carl3


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  82. Travis,

    This comes to the same point about delicateness. Put a very high cutoff on the energy spectrum appearing in the coherent state. This will be undetectable to any physicist whose measurement apparatus cannot resolve arbitrarily small distances. When two stars collide, we don't have to worry about what their wavefunctions look like at arbitrarily short distances, they will form a black hole as long as their wavefunctions at macroscopic scale describe sufficiently dense energy distributions. So cutoff coherent state wavefunctions will certainly form black holes as far as we know. Do you want to claim otherwise?

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  83. Carl3,

    My reference to Weinberg is meant as a pointer: please look there for the more precise statements. You can take the \psi_n to be, for example, the eigenstates of some Hermitian operator.

    I am specifically refusing to get into a discussion of what precisely constitutes a "measurement" because I don't think it has any bearing on the discussion. The kind of measurements we are talking about here are no different than those an astronomer would make when collecting photons from a star. In such cases, one doesn't have to worry about what are the subtleties of defining measurement. You just do it. The questions of whether information comes out in the Hawking radiation and what constitutes a measurement are pretty unrelated, and squashing them together is not going to be helpful.

    Don't worry, I have been familiar with Bell's writings long before this discussion.

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  84. Here's where I see the argument standing right now:

    The idea that disconnected surfaces can't be allowed in the Hilbert space because it would make the Hamiltonian degenerate has no force unless we're talking about an energy eigenstate with a disconnected surface, since that is the only instance where it makes sense to talk about "the" energy of the state. So we know that there are no such energy eigenstates in AdS/CFT, and the question is whether there are disconnected surfaces which aren't energy eigenstates. We have no reason to conclude that there aren't.

    The idea that splitting spacetime would imply an infinity of states below a certain energy only has force if we can define a state which retains an interpretation in terms of classical geometry as time goes to infinity. There is no reason to think that we can do that without describing a coherent state, and in that case there is no problem with it evolving into an infinity of other states since a coherent state is a superposition of an infinite number of energy eigenstates.

    So all in all, I don't see anything that rules out Tim's scenario.

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  85. Tim,

    In all the exchanges in this thread between persons A and B, the only ones that have turned nasty are the ones where A or B is named Tim Maudlin. As I have done with others, it is possible to disagree sharply without being condescending and insulting. Try it, and you will see that I and others respond in kind, and the whole discussion will be much more productive.

    Which of the rules have you violated? First, it would be extremely helpful if you would summarize your proposed rules in one place so we can examine them. Here is where I say that you violate the standard rules (as in Weinberg). Physical states are supposed to be in 1-1 correspondence with states (rays really, but let's keep this implicit) in a Hilbert space, and measurement probabilities are |<\psi_n|\Psi>|^2, where \psi_n and \Psi are elements of the Hilbert space. In your case, as I understand it, the probability to find the radiated particles in certain wavepackets in the external region cannot be expressed in this form, since there is no state in your Hilbert space representing external particles in definite wavepacket states.

    You say that you have answered every single one of my questions. Where did you answer my question about energy eigenstates? I will ask it again. Are energy eigenstates in your Hilbert space? As far as I can tell the answer is no, because a candidate energy eigenstate containing disconnected regions fails your criterion of evolving back in time to a purely connected region. Please answer this question.

    "Do you agree that every energy eigenstate is a static solution? And do you agree that n particles interacting under gravity will not be static? If you agree to both of these, then what is it you are calculating? If you don't, which do you reject?"

    An energy eigenstate has a time dependence e^{-iEt}, so it is not a "static solution" of the Schrodinger equation. Expectation values in such a state are time independent; is that what you mean? Same thing for n particles. The eigenstates will have time dependent e^{-iEt} and expectation values are time independent. Frankly, I can't make sense of your question. Perhaps you will find it helpful to review the single particle in a box: the classical solutions correspond to a particle bouncing around, which is nonstatic, but in QM we can compute the energy eigenstates, in which expectation values are time independent. Can you clarify your questions please?

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  86. Travis,

    Since you seem to think Tim's scenario is viable, could you do me the service of stating what the rules are in this scenario? Specifically, what is the rule for determining the physical Hilbert space and computing measurement probabilities? Are energy eigenstates in the Hilbert space?

    The rules as I understand them conflict with AdS/CFT, but perhaps we are talking past each other. If you would write down the rules then we can proceed from there.

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  87. BHG—

    Yes, the bad tone has always emerged with me because—if you hadn't noticed—until Carl3 and Travis started posting I was alone here while you and dark star and physphil adopted your condescending comments. I will defend what I have posted and continue to post under my actual name. If you are proud of how you behave and willing to defend it, then say who your are, OK?

    How do I point out in a polite way the following fact: You post as a "rule" of quantum theory that "Physical states are represented by rays in Hilbert space". Then when I say that Energy eigenstates are static you quite pompously reply "An energy eigenstate has a time dependence e^{-iEt}, so it is not a "static solution" of the Schrodinger equation." How do I point out that you seem not to understand the very "rules" that you assert every physicist knows? Just to be clear: *Vectors in a Hilbert space that differ only by a phase belong to the same ray*. So energy eigenstates are, exactly as I said, static: the physical state—the ray—does not change with time.How can I make clear that all of the calculating that you advert to, as if it shows that you "really know" quantum theory, has not prevented you from making such a straightforward error, and one that you only even fell into trying to show a flaw in what I had posted? How can I show more clearly what I have said over and over: there is a field of foundations of physics that the average physicist knows nothing at all about, is not trained in, and cannot do well?

    Now that you understand what a static solution is, I repeat: do you contend that there are non-vacuum, n-particle gravitational states in AdS that are static? I don't. Therefore, I do not believe that there are any non-vacuum states that are energy eigenstates. If you disagree, please specify how such a thing is possible. This has nothing to do with disconnected spatial regions.

    And if you want to improve the tone, how about just admitting that you were wrong about "compact" above and are wrong about "static" just now.

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  88. BHG,

    It's a little pedantic to argue that energy eigenstates are not static because they have an e^{-iEt} term... that term clearly only matters if you're looking at a subsystem of a larger system and so need to keep track of relative phases, which is not the case here.

    Anyway, you ask:

    "Specifically, what is the rule for determining the physical Hilbert space and computing measurement probabilities? Are energy eigenstates in the Hilbert space? "

    I don't know the answers to the first and third questions because I don't know that much about AdS/CFT, but the burden of proof is not on me or Tim. You're claiming that there is some way that we can know that Tim's scenario is not realized within AdS/CFT, and I'm saying that it seems to me that no argument you've put forward conclusively demonstrates this. If you're going to argue that Tim's scenario is not realized in AdS/CFT, then you should define the rules for what goes in the physical Hilbert space and then we can examine them and see if Tim's scenario is included. You've said before that anything that obeys the constraints, has finite norm, and has finite energy is allowed; if this is the case, then Tim's scenario is not ruled out because disconnected surfaces satisfy all these conditions.

    As for the question about computing measurement probabilities: you were trying to rule out Tim's scenario by saying that a measurement on the exterior would collapse the wave function into a product of a state on the interior and a state on the exterior, and that this contradicts the nondegeneracy of the Hamiltonian. Since we're talking about unitary quantum mechanics, we have to view such a description in terms of effective collapse, i.e., there is some piece of the wave function that represents an observer and the wave function is initially in some state

    (a|interior 1>|exterior 1> + b|interior 2>|exterior 2>)|observer ready to observe>

    and then the wave function effectively collapses to some state

    a|interior 1>|exterior 1>|observer sees exterior 1> + b|interior 2>|exterior 2>|observer sees exterior 2>

    and the probability that the observer sees exterior 1 is |a|^2 and the probability that the observer sees exterior 2 is |b|^2.

    The fact that the wave function evolves into product states like these only has a bearing on the degeneracy of the Hamiltonian if these product states are energy eigenstates, but we know that they can't be because if they were then they never would have (effectively) collapsed in the first place! (even putting aside the issue that these states are almost certainly not *exactly* product states). So I don't see how invoking measurement probabilities is going to help.

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  89. Tim,

    You say that my reply was pompous and you are clearly insulted. No offense was intended, I assure you, and I ask that you not assign to me the worst possible motives. I am honestly confused about what your point is here and am seeking clarification. So let's agree, as you say, that a solution of the Schrodinger equation with time dependence e^{-iEt} is "static". Now, you claim that there are no n-particle gravitational states in AdS that are static in this sense. Here you lose me, since there is no obstruction to constructing these explicitly. Consider n=2, and let's construct these states explicitly in perturbation theory. The free particle wavefunctions take the form \psi(r,theta,t) = (product of hypergeometric functions)*product of spherical harmonics*e^{-iEt}. The computation leading to this can be found in intros to AdS/CFT or any of the texts on the subject. The corrections due to gravitational interactions are then introduced via standard QM perturbation theory. The explicit formulas get a bit messy, but there is certainly nothing mysterious here. We can get into more detail here if you want. The resulting energies then agree with the energies computed in the CFT. You have to help me out here: what is the obstacle you claim and what is the straightforward error I am supposedly making? I brought up the simple issues of a particle in a box and the possible meaning of the word "static" not to insult or because I am confused about such things, but rather because I am totally flummoxed by your question and am trying to first establish the "basics".

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  90. BHG—

    Without doing any calculations, here is the thought arising from the character of the Newtonian gravitational theory we that should arise in the weak-field limit plus basic features of QM, plus some historical remarks.

    You claim that the formulation of AdS you are working with can handle a pure gravitational theory, i.e. a theory with no other "forces" than gravity. So you have only an attractive force and no possibly countervailing repulsive force. From a Newtonian perspective, how could any non-vacuum solution be static? In an infinite space, one could try to place massive particles at mutual rest symmetrically so the net gravitational force is zero on each one, but that requires infinite mass. In plain GR that will not be static, but lead to a Big Crunch. That is why Einstein put the cosmological constant in the field equation in the first place: so that some static non-vacuum solution would exist. He did not realize that the static solution he managed to allow was an unstable equilibrium, and so infinitely fine-tuned (BTW, I have read that AdS is like that too: there is a static AdS solution in GR, but it too is unstable and would have to be infinitely fine-tuned. If so, that is probably not a coincidence).

    As I pointed out, in plain non-Relativistic free QM there are also no energy eigenstates except the vacuum. The infinite square well, which you keep bringing up, is obviously non-physical. Indeed, in a truly universal theory of the sort that could be a candidate for a fundamental theory, there can be no such free-floating potential well: the V must arise from particle interactions. And in a pure gravity theory the only interaction is gravitational.

    What about your perturbation theory? Offhand, I would wager that these are only approximately eigenstates, not actual eigenstates. You are not going to get exact solutions out of the perturbation theory. What if I am wrong about that? OK, then I'm wrong. But as far as I can tell, there is not a thing in the world preventing the use of the very same techniques in my solution, so then you have answered the question yourself. What about the states with disconnected surfaces? If you are perturbing off a state on a connected surface, then these will not lie in the range of the perturbation theory. So your calculation connecting the AdS spectrum with the CFT spectrum will go through as usual. So I see nothing in this result that bears on my solution.

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  91. Tim,

    You say:

    "So you have only an attractive force and no possibly countervailing repulsive force. From a Newtonian perspective, how could any non-vacuum solution be static?"

    This just isn't a good argument. Neutron stars have energy eigenstates. "Atoms" which only feel the gravitational force have energy eigenstates. Arguments about classical stability just aren't helpful in quantum mechanics. How is this different from asking, if the universe consists of only one electron and one proton, how there can be any static non-vacuum solutions since there is only an attractive force and no repulsive force?

    (Also, Sabine: never mind, all of my comments have posted so far).

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  92. Tim,

    Thank you for clarifying. As Travis said, I believe you are inappropriately applying classical reasoning to a quantum problem. Loosely speaking, just like in the hydrogen atom, there is no obstacle to obtaining energy eigenstates with a purely attractive interaction. For the same reason, there is no obstacle to having energy eigenstates of two particles with an attractive interaction in AdS. For the full gravitational interaction it may be hard to get an exact analytic solution, but there is no doubt it exists: you can either use perturbation theory or show this numerically. The solutions are definitely there.

    I bring up the infinite square well because it shares a key fact with AdS in that energy eigenstates of, for example, a single free particle, are normalizable. I mean they have strictly finite norm, not an infinite delta function. AdS is quite beautiful in that shares the property of Minkowski space of being maximally symmetric, and also shares the nice normalizability property of a box.

    You mention the instability of AdS. I can explain more if you want, but this is much more benign than I suspect you are imagining. There is no sort of decay or catastrophic runaway behavior or anything like that. And it definitely does not conflict with the existence of energy eigenstates in any way that I can see.

    I didn't understand your last paragraph. I thought you had stated a rule saying that states are in the physical Hilbert space only if they can be time translated backwards to a time in which the surfaces appearing in the wavefunction are connected. If so, then this implies that there are no energy eigenstates in the Hilbert space which have disconnected components, right? (i.e. since energy eigenstates have time independent properties, they would have disconnected surfaces for all times). So is that your claim: all eigenstates of the Hamiltonian involve just a single connected component?

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  93. Travis,

    It's a fair comment, but the stability of a neutron star depends critically on there being more than just gravity. Without electromagnetism and chromodynamics there would not be such things with multiple energy eigenstates, yes? If AdS can be used for a pure gravity theory, and that theory turns out not to have many, or more than one, or even any energy eigenstates *in the space of physical states*, so what? As I have pointed out, the free non-relativistic theory has no energy eigenstates, but we are still free to use Fourier analysis.

    The only relevance this issue of energy eigenstates in the space of physical state has, as far as I can tell, if that if you allow disconnected hypersurfaces then the spectrum of the Hamiltonian becomes degenerate. But that observation has no implications at all for the solution I am defending. BHG said explicitly that such disconnected hyper surfaces fall within the ambit of AdS. If so, then AdS allows for massive degeneracy.If you are doing perturbation theory off a state that is not disconnected, then you will not access those states. And again, so what? Why expect black hole formation and evaporation to be accessible by perturbative quantum theory done in that way?

    In short, the whole issue of energy eigenstates in the space of physical states strikes me as a red herring.

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  94. Travis,

    First, about computing probabilities, what you are describing is not my objection, and I have said over and over that I don't want to get into any issues about (non)collapse. I am just asking for the abstract rule to compute probabilities. We at least need to establish those. According to the rules of QM (a la Weinberg) we compute these from the inner product of two states in the Hilbert space. The scenario on the table is not compatible with this because, e.g., states with radiated particles in specified wavepackets in the external region are not in the Hilbert space as Tim has defined it. But let me say that this objection is less sharply problematic than the energy eigenstate problem now under discussion.

    More generally, I don't agree that the "burden is on me" in the way you suggest. Tim's claim as I understand it is that you can just apply vanilla rules of QM and gravity in the spacetime implied by the standard Penrose diagram, and there is no conflict with AdS/CFT in doing so. Now the ordindary rules of quantum gravity include allowing all states that obey the constraints, have finite norm etc. I think we've all agreed that this doesn't work since it yields the energy degeneracy. So one has to do something else, and there are an infinite number of ways one could conceivably proceed. It is not unreasonable for me to ask the most basic of questions, like "what is the rule for determining the physical Hilbert space and computing measurement probabilities? Are energy eigenstates in the Hilbert space?" These question have very little to do with AdS specifically; they are just the most basic structural questions one needs to address to have a scenario that is even remotely well defined. If you're asking me to rule out all conceivable scenarios involving all possible modifications of GR and QM , then I am afraid I am not up to the task.

    "You've said before that anything that obeys the constraints, has finite norm, and has finite energy is allowed; if this is the case, then Tim's scenario is not ruled out because disconnected surfaces satisfy all these conditions."

    No, I am saying that the above is the usual rule for quantum gravity, and following it rules out Tim's scenario. That's why Tim is proposing not to follow this rule. In AdS/CFT, the statement is that the CFT describes a consistent sub Hilbert space of the above, namely the Hilbert space with only connected components. The precise claim is that surfaces never split dynamically. This is what comes out of the CFT in that if we compute the spectrum of energy eigenstates in the bulk under this assumption we get agreement with the CFT spectrum.


    I also want to point out how radical your claim is that if we impose an arbitrarily high energy cutoff on a coherent state then we won't form a black hole. In the presence of such a cutoff we can still describe two colliding stars that GR says would form a black hole, so its certainly a strong claim that something goes awry.

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  95. BHG—

    This entire line of reasoning about eigenstates of the Hamiltonian has really absorbed too much time over nothing, as we can now see.

    1) No such eigenstate can possibly lead to the formation and evaporation of a black hole, so they a completely irrelevant for our immediate purposes.

    2) If we allow for the possibility of states in the physical Hilbert space of exactly the form you endorsed, namely states defined on disconnected hyper surfaces with one component AdS that the rest not, or the rest closed, then the spectrum of the AdS boundary Hamiltonian is obviously massively degenerate, as it should be.

    3) This observation in no way conflicts with any claims you have made about AdS/CFT. As you say, the computation of the spectrum of the Hamiltonian in AdS is done via perturbation theory, the perturbation being off a state with just a single connected component. But starting from there, perturbation theory will never get to states with disconnected components, even if they arise via the dynamics. So the observation that the computed spectrum is non-degenerate is simply not relevant to any issue at all. Even if my solution is 100% correct, the computations will not be any different. And neither you nor anybody else has ever done any of the computations non-perturbalively.

    So all of this focus on the eigenstates of the Hamiltonian and the spectrum are completely beside the point. As are all of the numerical calculation etc. None of it has any bearing at all on my proposal.

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  96. BHG,

    You say:

    "I am saying that the above is the usual rule for quantum gravity, and following it rules out Tim's scenario. That's why Tim is proposing not to follow this rule."

    But you are also proposing not to follow this rule! You're also ruling out some states, namely all disconnected surfaces, whereas Tim is proposing only ruling out some disconnected surfaces. Neither of you are giving the explicit rule that you're using; you're both using arguments about why certain states should be in the Hilbert space and not others. But your argument doesn't seem like a good argument. It depends on the idea that a disconnected surface must contain disconnected energy eigenstates, one piece of which is degenerate. And we have no reason to think that. You are imagining that there is some energy associated with just the exterior, and this doesn't have to be the case. It's possible that all energy eigenstates don't look anything like a disconnected surface, and yet certain superpositions of them do. And given that a disconnected surface is supposed to only arise from a black hole, and therefore can't be a static state, it's entirely consistent with Tim's scenario that this would be the case.

    I just don't get the problem with probabilities. I can't speak for what Tim is trying to do here, but I agree with you that you should in principle be able to calculate the probabilities using only states in the physical Hilbert space. But why can't that be done here? If there were some superobserver existing outside of the entire AdS system, and he wanted to measure the entire energy of the system, then he can calculate what the probability of getting a particular value would be by looking at the density matrix of the state written in the energy basis. What is the issue?

    My claim is not so radical; I'm not saying that no coherent state with a high energy cutoff can form a black hole. I'm merely saying that no coherent state with a high energy cutoff can form a black hole an infinite number of times, which seems like a reasonable claim. This claim undercuts your argument that there are an infinite number of states below a given energy.

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  97. Travis,


    We seem to be talking past each other. I will repeat myself once more and then let you have the final word. Let's say we agree that the "full" Hilbert space in the bulk consists of all wavefunctions that obey the constraints, containing connected and disconnected surfaces. Then, there are two dynamical possibilities which are possible a priori: a) surfaces can split/join under time evolution b) they cannot. Under (b), the wavefunctions built on a single connected surfaces yield a Hilbert space by itself that is preserved under time evolution, and hence we can define a QM theory that is restricted to that sector, and this QM theory will obey all the normal rules. All evidence suggests that the CFT is precisely that QM theory. So there does seem to be a theory of quantum gravity realizing (b). Under (a), one either needs to accept a huge energy degeneracy (in which case this scenario cannot arise in AdS/CFT) or else modify the rules of to toss out most of the states. Tim is proposing the latter, and I am arguing that his way of doing this is inconsistent. So the situation are totally different. In the AdS/CFT scenario, nowhere am I departing from the standard rules of quantum theory.


    Finally, I stick to my claim that your coherent state energy cutoff is radical. With some very large but finite cutoff, you run into a problem after a finite number of black hole formation evaporation events. SO you will eventually find yourself in a situation in which you have a cloud of matter and there is some mysterious effect preventing you from collapsing it into a black hole. Similarly, you will be unable to verify or disprove unitarity of the S-matrix, because you will not be able to repeat experiments an arbitrary number of times. Again, I am not saying that this means it wrong, but it does imply a surprising breakdown of seemingly trustworthy principles and approximations.

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  98. Tim,

    The energy eigenstate issue now seems like the key showstopper in terms of trying to realize your scenario in AdS/CFT, so I certainly think it's highly relevant. Are you still committed to your rule that states are in the physical Hilbert space only if they time evolve back to a purely connected surface? If so, then I really need to press you on whether energy eigenstates in your scenario involve disconnected surfaces or not. Without this, I just don't have enough information to assess your proposal one way or the other. I of course agree that energy eigenstates don't "form" black holes, but if we want to see how your scenario is realized in a full quantum theory we need to understand this question. What might not be apparent is that I am trying to be constructive here...

    I want to stress again that the CFT has a nondegenerate spectrum, and the way this is understood in the bulk is that there is a consistent sub Hilbert space of connected surfaces that is preserved under time evolution. The CFT describes precisely this Hilbert space and nothing more. In AdS/CFT all the standard rules of QM apply in their usual form (i.e. as in Weinberg).

    Also, you wrote "And neither you nor anybody else has ever done any of the computations non-perturbalively." , but this is false. There is a big literature on computing non-perturbative energy levels on both sides of the AdS/CFT correspondence using cutting edge mathematics and the agreement is mind blowing. (I can give refs of course). It's pretty much impossible for anyone who has seen these results not to accept that AdS/CFT is "correct" in some very deep sense.

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  99. bhg,

    I suppose you are referring to specific examples for the correspondence here?

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  100. Bee,

    Yes, I am referring to methods based on supersymmetric localization, which do of course function within a specific theory. Namely supersymmetric string theory in the bulk and supersymmetric gauge theory on the boundary.

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  101. BHG—

    You are the one who insisted that the criterion for a "physical state" not involve reference to time evolution. And you are the one who said that the folium of physical states "in AdS" could include states on disconnected hypersurfaces, so long as only one was symptomatically AdS. So I am just following your own set-up here.

    Of course energy eigenstates do not evolve from connected surfaces: they don't evolve from anything. They are static.

    So your choice here is straightforward. Either you stick with the conditions you yourself laid down, in which case the spectrum of the Hamiltonian is massively degenerate, and by your own criteria you have disproven AdS/CFT, or you artificially restrict the states on the AdS side to states on connected hypersurfaces, in which case the whole argument begs the question at step 1, and all of these calculations are irrelevant.

    Straight question: are you claiming that the calculations of the spectrum in AdS included states on disconnected hypersurfaces?
    If they did not, then they are irrelevant and if they did then they are wrong.

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  102. Tim,

    "You are the one who insisted that the criterion for a "physical state" not involve reference to time evolution. And you are the one who said that the folium of physical states "in AdS" could include states on disconnected hypersurfaces, so long as only one was symptomatically AdS. So I am just following your own set-up here."

    Yes, those are the rules I think one should follow, and I am certainly doing so. Are you saying you know renounce your rule about time evolving states as a diagnostic to see if they are physical. That's I how read the above.

    "Of course energy eigenstates do not evolve from connected surfaces: they don't evolve from anything. They are static."

    Yes, and who claimed otherwise? Not I. I will ask the pivotal question one more time then give up: in your scenario, does the wavefunction for an energy eigenstate have support on both connected and disconnected surfaces?

    " or you artificially restrict the states on the AdS side to states on connected hypersurfaces, in which case the whole argument begs the question at step 1, and all of these calculations are irrelevant."

    There are no artifical assumptions. The CFT is a precisely defined object and its predictions are unique, though not always easy to translate into bulk language. I am saying that all evidence appears to show that surfaces never split. So while the Hamiltonian in the "full" bulk Hilbert space may be massively degenerate, the CFT consistently describes the connected component sub Hilbert space in which the Hamiltonian is non degenerate. But if surfaces split as you claim, then this obviously breaks down and there is a clash with the CFT..

    As for your last question: as I say, all evidence is that there are no disconnected surfaces required to match the CFT. Of course this is highly relevant, since it says that the splitting of surfaces appears to be ruled out by the CFT, and not by some added assumption.

    Anyway, I really do hope that you will take a stab at my energy eigenstate question, since I do think this is the most concise way to see the basic clash with AdS/CFT.

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  103. BHG,

    You're trying to draw an asymmetry between Tim's proposal and your proposal by saying that yours follows the standard rules of QM while Tim's does not. This is where I disagree, or at least where I don't think you've sufficiently demonstrated that this is the case. So I will try to rise to your challenge and describe how to build all of the states in Tim's scenario, at least from my perspective:

    -Start with a connected background AdS spacetime
    -Place quantum fields on the background spacetime, including the metric tensor
    -Find all energy eigenstates of the system using the normal rules of QM. The spectrum will be nondegenerate up to symmetries.
    -All of the energy eigenstates built in this way are part of the Hilbert space, along with any superpositions of them

    Is this any different from how you're proposing to build the Hilbert space? I'm genuinely asking.

    Now that we've built the Hilbert space in this manner, the question is: do any of the superpositions of energy eigenstates look like disconnected surfaces, in the sense that there is an approximately classical metric tensor which describes a disconnected surface? It's not clear how we can say that the answer is no (your original argument, that we can just operate on the interior without affecting the energy of the exterior, only works if we assume that the disconnected surface is an energy eigenstate and that operating on the interior brings you to another energy eigenstate, and there's just no reason to think that's the case). And given that there are some states that look like disconnected surfaces, we can ask what happens to them when we evolve them forward or backward in time. Under Tim's model, we will find that going one direction in time will cause the surfaces to reconnect (the "backward" direction) and going the other direction will cause the surfaces to remain disconnected and eventually split again as another black hole forms (the "forward" direction).

    You say:

    "With some very large but finite cutoff, you run into a problem after a finite number of black hole formation evaporation events. SO you will eventually find yourself in a situation in which you have a cloud of matter and there is some mysterious effect preventing you from collapsing it into a black hole."

    It's not at all clear that you will end up in a situation in which you have an approximately classical spacetime with a cloud of matter sitting in it and refusing to form a black hole. The quantum states here involve configurations of the metric tensor as well as configurations of the matter fields; so what will actually happen is that eventually, if you don't have a full coherent state, you will just end up in a situation that can't be described by classical geometry at all and it's unclear whether we should expect anything particular to happen with such a state (just like we don't expect anything *at all* to happen if we have a state that is a "superposition" containing just one energy eigenstate). Here's another way of putting it: with one energy eigenstate, your argument clearly doesn't work; we don't expect that state to eventually collapse to a black hole because we expect it to do nothing at all, and this is not some "mysterious effect". With two energy eigenstates, the state will just oscillate back and forth between them with a period of (E1-E2)/2pi, and it seems unlikely that somewhere in that oscillation there is a state which describes a black hole collapse. With three, still unlikely. So even once we get to a large enough number where we can reasonably say that we're at some point describing approximately classical geometry which forms a black hole, it's highly likely that the state will also at some point evolve to one that looks nothing like classical geometry and hence the whole question of "how many times has a black hole formed" becomes a nonsensical question.



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  104. BHG—

    Following your rules some energy eigenstates have support on a connected service and others on disconnected surfaces. That has nothing at all to do with anything in my solution. That just follows from your own definitions. So if its problematic, its your problem. What do you think follows of interest from this, because whatever it is you are stuck with it.

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  105. Tim,

    I am asking you what the situation is according to *your* rules, not mine. So, what are your rules for defining physical states, and do the wavefunctions for energy eigenstates have support on both connected and disconnected surfaces, according to your rules? Why are you refusing to answer these straightforward questions, which I need in order to understand your proposal?

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  106. BHG—

    I have never been very interested in your game of defining a "physical Hilbert space" because I could never see any point to it. As you define it, it is not anything I have thought much about, and I really don't care. And if I were to make up my own definitions, you would presumably not be in a situation to say anything about them, since if they are tied to the evolution of a state you need to solve for that evolution, which cannot be done.

    But since you have constantly been wanting to make some sort of point using that concept, let's use yours, exactly as you define it. By your own definition, the Hamiltonian in AdS is massively degenerate. Now what? Or do you deny that?

    I really can't see why you are pressing me for *my* definition. Surely, you can understand the situation better in terms of *your* definition! If there is some obscurity in my position just ask about it, and I will try to clear it up.

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  107. Travis,

    I said I would give you the last word, but since you asked me a direct question I will respond. I don't think you're giving an accurate description of how we quantize the bulk. We don't first find all the energy eigenstates and then ask if any superpositions look like disconnected surfaces. This has it backwards. We are asking about the space of wavefunctions \psi[g_{ij}(x)] that obey the grav. constraint equations. At the outset, we have a topological choice to make as to whether the 3-geometries on which \psi has support are all connected or not. This has nothing to do with any classical approximation. Suppose we make the natural choice that all surfaces, connected or not, are allowed. Then the Hamiltonian is highly degenerate. Next, there is the *dynamical* question of whether surfaces can split under time evolution. If not, then we can consistently restrict the Hilbert space to the connected sector and talk about a QM theory governing that restricted Hilbert space. The Hamiltonian has a nondegenerate spectrum on this smaller Hilbert space, and it is the one dual to the CFT. So I have no idea what you are talking about when you say that we find all the energy eigenstates using the normal rules of QM without saying anything about the topology of the allowed configuration space.

    Regarding the forming and evaporating black holes, again I think you are imagining mystery where there is none. This non-classical state you talk about resulting from the evaporation of many black holes is just a diffuse gas of gravitons, photons etc. That just follows from entropy considerations. There is nothing to stop us from collecting this radiation and collapsing it to form a new black hole, at least according to the usual laws of physics. It seems you are imagining the outcome of black hole evaporation to be some exotic final state, but viewed from the outside it is totally boring: just a diffue gas of thermal radiation on top of a background AdS space.

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  108. Tim,

    According to my rules (which I am arguing are the one that seem to follow from matching to the CFT) surfaces never split, hence one can self-consistently restrict to a single connected component, and the Hamiltonian is nondegenerate in this Hilbert space. But forget my rules, I am trying to understand *yours* to see if they are compatible with a CFT description.

    Unfortunately, I no longer have the foggiest idea what it is that you are proposing, especially you refer to my sincere request for clarification about what you take the Hilbert space to be a "game". I don't see how you can claim to have a concrete proposal for how black hole evaporation is described quantum mechanically if you refuse to answer such basic questions as what is the physical Hilbert space, and on what geometries do energy eigenstate wavefunctions have support.

    Maybe this is a good place to quit before we drive each other nuts.

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  109. BHG—

    As I said from the very beginning, and have repeated many times, my view of theory construction is that you start with a *kinematical* Hilbert space and then put restrictions on it to get a *solution space*, which is a subspace of the original Hilbert space (and therefore inherits an inner product from it). Some of the restrictions can be non-dynamical constraints. Some of the restrictions arise from demanding that the solution obey a Schrödinger equation. In a quantum theory of gravity, the dynamics of the bulk gets implemented also through a constraint rather than through a Schrödinger equation, viz., the Hamiltonian constraint. Once I have the solution space, I am done. What you want to do is interpose some "physical" Hilbert space that is a restriction of the kinematical space, but still wider than the solution space. It is that part of the game I can't see any point to. But it's your game not mine, and I have answered your question: in the physical Hilbert space of AdS as I understand you to define it, the spectrum of the Hamiltonian is massively degenerate. That is just what you said to Travis. So I take it we all agree about that.

    So what is the issue? You say the issue is whether the hypersurfaces on which the states of the system are defined can split, i.e., can dynamical evolution take one from a connected to a disconnected surface. I say yes. You say "Next, there is the *dynamical* question of whether surfaces can split under time evolution. If not, then we can consistently restrict the Hilbert space to the connected sector and talk about a QM theory governing that restricted Hilbert space. The Hamiltonian has a nondegenerate spectrum on this smaller Hilbert space, and it is the one dual to the CFT." That's fine, predicated on "if not". But I say indeed they can split. Why? Because I read that straight off the Penrose diagram. I guess you want to say "No, surfaces *cannot* spilt under dynamical evolution. But what possible grounds do you have to assert this? Where is your assurance about that coming from? Not from calculating what actually happens in the bulk!

    My point all along is that the standard Penrose diagram *does* imply that the surfaces can split, and indeed *requires* that they split in order to secure the standard properties of quantum theory such as information preservation. To oppose this you have to 1) change the usual presentation that everyone already accepts (i.e. the usual Penrose diagram) for a new one and 2) solve the information loss problem. Where does the "stray" information end up? My solution gives a straightforward answer to that. How does your solution even approach the information loss problem?

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  110. Tim,

    you cannot treat external measurements as projectors because the state after projection does not exist according to you. You also cannot treat measurements as unitary operators that act only on external part of the space. Those operators do not preserve the solution space you keep talking about. If you act with one of them and then evolve back in time you do not get a state in original space that would form a black hole. You get a state in larger Hilbert space, the same space you get by quantizing the disconnected geometry in normal way.

    So you cannot treat measurements at all. You cannot treat them as projectors. You cannot treat them as unitary operators. You cannot compute probabilities for measurements, like bhg keeps saying.

    QM without measurements and without probabilities is not QM.

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  111. BHG,

    When you say "The Hamiltonian has a nondegenerate spectrum on this smaller Hilbert space [the space of connected geometries], and it is the one dual to the CFT", this is of course what the whole discussion has been about. We're trying to figure out whether it's actually the case that the smaller Hilbert space that is dual to the CFT contains only connected geometries, and our only clue is that it must be nondegenerate. The only argument you've given is that we can operate on one of the disconnected pieces without affecting the energy of the other. As I've pointed out, this only works as an argument if we have an energy eigenstate before and after operating. Is it a known fact in quantum gravity that, if you have an energy eigenstate which has support on only connected geometries, that the same state with disconnected pieces added to all of the geometries will also be an energy eigenstate with the same energy? If so, then your argument is fine.

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  112. Physphil,

    I hope that you will reflect very, very carefully about what you have written here, because it precisely illustrates the actual situation once as BHG complained that I had not provided a "measurement theory". By that, one can mean one of two completely different things.

    One is just some reasonably well-defined method for producing probabilities for various possible "measurement outcomes", in the way that "standard QM" does all the time. All one is asking for is a prediction-making recipe, and the mathematical resources used to specify that recipe can be whatever you like. Demanding, for example, that the only mathematical objects employed in the recipe be mathematical representations of "physical states" is completely unmotivated and unjustifiable. It is, to repeat yet again, like forbidding someone doing standard non-relativistic QM from using Fourier analysis because the sines and cosines are not in the Hilbert space. Who cares? As long as the math gives the probabilities, you are fine. In particular, *there is no pretense to actually model the physics of the experimental situation*! All of S-matrix theory is like this: the S-matrix does not purport to specify the physics of an interaction, but just to summarize the statistics of the outcomes of the interaction. So for this purpose, I can use whatever mathematical techniques anyone else does in the service of grinding out predictions.

    The other thing you could want is a *physical account of the experiment*, i.e. an analysis of the interactions from a completely physical perspective, from which the predictions about the outcomes follow. This require solving the measurement problem and signing on to a particular solution. And that means confronting the measurement problem head-on. I have no issues about doing this: my preferred solution here (and the only one relevant since we are assuming that information is preserved) is a Bohmian solution. And a Bohmian solution to the measurement problem makes no use of projectors, no use of "unitary" (Hermitian) operators at all. All you need is the dynamics of the wavefunction and the guidance equation for the local beables. None of the usual quantum apparatus is employed at all. So I am more than happy to go this direction. The question is: can you?

    Do you have a solution to the measurement problem that you are using? Which is it? If you are using projectors, then presumably you are thinking of a collapse theory, but not along the lines of GRW. How do you deal with experiments that are analyzed using POVMs rather than projectors? I am really, really dying to know.

    So if you want 1, I can use whatever you can. If you want 2, I have at least said how I would go about getting that. Of course, I am not proposing any precise theory—just a generic way to think about (and solve) the so-called "Information Loss Paradox". But absolutely nothing about that solution interferes with a Bohmian solution to the measurement problem.

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  113. Travis,

    " Is it a known fact in quantum gravity that, if you have an energy eigenstate which has support on only connected geometries, that the same state with disconnected pieces added to all of the geometries will also be an energy eigenstate with the same energy?"

    I am having trouble converting these words into mathematics, but perhaps the following answers it. The Hamiltonian is a boundary term, so compact surfaces do not contribute to the energy. Suppose we have support on two disconnected components. Then we can write down a basis of wavefunctions of the form \psi_i,in x \psi_j,out, and choose \psi_j,out to be eigenstates of the Hamiltonian. So if \psi_i,out is an energy eigenstate in the single component sector, then \psi = ( \psi_in) x \psi_j,out is an energy eigenstate in the two component sector, for any \psi_in. So in this sense, yes you can "add disconnected pieces" while maintaining that the state be an energy eigenstate. It seems to me that the only way you can avoid a degeneracy is to do something trivial like say that the \psi_in Hilbert space is 1-dimensional. Are you claiming otherwise, and if so please elaborate.

    More generally, since the CFT Hamiltonian is non-degenerate we need to find a bulk QM theory that respects this. The simplest way this can happen is if only connected geometries appear in the bulk, and this appears consistent with all of the many computations in this subject. But I don't have a "proof" of this statement, and all I can do is examine counterproposals that do include disconnected surfaces. These proposals have to include enough detail that they can analyzed as well defined QM theories. At the moment, I don't see any such proposal on the table, but if you do please bring it to my attention. Tim refuses to answer the key questions that are needed to yield a well defined proposal, questions like "what is the Hilbert space?", so there is nothing to even argue against and I have given up trying to get him to clarify.

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  114. Tim,

    "Once I have the solution space, I am done."

    Great, let's suppose for the sake of argument I accept everything you have said. Now I ask, according to your rules, are energy eigenstates in your "solution space" and if so do their wavefunctions have support on both connected and disconnected geometries?

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  115. You wrote "So for this purpose, I can use whatever mathematical techniques anyone else does in the service of grinding out predictions." Then to get predictions you must use the states that do not exist according to you because to compute inner products you need them. To compute those inner products you have to calculate those states. So you agree that your proposal requires calculating the states in the big degenerate hilbert space that bhg keeps talking about. That hilbert space does not exist or make sense in CFT. It is not part of the CFT. So your proposal is not compatible with AdS/CFT. If AdS/CFT is true you can calculate the probability of any measurement using only CFT states.

    About the rest of your comment. I cannot use projectors to represent measurement. I cannot use unitary transformations. That is not QM.

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  116. Tim,

    There's a difference between being *able* to compute probabilities using states which are not physical states (like your sines and cosines example) and being *required* to. In standard QM, you should be able, in principle, to compute everything using only the physical states and their inner products.

    BHG,

    Ok. So I was thinking before that you could make a disconnected surface out of a superposition of connected ones, and if it's true that that's impossible then I'm pretty fairly convinced that Tim's scenario can't work. But it seems like it should be possible to get arbitrarily close to a disconnected surface using only connected ones, in the same way that it's possible to get arbitrarily close to a position eigenstate using momentum eigenstates. Is this not the case? And arbitrarily close is all we need, since it seems unfair to expect Tim's model to be describing an exactly disconnected surface since that would require having classical geometry all the way down to the spacetime point level. Or would it just be considered a remnant solution if it's not perfectly disconnected? If so, then it seems like we should have always viewed Tim's solution as basically a remnant solution because of course it's not going to be literally a disconnected surface as you go all the way to the infinitesimal level; if it was, then it would be extremely weird that running time backwards would reconnect them.

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  117. BHG—

    Well, if it were me trying to define what "AdS" is (recall, it is not a theory but a restriction of a theory to a subspace of solutions) I would do it this way: among all full solutions (these are full 4-d solutions) collect together all and only those that have a maximal hypersurface that is both AdS and connected. Whether these include any eigenstates of the surface Hamiltonian is something I cannot prove, however I imagine there will be some. But for sure, if there are any energy eigenstates at all, they are never on disconnected surfaces in this set of solutions for the simple reason that they are static. There would be energy eigenstates in the complete set of solutions that are energy eigenstates such that every maximal hypersurface is disconnected, but those would not show up what I would call AdS. And the AdS set should include solutions where the space-time "disconnects" or "branches".

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  118. Travis,

    I wouldn't say that you can make a disconnected surface out of a superposition of connected ones, but I agree that you can make a connected surface approach a disconnected one by "pinching it". And as you say, if such surfaces are the end result of black hole evaporation then you are in the usual remant scenario. Of course, remnants run into essentially the same conflict with CFT; although they don't imply a strict degeneracy in the energy spectrum, they do imply a huge number of low energy states (essentially an unbounded number, if you suppose that an arbitrarily large black hole radiates down to a remnant with Planck size horizon). Again, the basic physical obstacle one runs into is that by examining the emitted radiation one can "account" for all of the energy, and if upon doing so one is still missing the "information", then it must be that a lot of information can be stored somewhere at almost no energy cost. But in a CFT this is not possible because there are only a finite number of low energy states.

    I should also clarify that at a fundamental level the notion of connected and disconnected components probably has no precise meaning, in that the very idea of geometry is probably an emergent one. To draw a parallel, in a free quantum field theory one can label states in the Hilbert space by how many particles are present. But in an interacting QFT this makes no sense unless the particles are very widely separated. For a localized blob of energy there is no well defined notion of particle number, and so one can go continuously between a 1-particle state and a widely separated 2-particle state. Maybe one could even use this in some way to modify Tim's scenario so as to make it jibe with AdS/CFT. However, I must say that I don't see how this could get around the physical obstacle noted above.

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  119. Physphil—

    I don't see why you are having so much trouble with this. If all you want are predictions, I can use whatever mathematical techniques I like. They need have no physical significance. Nothing follows from what I use.

    If you want a principled solution to the measurement problem, and for me to derive the predictions from that, I can do it without ever taking an inner product.

    You claim that you cannot use projectors and cannot use unitary transformations because "that is not QM". OK, I'll bite. What, in your estimation is QM? I can't make a lick of sense out of this claim. How about explaining what your "measurement theory" is?

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  120. Travis—

    Where is this claim about what you are "able, in principle, to compute" in QM coming from? All of you keep throwing around these supposed requirements without even hinting at a justification. Probabilities in Bohmian mechanics are calculated, in principle, without use of any "observables" or PVMs or POVMs. And there you have a complete solution to the measurement problem. Please explain what you mean by "QM". Does it solve the measurement problem? If so, how? If not, then you don't even have a physical account of how any measurement outcome occurs, much less the probabilities for various possible outcomes. BHG has laid out the rules for a sucker's game, and I refuse to play. If you can justify these rules, please do.

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  121. BHG:

    "Again, the basic physical obstacle one runs into is that by examining the emitted radiation one can "account" for all of the energy, and if upon doing so one is still missing the "information", then it must be that a lot of information can be stored somewhere at almost no energy cost." All of this talk of "accounting for energy" and "low energy cost" is, of course, empty in the situation we are discussing. You have no notion of energy in the bulk, and no conservation law that can be applied there. I assume you know this, but if not I will explain. Think Noether's theorem.

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  122. Physphil

    One more comment. You don't seem to understand what the big deal about the AdS/CFT duality is supposed to be. Just because I go through a certain calculation in the AdS, it does not follow that I make anything like the same calculation for the dual situation in the CFT. That's the whole point: calculations that are hard on one side of the duality can become easy on the other side, and the duality is supposed to guarantee that the results will match. So absolutely nothing follows about the mathematics needed on the CFT side from the mathematics needed to do the dual calculation on the AdS side. The fact that "that hilbert space does not exist or make sense in CFT. It is not part of the CFT." is neither here nor there.

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  123. bhg,

    You still seem to be referring to energy states as if they were units of information. They are not.

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  124. Tim,

    "If all you want are predictions, I can use whatever mathematical techniques I like. They need have no physical significance." I do not agree. But that is not my point. If AdS/CFT is true, the CFT contains everything you need to make all predictions. The CFT does not contain the states you get from quantizing the disconnected internal part. It only contains the states you get from quantizing the external part. So, no disconnected states are necessary to make predictions. But you say you must use those states to calculate probabilities. So, your idea contradicts AdS/CFT.

    "So absolutely nothing follows about the mathematics needed on the CFT side from the mathematics needed to do the dual calculation on the AdS side. The fact that "that hilbert space does not exist or make sense in CFT. It is not part of the CFT." is neither here nor there."

    It is you that does not understand. The CFT is identical to quantum gravity in AdS. They are the same theory. The Hilbert spaces are the same. The states are the same.

    You say the extra states from quantizing internal space are not physical. Now you say you do not need them to calculate anything, you can use the CFT. They are not physical and not necessary to calculate anything. When will you admit the internal space does not exist?

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  125. For this: "You claim that you cannot use projectors and cannot use unitary transformations because "that is not QM". OK, I'll bite. What, in your estimation is QM? I can't make a lick of sense out of this claim. How about explaining what your "measurement theory" is?"

    Again you do not understand me at all. According to you, I cannot use projectors to find the state after measurement because that would leave me in state that does not exist. Also I cannot use unitary transformations acting on the exterior to find the state after measurement because that would also leave me in a state that does not exist. So I cannot use any standard QM treatment of measurement, according to you. QM is a rule for how state changes plus a rule for measurements. You have taken away 50% of that. What is left is no longer QM.

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  126. Tim,

    POVM's only matter if you are going to entangle the system under consideration with some larger system. If you're going to allow the system to always remain isolated (which of course the entire universe is), then you only need PVM's. So any possible probability should be able to be computed by projecting onto some subspace of the physical Hilbert space.

    I agree of course that it's problematic to actually allow the state to collapse to some subspace, which is why what we should have in mind is some superobserver existing outside the entire universe and think about what would hypothetically happen with various probabilities if he were to measure the entire universe, knowing that he never will. I think this is the closest to "standard QM" we can get without bringing the measurement problem into the picture. Keeping this in mind, there is no issue of needing the exterior and the interior to be exactly product states because no such superobserver exists, and once you start talking about actual observers existing *within the spacetime*, then it's much more complicated what the effective state of the universe will look like to that observer, and there's certainly no duality between CFT and the state of the rest of the universe as seen by that observer.

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  127. Bee and Tim,

    About energy/information, let's perform the following experiment (actually repeat it many times). Surround the black hole with a giant spherical particle detector and use it to collect as much of the Hawking radiation as you can. So we are eventually left with an external region that is empty AdS, plus a few stray gravitons etc. that we failed to detect. According to Tim, the emitted radiation is in a mixed state with some large von Neumann entropy S_rad. On the other hand, the full state of the world is supposed to be pure, so there must be some other Hilbert space of dimension at least e^(S_rad) that purifies it. But since we are left with essentially empty AdS these states must have essentially zero energy. Hence the large degeneracy that clashes with the CFT.


    Even at the handwaving level I don't see that Tim has put forth any counterargument to this. Did I miss it?

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  128. Tim,

    "You don't seem to understand what the big deal about the AdS/CFT duality is supposed to be. Just because I go through a certain calculation in the AdS, it does not follow that I make anything like the same calculation for the dual situation in the CFT. That's the whole point: calculations that are hard on one side of the duality can become easy on the other side, and the duality is supposed to guarantee that the results will match. So absolutely nothing follows about the mathematics needed on the CFT side from the mathematics needed to do the dual calculation on the AdS side. The fact that "that hilbert space does not exist or make sense in CFT. It is not part of the CFT." is neither here nor there."

    What you say here is incorrect, and in fact this explains a lot of the underlying confusion. Like other QM dualities, AdS/CFT is an exact equivalence between Hilbert spaces and operator algebras. The prototype is boson-fermion equivalence in 1+1 dimensions: any fermionic operator can be constructed out of bosons and vice versa, and operators that are simple on one side are complicated on the other. So this is the reason why your proposal to compute probabilities using states outside the Hilbert space is incompatible with AdS/CFT. I should also say that notion of equivalence is what is shown explicitly in the low energy sector of the theory; i.e. a collection of weakly interacting particles in AdS is mapped to a specific state in the CFT, and similarly for the operators acting on these states.

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  129. Tim,

    "among all full solutions (these are full 4-d solutions) collect together all and only those that have a maximal hypersurface that is both AdS and connected"

    I appreciate you providing an answer to my question here, but unfortunately the above is a bit too vague, since it sounds like you are thinking of the solutions in classical term. We are dealing with t-dependent quantum wavefunctions that assign amplitudes to various 3-geometries and matter configurations. In these terms, what does it mean to say that a wavefunction "has a maximal hypersurface that is both AdS and connected"? Do you mean that at some specific time t the wavefunction is required to assign some nonzero amplitude to such a surface? Or can this be true for any time? Or does the wavefunction at some (or any?) time need to be strongly peaked about such surfaces so as to approximate a classical state. I really can't tell what you mean here, so I ask that you clarify. If I hand you a wavefunction, what is the precise criterion I am supposed to apply to determine whether it is in your "solution space".

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  130. Phyzphill writes: "According to you, I cannot use projectors to find the state after measurement because that would leave me in state that does not exist. Also I cannot use unitary transformations acting on the exterior to find the state after measurement because that would also leave me in a state that does not exist."

    No, no, a thousand times no! You are asking Tim for his recipe for calculating measurement probabilities, and he has given you that. It is. NOT a recipe for saying what the total state is post-measurement! So it is not a problem for Tim that the "collapsed" statesyou refer to are not in his physical Hilbert space. It's only a problem for those who think real collapse happens, which as far as I can glean from the discussion is only you, Physphill. Tim's proposal may founder for some other reason, but not for the one you keep giving.

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  131. BHG,

    If we all agree that at the fundamental level there is no concept of connected or disconnected, then I think the scenario I am describing is what must be under consideration: that is, we start with all of the possible energy eigenstates that exist on a connected background AdS spacetime, and then our Hilbert space is the space of all states that are superpositions of those energy eigenstates. Then we can ask whether there are *approximately* classical disconnected surfaces, and it seems that there will be.

    One point that I just thought to bring up is this: in the standard explanation of black hole evaporation, the radiation is thermal radiation which contains every energy. So to the extent that black hole evaporation happens according to the standard explanation, a state with an infinite number of energy eigenstates is needed. And in fact, it is not implausible, as I pointed out before, that the more your state resembles a classical solution, the more energy eigenstates you will need in your state. So I think if we interpret Tim's model as being what happens in the limit that the state under consideration approaches an approximately classical state, then everything should be fine.

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  132. Physphil,

    If you have been following this exchange from the beginning, you will recall that I spent quite a bit of time early on asking BHG over and over and over "What exactly does AdS/CFT even claim?". And instead of answering that question right away, he ignored it and ignored it and ignored it, and finally said "AdS/CFT is a work in progress". I can supply the exact citation if required. Since that time I have had to try to carry on this discussion without any clear account of what AdS/CFT even asserts. But one thing sure: BHG can't believe that it asserts that "The CFT is identical to quantum gravity in AdS. They are the same theory. The Hilbert spaces are the same. The states are the same." That is a quite direct and simple claim. BHG, as I infer, does not believe it. And for just the reason I gave—that the calculations of the dual quantities in the two theories are supposed to be done in completely different ways—I don't believe it either. So please provide a citation to some text that makes this assertion. If you want to defend this claim, fine. But be prepared to answer the obvious first questions about how theories set in different dimensional space-times could possibly be "the same theory" in any literal sense.

    You continue to refuse or neglect to answer my questions. I have both asserted and illustrated how standard QM does not require that calculations of probabilities only use mathematical representatives of "physical" states. Just the opposite. So your restriction to "physical" states in the predictive recipe is simply not part of what is usually meant by QM. But you think a) there is such a restriction and b) QMs "measurement theory" abides by the restriction. So yet again: please say, in plain language, what you think QMs "measurement theory" is. Does it include a solution to the measurement problem? If so, which solution? If not (that is , if there is no physical account of the sorts of experiments we call "measurements") then how can it possibly contain some restriction to "physical states"? Just give us a straightforward account of what you mean here, as I can give you a straightforward account of how the measurement problem is solved in Bohmian mechanics and therefore how a completely principled physical account of the predictions goes. That account makes no use of "observables", as is appropriate, and no use of inner products of states.

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  133. bhg,

    I am saying that the states you are referring to merely by energy aren't unique and are indeed largely degenerated. You can use any unitary operation on these states that changes the entanglement between some regions of space and the resulting state will still have the same energy but a different information content. You can do this already for the vacuum, and this degeneracy certainly does not get smaller for excitations. The only way I can see to avoid this is to simply forbid all states that have spatial entanglement, but that would include forbidding Hawking radiation (which kinda solves the problem, arguably, but not in a very interesting way).

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  134. BHG,

    Once again, you are trying to sucker me into a mug's game. I have denied that AdS is a theory. This is obvious. The relevant theory is N =8 supergravity, or some such, which better have piles of non-AdS solutions if it is to have any physical significance at all. When people talk of "AdS" they mean something along the lines of "N = 8 supergravity as it applies to situations where the emergent space-time is AdS". And now you are asking *me* to be perfectly precise about what "where the emergent space-time is AdS" means! Well, how about you go first! I wrote a paper about the general form of an obvious solution to the "Information Loss Paradox". I never so much as mentioned AdS because its not a physically realistic model of the actual world in any case. First you try too attack this general solution by trying to apply it in this non-realistic case, then you refuse to say exactly what the AdS/CFT duality is supposed to be, but we are nonetheless required to take the truth of the unspecified hypothesis for granted, and now you are asking me to be more precise about the exact content of this unspecified hypothesis than you have ever been! How in the world have I been cast in the role of making clear the precise meaning of your unproven and indeed unspecified hypothesis? How about you do the work of clarifying first what "AdS" means, and then clarifying what "duality" means in the sort of precise terms that you are holding me to?

    The irony of this situation cannot be overstated. You attempt to criticize my solution by importing a physically unrealistic and vague unproven hypothesis which we are supposed to accept because some people did some calculations, and then hit me up with the task of clarifying then terms in which your own vague and unproven hypothesis is stated, while discussing physical states (energy eigenstates) that are also completely irrelevant to the physical problem (evaporating black holes) that are at the center of the entire issue! You ask for a "measurement theory" which I am happy to sketch along Bohmian terms while you offer no measurement theory at all. You now talk about the quantum states as functions over *all* of superspace, not just the AdS sector of superspace, which is fine, but then demand that *I* specify exactly which quantum states count as "AdS".

    If you really expect me to answer all this—which has zero bearing on my paper!—have the good graces to first provide a model of answers at an acceptable level of precision by providing your own answers. I have enough on my plate making sure my own statements are clear without being burdened with making yours clear as well. So: what is it that *you* mean by the "Hilbert space of AdS"? Take the very questions you just asked of me in order to make my proposal clear, and answer them. My guess is that I will be happy with whatever you deem acceptable in your own case here. Bon chance.

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  135. BHG

    "According to Tim, the emitted radiation is in a mixed state with some large von Neumann entropy S_rad." I don't recall ever saying anything to that effect. So I'm reasonably certain I didn't. Citation please.

    And please also be prepared to answer quite specific questions about this new thought experiment fo yours. That should be illuminating.

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  136. Bee,

    I described a specific implementation in terms of particle detectors surrounding the black hole. Is your comment meant to refute this? -- I can't tell.

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  137. Tim,

    " "What exactly does AdS/CFT even claim?". And instead of answering that question right away, he ignored it and ignored it and ignored it, and finally said "AdS/CFT is a work in progress"

    OK, let me clarify. AdS/CFT is the statement that CFT and quantum gravity in AdS are the same theory: the same Hilbert space and the same operator algebras. The two sides are just expressed in terms of very different "variables". Of course, for this to be at all nteresting or useful one needs to know something about the dictionary that translates between one set of variables and the other, and that's the part that is a work in progress. In some cases, like for weakly interacting particles in an AdS background, we understand this dictioary well. But when black holes are present our understanding of the dictionary is much more rudimentary.

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  138. Carl3,

    "So it is not a problem for Tim that the "collapsed" states you refer to are not in his physical Hilbert space. It's only a problem for those who think real collapse happens, which as far as I can glean from the discussion is only you, Physphill."

    It is me and >50% of all physicists. But I gave two options and you ignored the other. The other way to treat measurement is part of Hamiltonian evolution, like von Neumann did. An measurement of external particles is a unitary transformation that acts only on exterior state. Just like projection, that unitary produces a state that is not in the set Tim says exists. It is just as big of a problem as projection.

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  139. Tim,

    "But one thing sure: BHG can't believe that it asserts that "The CFT is identical to quantum gravity in AdS. They are the same theory. The Hilbert spaces are the same. The states are the same." That is a quite direct and simple claim. BHG, as I infer, does not believe it."

    You are very sure of many things. This one is wrong also. bhg writes "Like other QM dualities, AdS/CFT is an exact equivalence between Hilbert spaces and operator algebras. The prototype is boson-fermion equivalence in 1+1 dimensions: any fermionic operator can be constructed out of bosons and vice versa, and operators that are simple on one side are complicated on the other."

    "But be prepared to answer the obvious first questions about how theories set in different dimensional space-times could possibly be "the same theory" in any literal sense."

    That is trivial, Kaluza-Klein is one easy example. AdS/CFT is not trivial. But, I only say that if it is true it contradicts your idea.

    "So yet again: please say, in plain language, what you think QMs "measurement theory" is."

    I quoted Landau when you asked this before. We go in circles. State after measurement is projected, or, it is related by a unitary. Probabilities are squares of inner products. No more is needed for this.

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  140. bhg,

    Yes, my comment was a response to your proposed thought experiment, pointing out (as I have done several times previously) that you assume without justification that few energy states imply few ways to encode information. I have told you various times that this just isn't correct. The vacuum state already has a large degeneracy, and adding excitations on it certainly doesn't remove that degeneracy.

    Let me also repeat the best-known example for this (which I had already mentioned ~700 comments earlier) that a unitary operation can remove the entanglement of the Hawking-radiation across the horizon without changing the spectral energy density. Indeed you must do this if you want to avoid a firewall and have the BH entropy. The states before and after the operation are not the same because the entanglement between subsystems is not identical, but they do have the same energy. There is an exponential number of such operations that you can use to entangle or disentangle multi-particle states in all existing subsystems - the numbers quickly pile up with the number of particles and the number of subsystems. Each of these operations will result in a different state, but keep the energy the same. Each of these states has a different information content.

    (I don't know why Tim thinks the out-state isn't mixed in his scenario.)

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  141. Physphill,

    If we have a theory, as we have here, which is purporting to describe the entire universe, do you really think that >50% of physicists believe that it will at some point collapse to a projected state? Or more generally, do you think that >50% of physicists believe that any completely isolated quantum system will ever collapse to a projected state? Unless you are going to try to argue that an entire AdS spacetime is not an isolated system, you can't seriously believe that actual collapse will ever occur, unless you believe in a GRW-type theory which would make the BH info problem irrelevant in the first place.

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  142. Bee,

    " The vacuum state already has a large degeneracy, and adding excitations on it certainly doesn't remove that degeneracy. "


    I am talking about AdS/CFT here. As you doubtless know, the CFT vacuum is unique hence so too is the AdS vacuum -- no degeneracy. If the collected Hawking radiation have some large entropy S, but the full state is pure, it follows that there must be some other Hilbert space of at least dimension e^S that purifies the state. And since the external region is essentially empty after the Hawking radiation has been collected, there thus must be at least e^S states of essentially zero energy, in dramatic contradiction to the CFT. So in fact the Hawking radiation must be in a pure (or nearly pure) state in any AdS/CFT compliant scenario. I am just repeating a classic argument.

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  143. bhg,

    I gave you a concrete example for a transformation. How does AdS/CFT forbid this? I think you should be more precise about what you mean with the vacuum being unique.

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  144. BGH and Physphil—

    Both of you have said things like this:

    "OK, let me clarify. AdS/CFT is the statement that CFT and quantum gravity in AdS are the same theory: the same Hilbert space and the same operator algebras. The two sides are just expressed in terms of very different "variables""

    Now I take it that you both know that all infinite dimensional Hilbert spaces with countable bases are isomorphic, so citing the fact that the Hilbert spaces are isomorphic is, shall we say, less than compelling as evidence that you have "the same theory". What about the isomorphism of the operator algebras? Well, can you specify exactly which operators you are talking about? I have no clear notion about how strong a constraint this is. But I am very, very suspicious of the "just" in "just expressed in terms of very different variables". I can certainly imagine cases where by any reasonable criterion these count as different theories, even though there may be some highly abstract isomorphism between them. Indeed, I have no clear notion at this point of what exactly is supposed to be trivial and what is non-trivial. The isomorphism of the Hilbert spaces is, as I said, trivial. Is the isomorphism of the operator algebras something proven or just conjectured? If proven, then apparently all the real action is in the dictionary, in the way the "variables" of the two theories relate to each other. However they do, it better be non-trivial or else, as I said, the idea that a problem in Ads could be hard to solve but the dual problem in the CFT easy makes no sense.

    In any case, a difference in the variables can mean that these are different theories in any physically reasonable sense. And the absence of any dictionary taking us from states in the CFT to conditions in the middle of the bulk of the AdS means that the "just" in "just different variables" does not indicate a trivial problem.

    Here is a simple question: in what sense is the boundary Hamiltonian in the AdS "dual" to the Hamiltonian in the CFT? How do you know it is dual?

    Here is another question. As I have said, AdS is not a theory, it is rather some sort of a restriction of a more general theory to a sector which counts as the AdS sector. Is the CFT similarly the restriction of a more general theory? Can one state the nature of the restriction in the language of the CFT? Are either of you going to explain exactly what counts as a state being an "AdS" state? These are simple questions that need to be answered to begin to understand what you are claiming.

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  145. Physphil,

    "But I gave two options and you ignored the other. The other way to treat measurement is part of Hamiltonian evolution, like von Neumann did." What are you referring to here? von Neumann's classic presentation is a collapse theory (Process 1 and Process 2), not a theory with a unitary evolution. A collapse theory, as we have been saying over and over, loses information all the time, so there can be no "Information Loss Paradox". And a non-collapse theory is either a "hidden" variables theory or a Many Worlds theory. Which did you have in mind?

    I referred you to Bell concerning Landau and Lifshitz. Did you read it? Once again, the idea that there is some "standard" account fo the measurement problem to be found in all textbooks is a fiction.

    The Kaluza-Klein theory was a theory that postulated a fiber bundle, awkwardly and misleadingly presented as if it were a theory in a 5-dimensional space-time. Are you suggesting that this is how to think of the bulk theory? If so, then is just is not a theory of gravity.

    One more time: the "physical" Hilbert space should, by any reasonable definition, only need to contain states that can arise by physical evolution. No collapses via projectors can, unless you have non-Hamiltonian evolution. I assume you are not advocating that.

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  146. BHG—

    Sabine is perfectly correct. You have said that the AdS is a theory of gravity and perhaps other fields. Entanglement swapping will not change the ADM mass (or whatever the analog is in AdS) of a state. So, as she asserts, there will be a massive degeneracy of the boundary Hamiltonian spectrum.

    And one more time: you yourself said that AdS admitted states on disconnected surfaces, provided that only one of them was AdS and the others were closed. But the state on the closed part can contribute nothing to the ADM (or whatever) mass of the AdS part. So by your own definition, the spectrum of the boundary operator in the AdS part is massively degenerate with respect to the possible physical states of AdS. If that conflicts with AdS/CFT then it does: nothing to do with me! As I keep saying, you seem to be operating under rules that imply the falsity of AdS/CFT, then somehow want to blame that on my solution when it is rather the product of your own rules. If AdS/CFT is inconsistent all by itself then you can't criticize my theory by pointing out that conjoining it with AdS/CFT yields a contradiction! The contradiction was already there.

    As for your spherical detector: can you walk through what the argument is supposed to show more clearly, and in particular why you insist that the experiment has to be repeated? I can't follow at all what you are claiming. What we know is that the *expectation value* of the ADM mass in AdS is massively degenerate in its spectrum. Again, that has nothing to do with me.

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  147. Travis,

    "Unless you are going to try to argue that an entire AdS spacetime is not an isolated system,"

    It is no more "isolated" than any experiment in a physics lab. AdS boundary has infinite volume and infinitely many states and can have big classical detectors or different boundary conditions. The black hole can evaporate in one nanosecond inside a lab or big spherical detector inside Ads. All that can be much smaller than Ads radius.

    And, you ignore the other possibility, that exterior measurements are unitary transformations acting on exterior. That is as bad for Tim because that also gives a state he says does not exist.

    I have never mentioned or even thought of GRW.

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  148. Tim,

    "von Neumann's classic presentation is a collapse theory (Process 1 and Process 2), not a theory with a unitary evolution."

    Von Neumann's process 2 is unitary evolution. To be sure of nomenclature I checked von Neumann's book. It is very simple and clear. You are an expert on QM foundations?

    "A collapse theory, as we have been saying over and over, loses information all the time, so there can be no "Information Loss Paradox"."

    This is false, as I said and said. If true no one could check unitary evolution in any QM system or no QM computer would work. Black hole formation/evaporation is not in principle different. It could take place in one nanosecond in a lab in a vacuum chamber. You just ignore this.

    "And a non-collapse theory is either a "hidden" variables theory or a Many Worlds theory. Which did you have in mind?"

    Process 2 is something like Many Worlds. It is just as bad for you as projection because unitary acting on exterior does not preserve your states.

    "The Kaluza-Klein theory was a theory that postulated a fiber bundle, awkwardly and misleadingly presented as if it were a theory in a 5-dimensional space-time. Are you suggesting that this is how to think of the bulk theory? If so, then is just is not a theory of gravity."

    What? KK showed how a theory in 5D with periodic boundary conditions is exactly same as a theory in 4D with extra states. You thought that was impossible or hard to show. It is not.

    I am tired of this conversation. I do not think it goes anywhere. bhg maybe has more patience with you.

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  149. Physphill,

    We have to consider the entire AdS spacetime as being our quantum system since that is what is dual to the CFT. If you look at only some small portion of it within a lab or use a big classical detector then it will not be dual to a CFT and hence no arguments here can have any force. The entire AdS spacetime is indeed more isolated than any experiment in any physics lab has ever been. It is trivially true that the only unitary operation that will ever be performed on it is time evolution since measurement is something that must be imposed from outside a system, hence an isolated system will never be measured.

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  150. Tim, physphill,

    Not sure what KK has to do with that but the whole point of the construction is that Lie-Algebras have manifold-structures, hence the U(1) fibre-bundle can be realized as an extra-dimension. Indeed you can do the same thing with the rest of the standard model using the canonical metric induced by the Lie-brackets on SU(2) and SU(3). It's interesting until you notice it doesn't work for the fermions.

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  151. Physphil,

    I said "von Neumann's classic presentation is a collapse theory (Process 1 and Process 2), not a theory with a unitary evolution." and you respond

    "Von Neumann's process 2 is unitary evolution. To be sure of nomenclature I checked von Neumann's book. It is very simple and clear. You are an expert on QM foundations?"

    Yes, and Process 1 is a collapse! This is why I wish you were posting under your real name. Maybe you would be a little more careful not to write complete nonsense and finish it off with an insult. Of course "with a unitary evolution", in that context, means "with *only* a unitary evolution" as anyone paying any attention and not just trying to be obnoxious could see. You evidently don't even know what the phrase "a collapse theory" means, and you are so set on being dismissive of someone who knows a lot more than you do that you grasp at the nearest straw, only to embarrass yourself. If you are not going to be serious, please do stop posting. We can certainly now all infer and remember that physphil, whoever that is, does not know the meaning of "collapse theory" well enough to recognize that the single most famous and clear collapse theory in the history of QM is one.

    Every "point" in your post above is just as bad as this one, and I won't waste more of my time and bore everyone else (who I imagine sees this just as well) going through all of them. I cannot recall ever seeing such a display of manifest error put forward as a show of knowledge. Your ego is engaged in this in a way that makes the entire exchange tedious, pointless and annoying. Look back over what you wrote, and if you really cannot sees what complete nonsense it is then either go do some studying or be prepared to have the rest of your claims above thoroughly debunked. The only patience I have left for you is to show in more detail how ignorant you are about the matters you think you know something about.

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  152. Sabine:

    I'm not sure that we are referring to the same thing by "KK". The original paper was presented as introducing a 5-dimensional space-time with one compactified dimension, but when you look at the details the whole thing is done in a coordinate-dependent language and the allowable coordinate systems are restricted in a way that makes no sense for a truly 5-dimensional space-time. In particular, the coordinate curves of the fifth coordinate are restricted so that the have to be the same in all coordinate systems, or in other words there is a preferred foliation of the fifth dimension. From a modern perspective, this is an awkward way to indicate a fiber bundle with a U(1) fiber associated to each point in at 4-dimensional base space, not a way to produce a theory on a 5-dimensional space-time manifold. So that's what KK actually did. There may for some other construction called "Kaluza-Klein theory" in contemporary usage that is not the theory one finds in the original papers. If not, then I think people are being sloppy: there is a fundamental difference between a fiber bundle with an N-dimensional fiber over an M-dimensional space-time manifold and a N + M dimensional space-time manifold.

    I'm not surprised it does not work for fermions. The anti-commutation relations would kill it.

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  153. Bee and Tim,

    I am simply not seeing how Bee's comments are pertinent to my argument. She is talking about correlations across the horizon, whereas I am describing a physical experiment that doesn't even make reference to a horizon. Once again, in a bit more detail:

    Consider a radiating object, formed from a pure state, in the center of AdS, perhaps a star or perhaps a black hole. At some very large distance, we surround the object with a spherical particle detector and we measure the energies, spins, charges etc of all radiated particles. If we repeat this experiment many times, starting from the same initial state, we can reconstruct the density matrix \rho_rad of the detected particles. I again stress that we are considering these particles very from the location of the radiating object. Now we can compute the von Neumann entropy of this density matrix, S_rad = - \Tr(\rho_rad ln(\rho_rad)). In the case of a star we expect to get 0, while for a black hole in the info loss scenario we get a macroscopically large S_rad. Now, on the one hand, after the radiation has stopped what we see is essentially empty AdS. On the other hand the claim is that the total system is in a pure state, with \rho_rad arising from having traced out some set of states from the full Hilbert space. It a math theorem that these other states must span a space of dimensionality at least e^(S_rad) if they are to purify the density matrix as claimed. But the energy has already all been accounted for in the sense that \Tr (\rho_rad H_rad) = E, where E is the initial (presumably conserved) energy. So how are you going to avoid the conclusion that in order to purify the full state you need to introduce e^S states of zero energy (or of arbitrarily small energy if we allow for some measurement error)? That is the question.

    To put it in even simpler terms, suppose I tell you I have a pure quantum state whose energy is sharply peaked around some value E. The system is confined to two well separated boxes. In one box there are a bunch of particles described by a density matrix \rho_1 whose von Neumann entropy S_1 is huge and for which \Tr \rho_1 H_1 \approx E. It follows mathematically that \rho_2 must obey S_2 =S_1 and \Tr (\rho_2 H_2) \approx 0. So box 2 has a huge entropy but almost no energy.

    The disconnected surface scenario does of course allow for a pure state of the total system, but it does so precisely by using the fact that the state on the disconnected surface have zero energy as measured at the AdS boundary.

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  154. Tim,

    "And one more time: you yourself said that AdS admitted states on disconnected surfaces, provided that only one of them was AdS and the others were closed. But the state on the closed part can contribute nothing to the ADM (or whatever) mass of the AdS part. So by your own definition, the spectrum of the boundary operator in the AdS part is massively degenerate with respect to the possible physical states of AdS."


    What I have said, at least 10 times, is that even if the full bulk Hilbert space contains such states, if surfaces never split then there is smaller Hilbert space built on the connected surfaces alone. And there is then a self-consistent QG theory describing just these connected surfaces, and that is what AdS/CFT describes. So once more with feeling; the situation appears to be that the CFT is dual to a quantum theory of gravity in AdS that has only connected surfaces. The isomorphism between Hilbert spaces and operator algebras is between those two theories, not the larger bulk theory containing disconnected surfaces, if such a thing indeed exists. The other option, which is what we have been discussing all this time, is whether it could be possible that the CFT is dual to a bulk theory that has disconnected surfaces. I have argued "no" because of the energy degeneracy. So that is where things stand, and I ask that you at accurately represent this.

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  155. TIm,

    "Here is a simple question: in what sense is the boundary Hamiltonian in the AdS "dual" to the Hamiltonian in the CFT? How do you know it is dual?

    Here is another question. As I have said, AdS is not a theory, it is rather some sort of a restriction of a more general theory to a sector which counts as the AdS sector. Is the CFT similarly the restriction of a more general theory? Can one state the nature of the restriction in the language of the CFT? Are either of you going to explain exactly what counts as a state being an "AdS" state? These are simple questions that need to be answered to begin to understand what you are claiming."



    If you are genuinely interested in these issues I ask that you do some reading and then come back with questions. I am not inclined to present an introduction to AdS/CFT via blog comments. One place to look could be the nice recent lecture by Harlow, TASI Lectures on the Emergence of the Bulk in AdS/CFT. arXiv:1802.01040



    I can assure your that you are not going to find any trivial inconsistencies along the lines of saying that the duality can't work because the spaces have different dimensionalities. This is the kind of issue that you can easily resolve for yourself by working out the very simplest examples corresponding to a free scalar field in AdS.

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  156. Tim,

    "! I wrote a paper about the general form of an obvious solution to the "Information Loss Paradox". I never so much as mentioned AdS because its not a physically realistic model of the actual world in any case"


    This is getting pretty ridiculous, and I am about to give up. Now you are mischaracterizing your own published work. Above you say that you "never so much as mentioned AdS", yet p. 23 of your paper contains a section called "AdS/CFT and superscattering" where you do discuss AdS/CFT. Do you not know what is in your own paper?


    I am being very patient because I think that there is still a chance of making progress, but comments like this make me pretty pessimistic.

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  157. travis,

    "We have to consider the entire AdS spacetime as being our quantum system since that is what is dual to the CFT. If you look at only some small portion of it within a lab or use a big classical detector then it will not be dual to a CFT and hence no arguments here can have any force. The entire AdS spacetime is indeed more isolated than any experiment in any physics lab has ever been. It is trivially true that the only unitary operation that will ever be performed on it is time evolution since measurement is something that must be imposed from outside a system, hence an isolated system will never be measured."

    You can do anything you want to the CFT. You can make the Hamiltonian time dependent exactly how von Neumann described unitary measurement (process 2), acting on Hawking particles very near the boundary. Or you can eliminate all Hawking particles at the boundary by making the boundary conditions absorbing. You really can do any kind of measurement that could be done in lab. AdS is a very good box to hold black holes and experiment them.

    If you make boundary conditions absorbing the black hole will evaporate and energy is lost and you get back to close to the ground state of the exterior. But this does not affect the interior at all because it is disconnected. So, these final states would have a large degeneracy because the interior state can be anything. That degeneracy does not exist in the CFT. Like bhg says this is a simple way to see Tim's idea is not consistent with AdS/CFT.

    These are old, standard arguments. They are well known, not new here. AdS/CFT is not consistent with baby universe or remnants.

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  158. Sabine,

    about KK yes, I was giving simple example where some theory in 5D = another theory in 4D.

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  159. bhg,

    We're talking past each other. You are claiming that a state of distinct energy cannot have a large degeneracy of states. I am asking how do you justify this. In response you ask

    "So how are you going to avoid the conclusion that in order to purify the full state you need to introduce e^S states of zero energy"

    But I am not "avoiding the conclusions," I am asking how do *you* want to avoid these states given that you can change entanglement without affecting energy, something that indeed you must do in order to get the out-state to be pure because the Hawking-radiation ordinarily has a cross-horizon entanglement. I am just saying there's more than one way to purify a state and as a consequence you'll be left with an exponentially large number of states of almost zero energy (you better stop before reaching the Planck scale).

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  160. BHG,

    OK, so if I can respond to your earlier question with exactly as much specificity as you have employed yourself: a full (4-dimensional) solution to the QG theory counts as "AdS" if there is at least one connected maximal hypersurface such that the wavefunction support is on that hypersurface and the hypersurface is asymptotically AdS. All and only the states on maximal hypersurfaces through such solutions are in the "physical Hilbert space". If the maximal hyoersurfaces can split, as I propose, then some maximal hypersurfaces are disconnected, then there are both connected and disconnected hypersurfaces in the physical Hilbert space. And if the Hamiltonian applied to the states on connected maximal hypersurfaces is not massively degenerate, then neither is it on the full set of states, both connected and disconnected. So there is no conflict with the CFT.

    The disconnected surfaces provide for states that purify the global wavefunction. These are all states of zero ADM mass—obviously!—because they are not even asymptotically AdS. But the states on the disconnected parts alone are not in the physical Hilbert space. Obviously, again. So there is no argument to any degeneracy in the boundary Hamiltonian.

    Now: can you at least specify which operators are in the operator algebras that are supposed to be isomorphic according to the duality? As I said, saying that the Hilbert spaces are isomorphic is basically empty.

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  161. BHG

    By the way, all this stuff about building a spherical detector, etc is really just a distraction. If you want to talk about the von Neumann entropy of a subsystem just do that. Of course the global state is always pure, and its von Neumann entropy is zero. And if you are talking about a state on a disconnected maximal hypersurface, there will typically be entanglement between the two surfaces, and the von Neumann entropies of the reduced states on each connected component will be high. All of this can be said without reference to any detectors or repeated measurements of identically prepared states, etc.

    But so what?

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  162. Tim,

    It seems really off topic, but on the issue of KK. I might misremember this, but I believe Kaluza didn't originally even compactify the coordinate. Having said this, of course if you assume a direct product with a compactified space you are not dealing with a general 5-d manifold any longer. (Same thing of course for AdS/CFT, which has a direct product attached that no one ever mentions.) And yes, the radius in general isn't stable. There are many reasons this idea wasn't further pursued. All I am saying is that the originally idea hinges on making the Lie group as a manifold part of the actual physical space.

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  163. Sabine,

    Yes, it's too off topic. But just to ask, part of what I am concerned about are Penrose's arguments about the mismatch between the AdS and the CFT as he measures the sizes of the function spaces. Are you familiar with this? Do you have any opinion?

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  164. Tim,

    I have heard him make various arguments. One was about the number of degrees of freedom, but that ended up comparing infinity with infinity and didn't make much sense to me. Another one was about the possibility of exciting extra dimensions by spreading out the necessary energy over large areas in weak background fields, and at least I haven't been able to find a good explanation for why that shouldn't be possible (neither, for that matter, did I find anything in the literature). In a certain twist of irony, Penrose's argument is similar to Susskind's argument against black hole remnants, just that Penrose's doesn't suffer from the same problem as Susskind's. In any case, which argument did you have in mind?

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  165. Sabine

    Yes, the one about degrees of freedom. It is about comparing infinities, but then so was Cantor's work about comparing infinities. Before Cantor, if someone asked whether there were more integers or reals, you might well say: infinity is infinity, what do you mean? But Cantor showed a way to make sense of the question in terms of infinite cardinals through bijections. Of course, in terms of cardinality, there are as many points in a line segment as in a square or a cube, which also seems odd. That's because cardinality is insensitive to questions of continuity, but degrees of freedom are not. And despite having the same cardinality, the segment and square and cube differ in dimensionality, which shows up when you want to cover them with coordinates, for example. In other words, what if we require not just that a bijection exist between "equally big" infinities, but that it be a continuous function? For example, coordinate functions are required to be continuous....if they weren't, then you could coordinatize the segment, square and cube all with a single real coordinate. Penrose's notation is keeping track of that sort of thing. I don't know how rigorous it is, but intuitively what Penrose is saying seems perfectly correct. And to have a real isomorphism of structure, the degrees of freedom in two theories should match in this way. Anyway, I think what his notation is getting at is clear enough. Think of it as keeping track of the dimensionality of the space of degrees of freedom rather than just the cardinality of that space, which means putting a topology on the space of degrees of freedom. Anyway, I can follow what he's saying to that extent, but it sounds like it did not make any sense to you.

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  166. Bee,

    Yes, we do seem to be talking past each other, and I still don't get your point. I am saying that if the entropy of the emitted radiation, collected very far from the black hole, is large, but the full state is pure, then there must be a lot of zero energy states in the theory. The CFT has no such zero energy states, so this scenario cannot occur in AdS/CFT. In AdS/CFT the emitted radiation will be in a pure state on its own, so no zero energy states are implied. I am failing to see the relevance of your unitary transformation. My argument here is of course not novel -- it is the standard one in the literature that you have presumable seen many times.

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  167. Bee,

    Yes, we do seem to be talking past each other, and I still don't get your point. I am saying that if the entropy of the emitted radiation, collected very far from the black hole, is large, but the full state is pure, then there must be a lot of zero energy states in the theory. The CFT has no such zero energy states, so this scenario cannot occur in AdS/CFT. In AdS/CFT the emitted radiation will be in a pure state on its own, so no zero energy states are implied. I am failing to see the relevance of your unitary transformation. My argument here is of course not novel -- it is the standard one in the literature that you have presumable seen many times.

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  168. Tim,

    "All and only the states on maximal hypersurfaces through such solutions are in the "physical Hilbert space"


    Good, this is taking us in the right direction, and let's see if we can develop this in a constructive manner. Here are the rules as I see it. On the one hand we have the CFT which obeys certain non-negotiable rules; a) it obeys the usual rules of QM as stated by Weinberg b) it has unitary time evolution c) the energy spectrum is non-degenerate (up to symmetries). Now, we imagine "rewriting" this theory in "bulk variables", and we will then get some theory of gravity in AdS that obeys the rules (a,b,c), since we are just rewriting the theory. In perturbation theory around AdS we can carry this out very explicitly. A bulk field operator \phi(x), where x is some bulk point is expressed in CFT variables as a single trace operator smeared against the HKLL kernel. This then gets correcter order by order in the 1/N expansion, and this way one completely explicitly builds up the bulk gravity theory. If you go through this you will see how Penrose's argument is resolved, again in a completely explicit manner. A lot of smart people have put in a lot of work over the years to develop this formalism, which you dismiss with comments like "...vague unproven hypothesis which we are supposed to accept because some people did some calculations," I suggest you stop and think before writing such things.


    Now, how can the bulk theory obey the rules (a,b,c)? I claim that if the bulk Hilbert space consists of wavefunctions that have support only on connected surfaces, and have finite norm, and are asymptotically AdS, then we will get a QM theory that obeys the rules. And indeed we can verify this in perturbation theory as I said above. So if this is in fact the correct way that the CFT gets rewritten in bulk variables, then it follows that surfaces never split. The fuzzball/firewall scenario gives a proposal for what happens when matter collapses. Note that my rules for the bulk theory are in accord with all the Weinberg QM axioms, and the only real point of contention is that the wavefunction has zero support on disconnected surfaces. It is likely possible to embed this bulk theory in a larger theory that includes disconnected surfaces, but this is irrelevant, since the CFT maps to the sector with a single connected component. I hope that answers all your questions.


    Let's now turn to your proposal. Please confirm that the following is correct. Suppose I hand you a wavefunction at some time t. The wavefunction has support on a disconnected surface, one component of which is a "compact" (I think you know what I mean here) internal surface, and the other is asymptotically Ads. I ask you if this wavefunction is in the physical Hilbert space. If I understand you, your rule is that you should time evolve the state back to some specific time t_0, and then ask whether the wavefunction has support only on connected asymptotically AdS surfaces. If yes, the state is in the physical Hilbert space. If no, then it is not in the Hilbert space. Is that the rule? If you can't answer this clearly then all you have is a "vague unproven hypothesis". I hope you take the following comment seriously: if you cannot explain how your proposal obeys the rules (a,b,c) then your proposal will be completely ignored by physicists, and for good reason.

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  169. travis said: We have to consider the entire AdS spacetime as being our quantum system since that is what is dual to the CFT. If you look at only some small portion of it within a lab or use a big classical detector then it will not be dual to a CFT and hence no arguments here can have any force.

    Are you saying that any measurement performed within the AdS destroys the AdS/CFT duality, or that the duality can tell us nothing about such a measurement? If so, of what use is the duality? Surely the duality tells us something about measurements that can actually be performed within the AdS.

    bhg said: Consider a radiating object, formed from a pure state, in the center of AdS, perhaps a star or perhaps a black hole. At some very large distance, we surround the object with a spherical particle detector and we measure the energies, spins, charges etc of all radiated particles. If we repeat this experiment many times, starting from the same initial state, we can reconstruct the density matrix \rho_rad of the detected particles…

    I appreciate this attempt to make connection with experiments that could, at least in principle, be performed. Superficially it might seem to conflict with the comment from travis quoted above, but perhaps travis is referring to a single measurement, whereas bhg is envisioning a series of measurements “repeated many times” from “the same initial state”. One question is: How many times would we need to perform this experiment to accurately reconstruct the density matrix of the detected particles? Would it require some unimaginably large number of repetitions? Another question is: Are all these repetitions performed within a single AdS universe, in which multiple black holes are created and evaporate, and measurements are made on each black hole in turn within this single AdS, or do we need a new AdS universe (with the same initial state) for each trial? These may be dumb questions, but I’m trying to understand what is necessary to preserve the AdS/CFT duality in the light of travis’ comment that if we make a measurement on only part of the AdS it will no longer be dual to the CFT.

    I also see tm’s comment:

    All this stuff about building a spherical detector, etc is really just a distraction. If you want to talk about the von Neumann entropy of a subsystem just do that… All of this can be said without reference to any detectors or repeated measurements of identically prepared states, etc.

    I agree that we can talk about an unmeasured unitarily evolving wave function of the universe, and subsystems within it, but for empirical physics don't we ultimately need to connect this to some measurements that we can, at least in principle, perform (or observations we can make)? If we can’t do this, even in principle, then are we really talking about science? I think it's worthwhile to try to understand bhg's proposed measurements, and whether the AdS/CFT duality (and the information paradox itself) still applies in the context of those measurements being performed.

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  170. BHG—

    Regarding what you wrote. If you are perturbing around the vacuum, which I assume is what you meant, then you just won't get any states with black holes, and a fortiori with evaporating black holes, in the AdS. So the calculation no more supports not splitting than it does splitting. The perturbative results will be identical on both. So unless you are doing something else, this is no proof of anything. You can't figure out what will happen when a black hole evaporates without, well, figuring out what happens when a black hole evaporates.

    Your argument above first says "So if this is in fact the correct way that the CFT gets rewritten in bulk variables, then it follows that surfaces never split." Well yes, but that "if" is the nub of the matter isn't it? Then a few lines later you write "but this is irrelevant, since the CFT maps to the sector with a single connected component". But you never so much as tried to prove that: you just said "if it is the right way...". Your conjecture magically becomes a theorem without an intervening proof.

    A little more explicitly: you say that you can write some theory that obeys Weinberg's rules that lives on only the connected surfaces. Then you say that any theory that obeys Weinburg's rules lives on only the connected surfaces. The second does not in the least follow from the first.

    As for the way I put my theory, I certainly never made any mention of a "specific time t_0". Indeed, in the bulk, because it is GR, there is no canonical way to make sense of the "state at a given time", because of the diffeomorphism invariance. You don't "time evolve backward" in the bulk as you do with the surface term and with the CFT: in the bulk the time evolution is "pure gauge" and is implemented by the Hamiltonian constraint, not by a Schrödinger equation. The right way to think is in terms of full 4-dimensional solutions. And we are restricting the physical space to 4-dimensional solutions that are connected: no pairs of disconnected space-times allowed. Given such a complete 4-D solution, it belongs in AdS if it has a maximal hypersurface that is connected and asymptotically AdS. But there is no canonical way to assign a universal time to that hypersurface.

    I take it that that is perfectly clear.

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  171. Amos

    In any meaningful sense, BHG's experiment cannot be performed. To do what he wants (which is tomography) he requires doing exactly the same experiment *on precisely the same initial state* an unbounded number of times, and with the "detector" set in infinitely many different ways. Because he would somehow have to recover the exact statistics for *all possible* global measurements that can be done. So there is just no serious sense in which this is physically possible. If that's what you require for "science" then this isn't science. I myself think that your requirement is too strict. It is a decent question whether there was anything that preceded the Big Bang, but no experiment we can do can directly check that. What we are investigating here is how various general features of QM (particularly unitarity) can be implemented in an evaporating black hole scenario. That is a clear question to ask that may or may not lead to any empirically verifiable results.

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  172. bhg,

    As you keep saying the zero energy state is unique up to symmetry transformations by which I assume you mean unitary operations. All I am saying is that unitary operations aren't necessarily passive, but can be active and in many cases result in globally different states because they can and do change entanglement across boundaries of many-particle subsystems.

    Hawking radiation is normally entangled across the horizon. If you want to remove the entanglement to make the outgoing state pure you have to make sure to keep the spectral energy density the same, otherwise you create a firewall. But there are many ways of doing this because just fixing the spectral energy density doesn't tell you anything about the entanglement.

    Now please note that just because you have removed the across-horizon entanglement doesn't mean there is nothing behind the horizon. Of course you still have in-modes, it's just that these will no longer mix with the out-modes, hence no problem with a mixed states. But you can then of course also make a large number of entanglement swaps *inside the horizon* which will have no effect whatsoever on what goes on outside the horizon. None of these transformations will do anything to the energy.

    Hence, I am saying that equating one energy state with one unit of information doesn't make sense and you should be more specific what you are referring to.

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  173. Amos,

    I am saying that no "measurement" made within the AdS can possibly cause the universal wave function to collapse to a state different from what the time evolution would predict (I think everyone here except possibly Physphill agrees with this). If we want to talk about measurements performed within the AdS, then we first have to separate the universal wave function into an observer portion and an observed portion and obtain an effective wave function for the observed portion which will collapse upon suitable interaction with the observer portion. This effective wave function is not dual to any CFT. BHG's spherical detector would have to be set up within the AdS spacetime, so the quantum system that it would measure would be only a subsystem of the full system that is dual to the CFT, and the detector itself would be part of that full system, which greatly complicates things. I think BHG is hoping that the effective wave function of that subsystem will be close enough to the wave function of the full system for all intents and purposes, but that's a hope that needs to be justified somehow.

    You ask:

    "I agree that we can talk about an unmeasured unitarily evolving wave function of the universe, and subsystems within it, but for empirical physics don't we ultimately need to connect this to some measurements that we can, at least in principle, perform (or observations we can make)? If we can’t do this, even in principle, then are we really talking about science?"

    I mostly agree with your point here, but making the connections to actual measurements that *we* can perform, rather than some impossible god's eye observer, can often be difficult and it doesn't become not science just because we're taking the implications of a theory seriously outside of regimes that we could hope to measure. I know you used the qualifier "in principle", but it's unclear exactly what "in principle" means; if it turns out for instance that Bohmian mechanics is the best way to solve the measurement problem, there would still be a sense in which "in principle" we can't measure the positions of the particles in that theory and yet it wouldn't make the theory unscientific. In the case at hand, the question of whether or not *we* can verify what the universal wave function is like is a separate question from whether the structure of that universal wave function clashes with the structure of AdS/CFT.

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  174. Bee,

    I still don't get your point - sorry. I am intentionally phrasing everything in terms of observations far from the black hole, precisely so I can avoid any mention of the horizon (or lack thereof). I am then making a mathematical claim about the density matrix. As for my statement that there are no zero energy states in the CFT, as I am sure you know, in a CFT there is a unique vacuum state. What is your bottom line claim: are you claiming that having the emitted radiation be in a high entropy density, combined with having the full state of the system be pure, does not imply a large number of zero (or very low) energy states in the full spectrum? If so, I would really like to understand more about how you arrive at this conclusion.

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  175. Tim,

    The only time I heard about this was in a public lecture that Penrose gave and I was left with a rather fuzzy idea of what he's on about. Sure in some cases you can compare infinities with infinites but it requires effort. I don't see why the cardinalities should differ here.

    Leaving aside the issue of infinities, I don't see why you'd even want the both function spaces to be the same. As I have noted earlier AdS/CFT does select a subsection of functions in the bulk and the remainder isn't there qua assumption. Concretely, any function that cannot be expanded around the boundary or doesn't have a radius of convergence that covers the whole bulk can't be in the gravitational theory because that's how you defined the gravitational theory. If you eg do anything behind a horizon that's disconnected from the boundary, and remains disconnected for all times, the boundary theory cannot know anything about it. For all I can tell the assumption is simply that this cannot happen, that there must be some non-local correlation to prevent this or something.

    As I keep saying, I think it's consistent, but I am not very convinced it's physically meaningful. You're basically putting in what you want to prove already.

    While I am at it, let me add that you don't need AdS/CFT to achieve this. You could simply take ordinary GR and postulate that all functions must be analytic on the whole manifold. Then you can expand anything around any point and get the information about the whole bulk and voila no information can get lost. (Note that in the derivation of Hawking radiations you use functions that have support only in part of the space and are assumed to be identical to zero in or outside the horizon, which is why you can lose their information for good.)

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  176. Tim,

    The first three paragraphs of your message starting with "Regarding what you wrote.." are spot on, and I agree entirely. The point about perturbation theory is that this establishes that the CFT is able to correctly describe "low energy gravity", i.e. stars and planets, bending of light, gravitational redshifts, etc. And it does this in a framework that is mathematically well defined and obeys the axioms of QM. No other QG theory comes anywhere near this. This being the case, a big question is what does it predict for black holes? You are completely right that here is where we run into the boundary of our knowledge. My claim about surfaces not splitting is not based on any rigorous computation, but is rather mostly arrived at by exclusion, since I can't see how any scenario with splitting surfaces is compatible with the CFT properties, namely the non-degenerate energy spectrum. My whole reason for pressing you on details is that I am very interested to know if I am missing something here -- can it possibly be that the CFT is compatible with a scenario like yours? My scenario is problematic in the sense that it inevitably leads to something like a firewall scenario, which is a radical claim about the dynamics for which there is no independent evidence. I am emphatically not claiming to have a fully satisfactory picture of what's going on.


    Next you write

    "As for the way I put my theory, I certainly never made any mention of a "specific time t_0". Indeed, in the bulk, because it is GR, there is no canonical way to make sense of the "state at a given time" "


    Here I am speaking of a boundary time. In GR there is most certainly a canonical way to assign a boundary time; that is the whole point of Regge-Teitelboim. Namely, the boundary Hamiltonian is the uniquely defined generator of boundary time translations. This being the case, I need to press you a bit more. Is the claim that a wavefunction needs to have support only on connected surfaces at some specific time t_0 in order to be in the Hilbert space, or is it allowed to satisfy this criterion for any time? I actually see problems with both options, but I will wait for your response.

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  177. bhg,

    "are you claiming that having the emitted radiation be in a high entropy density, combined with having the full state of the system be pure, does not imply a large number of zero (or very low) energy states in the full spectrum"

    No, I am asking how you want to prevent the existence of this large number of low energy states given that there are unitary operations in the bulk that will generate them. These are local transformations, they are energy-preserving, and they are unitary *on the full state*. You can use them to create or remove subsystem entanglement. Since you can store information in the presence or absence of this entanglement, I count these as distinct microstates. Yet they do have the same energy. I am asking where do you account for these states? Are you claiming there is some reason you cannot create them? Are you claiming the CFT doesn't distinguish between them?

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  178. BHG—

    Yes, of course there is a time function at the boundary: you basically stipulate that the boundary region will have a (unique?) timelike Killing field. So you can use that to define a foliation at the boundary and assign the leaves a t value. But the point is that whether the complete maximal hypersurface is connected or disconnected is not determined by how it meets the boundary. There can be maximal hypersurfaces that overlap at the boundary but diverge in the bulk in such a way that one is connected and the other disconnected. In the standard Penrose diagram (non-AdS) this is obvious since all the Cauchy surfaces asymptote to spacelike infinity.

    The point is that there is no timelike Killing field in the bulk, and you have a free hand to specify the maximal hypersurface there. That's why the Hamiltonian in the bulk is implemented by a constraint rather than by a Schrödinger equation. And all of the various maximal hypersurfaces that agree at the boundary, and so are assigned the same boundary time, are regarded as "gauge equivalent": even if some are connected and some are disconnected! The easy way to understand what is going on is to think 4-dimensionally rather than 3+1 dimensionally in the bulk: a solution is the whole connected 4-dimensional space time. and a solution "at a boundary time" is the whole 4-dimensional Wheeler-Dewitt patch of events that are not time-like separated from the leaf at the boundary.

    In AdS, if you specify the boundary conditions, then there will be a set of full 4-dimensional solutions, and the folium of 3-dimensional physical states are all of the states relative to maximal hypersurfaces through these solutions. Some will be connected, others disconnected. And even having specified a boundary time, at least for some times, some will be connected and some disconnected. This is true at all times in the standard Penrose diagram, but in AdS early enough times will have all connected states, late enough time will have all disconnected states, and middle times (where the evaporation event lies in the Wheeler-DeWitt patch) will have both connected and disconnected states.

    Is it clear why that is so?

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  179. Tim,

    Unfortunately your message does not contain an answer to my question. Once more: if I hand you a wavefunction \psi[g_ij,t) at some specific boundary time t, what do I have to do to check whether it is in the physical Hilbert space according to your rules? I am looking for an answer at the same level of specificity of my answer, which is the following. The only rule which I think is compatible with the CFT is that one should demand that \psi has support only on a a single asymptotically AdS connected surfaces, and have finite norm and finite energy. That's it. This proposal is of course only self-consistent if surfaces never split under time evolution, and it can and should be questioned on those grounds. But it is a definite proposal! What is your answer? Please avoid ill-defined phrases like "the solution should contain ... ". without defining what you mean.

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  180. Bee,

    "These are local transformations, they are energy-preserving, and they are unitary *on the full state*. You can use them to create or remove subsystem entanglement. "

    How can local transformation change subsystem entanglement? I do not think that is possible. Partial trace of a state is not changed unless unitary is not local.

    Why would any unitary like that be energy preserving? I do not think it is. For example Rindler vacuum has zero entanglement across the horizon but it has singularity in quantum stress tensor.

    For vacuum state, it is unique in the CFT so if AdS/CFT is true it must be unique in gravity also. I think you are asking how that is consistent. CFT vacuum is dual to empty AdS. True, that can have a horizon in some coordinates. Then you can act with your unitary. But that unitary will increase the energy.

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  181. Bee,

    My reading of your message is that you agree with my original point: if (as in Tim's scenario) the emitted radiation is in a highly mixed state but the full state is pure, then the theory must contain many zero energy states, in contradiction to what we know about the CFT. So such a scenario cannot arise in AdS/CFT. Do you agree?

    You are then arguing (I think) that these considerations also rule out other scenarios where the emitted radiation is in a pure state. Sure, if you can find a unitary operator that takes an energy eigenstate to an orthogonal energy eigenstate with the same energy, then this clashes with AdS/CFT. But you'll have to explain in more detail what this unitary operator is if you want me to comment on it.

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  182. physphil,

    I don't know how it matters, but what I meant by local is that you can assume the transformation acts on a time-like curve. It's a boundary-condition, basically. Look at 't Hooft's recent papers for an example. Not sure what your remark about Rindler space is supposed to say. Yes there are states that are pathological for this or that reason.

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  183. bhg,

    I am asking what a change in subsystem entanglement in the bulk does to the state on the boundary and why I am either a) not allowed to make such a transformation or b) why these states are identical in information content (as you seem to claim). I can't see it. If the transformation is allowed, they induce, for all I can tell, a large degeneracy of vacuum states (or at least low energy states). If they are not allowed, what forbids them? As I have mentioned above, you need these transformations to get rid of the firewall to begin with, so it's not like you can just assume them away.

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  184. Bee,

    This is too abstract for me to make sense of. Can you provide a specific and concrete realization of what you have in mind? Also, you didn't answer whether you agree that my argument shows that Tim's scenario is incompatible with AdS/CFT.

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  185. Bee,

    I looked at some of 't Hoofts papers but did not see what you refer to. Have you a more specific reference? Local unitary operations can not change entanglement (it is because entanglement is not local).

    Rindler example shows that removing entanglement (with a non local unitary) does not preserve the energy. Instead, it creates a pathological state. All or most states with little entanglement across the horizon are similar.

    "If the transformation is allowed, they induce, for all I can tell, a large degeneracy of vacuum states (or at least low energy states)."

    Again, why do you think the transformations preserve the energy? They do not for my Rindler example. I think they will change the energy generally. Why would they not?

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  186. bhg,

    An example is the operation discussed in this paper. I do not think the rest of the paper makes much sense though. I used a similar example in this paper. See section 4.1 for a very simple example. 't Hooft uses a similar idea in this paper. In both 't Hooft's and my paper the transformation is unitary, local, and preserves stress-energy.

    It's the preservation of stress-energy that ensures there is no firewall. Hence my question, if these transformations do exist and you can use them (indeed must use them) to shovel around subsystem entanglement without ruining unitarity or changing the total energy, then how can it be that there is only one state of a given energy. If you insist on this you have to come up with a reason for why these transformations aren't allowed or why you can't measure the difference. I have a hard time seeing how this should be given that they are transformations that are actually used in quantum computing, so they must exist in some effective limit and I do not see why the result should be nonobservable.

    As to Tim's scenario. Well, he adds some point to the manifold. I don't think that's compatible with AdS/CFT because that's by construction unpredictable from the boundary.

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  187. BHG—

    I will be even more explicit, but everything is in the answer already given. But first: you don't mean what you have written. Noe every state on a connected hyperplane with unit norm (not finite, I assume) and "finite energy" (What is this, an expectation value?) is a physical state. It has to also obey the spacelike constraints. And the full solution, of course, has to obey the Hamiltonian constraint as well. You continue to focus on just the boundary bit, ignoring the interior, where all the action is.

    One more time: If someone just hands me a wavefunction, I can explain in principle what has to be done, but not how to do it in practice. Given the quantum gravity theory, we agree that it has a set of solutions—a set of models— yes? OK, take the set of models, Now cut down to the subset that are single connected space-times. Next cut that down to the set of spacetimes that are asymptotically AdS on some connected maximal hypersurface. Now identify the set of maximal hypersurfaces through those spacetimes. The quantum states relative to those hyoersurfaces form the set of physical states.Any given wafefunction is physical iff is it a member of that set. What is hazy or unclear about that? It is perfectly precise.

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  188. physphil,

    Entanglement can be created by local operations, otherwise we could never entangle anything. Consequently you can also remove it locally. I just put references in the previous reply to bhg.

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  189. Bee,

    "Entanglement can be created by local operations, otherwise we could never entangle anything. Consequently you can also remove it locally."

    This does not follow. It is like saying, "babies are created locally. Therefore all babies and parents are stuck together forever."

    To be clear, I guess you are talking about entanglement between two regions in space. It is true that such entanglement can be created locally by action in the past. It cannot be created locally in the present, because it is between two regions that are not local. Also, it cannot be removed locally.

    This is very easy to prove. Just write the outer product and take trace. It follows immediately. I do not know how to make math symbols here but it is really very simple.

    I will look at references when I have time.

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  190. Tim,

    Yes, i of course meant that the wavefunction has to obey the constraints -- just forgot to type that after having done so so many times. And by finite energy I mean that the Hamiltonian should have finite matrix elements.

    Anyway, I still can't translate your words into the precise mathematics I am looking for. For instance when you say "Next cut that down to the set of spacetimes that are asymptotically AdS on some connected maximal hypersurface." I don't know what you mean because a "spacetime" is a semi-classical concept that has no meaning for the general wavefunction \psi[g_{ij},t}. The point here is that I am interested in seeing if there could exist an isomoprhism with the states in the CFT, and the corresponding spectrum of the Hamiltonian, but for this one needs to phrase things not in terms of semiclassical bulk concepts but rather in more abstract mathematical terms, since only the latter notion is common to both the bulk and the CFT. Since further interchange on this issue seems unlikely to be fruitful, I will not ask for any more details and leave it at that. Overall, I am getting the feeling that we have quite a different idea of what it means to "solve" the black info. puzzle, and maybe I'll write more about that later.

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  191. physphil,

    I use the word local in the meaning of local interaction. You have now moved from doubting it's local to asking for a time, which is an entirely different question. Roughly speaking the answer is that you remove the entanglement the moment it would have been created so it's never created in the first place. In any case, just look at the papers I mention.

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  192. BHG—

    The problem here is that 1) you are asking for much more specific detail than is appropriate for the conceptual point I was making, which has to do with the inference from "The horizon of the black hole not longer exists" to "The interior of the black hole no longer exists", which is demonstrably an inference that Hawking made repeatedly and 2) your demand for precision is elastic and you stretch it and contract is as you like.

    If the notion of a complete 4-dimensional solution is too "semiclassical" for you, then the concept of a "connected hyoersurface" must be as well, but that is one you have used freely. So you are happy when you want to be and then raise objections when you prefer. I am taking perfectly sharp concepts from GR, which can be used to form the configuration space over which the wavefunction is defined. If that is not clear enough, then I will wager that you can provide no more precise connection between the bulk quantum state and any geometrical concepts.

    In any case, I think we have established some important points. One is that the whole notion of "AdS" is problematic. The fundamental theory is supposed to be a theory of quantum gravity, and the "AdS" is a restriction of the full theory to some subset of physical solutions. But since "AdS" is a term in classical GR, is seems as if it may be problematic to specify what it means to be part of "AdS" in the quantum theory. Another is that the AdS/CFT correspondence has been checked via perturbation theory, which seems to mean that the very situations we are concerned with—black hole evaporation—has not been tested.

    Furthermore, I am happy to accept that there are an infinite number of zero-ADM-Mass states, but these do not appear in the AdS physical space. Why should they? If there is always entanglement between the disconnected pieces then no isolated such zero-ADM mass state will appear in the folium of AdS physical states.

    Now: could you explain which operators are at issue for the ismorphism between operator algebras?

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  193. Bee,

    thank you for references. I looked at first one (Verlinde^2). They apply the CNOT operation. C means conditional. It is not local if the qbits are not in the same place. Action on one qbit depends on the other qbit. In the paper they say this is like QM teleport. That is true. In QM teleport Alice measures (local), then Alice sends the result to Bob (not local if Alice and Bob are not together), then Bob performs a different local unitary conditional (C) on content of Alice' message.

    I think this is clear. What is not clear is why you say these operations do not change the energy. Because we are discussing AdS/CFT what is relevant is the exact energy that includes all quantum and quantum gravity effects. I already gave an example where removing the entanglement changes the energy a lot when gravity is included.

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  194. physphil,

    Right, the transformation in the Verlindes' paper is not local. As I write in my paper (page 2) "In particular one may worry whether the swap can be achieved by any local operation or if at least the necessary non-locality can be confined to a region sufficiently close by the event horizon." If you go and read the rest of the paper you will find that it can be done locally.

    In general such a transformation will just do nothing if it acts on a particle, entangled or not, exactly because otherwise you'd violate energy conservation. But really the particular type of transformation isn't relevant. The relevant point is merely that such these operations exist. Even if they're not local the question remains where are those states?

    Yes, there are ways to change subsystem entanglement that do not conserve energy. But I am not aware of a theorem that proves any change in subsystem entanglement must necessarily violate energy conservation.

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  195. Bee,

    "Yes, there are ways to change subsystem entanglement that do not conserve energy. But I am not aware of a theorem that proves any change in subsystem entanglement must necessarily violate energy conservation."

    Earlier you said that these are energy preserving. Now you say that you do not know how to prove they are not. That is very different and I agree with the new statement, I also do not know how to prove that without assuming something.

    AdS/CFT tells us that the vacuum state is unique. Also, the energy spectrum is not degenerate (except for global symmetries that have nothing to do with entanglement). So, assuming AdS/CFT you can prove it. Maybe there is a way to argue that bulk entanglement should affect energy without using holography.

    "Even if they're not local the question remains where are those states?"

    For a black hole if you make a small change they may be the other nearby energy states. The entropy is very large so there are many.

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  196. physphil,

    "Earlier you said that these are energy preserving. Now you say that you do not know how to prove they are not."

    No, I said some of them are energy preserving and merely pointed out that the existence of some that are not energy preserving does not mean none are energy preserving. In any case, we seem to agree on the bottom line.

    "AdS/CFT tells us that the vacuum state is unique. Also, the energy spectrum is not degenerate (except for global symmetries that have nothing to do with entanglement). So, assuming AdS/CFT you can prove it. Maybe there is a way to argue that bulk entanglement should affect energy without using holography."

    Yes, you keep repeating this. Trust me I got the message. I have been asking could you please be more precise on the statement "except for global symmetries". But what's the formal statement? If you are merely referring to global unitary transformations that leave global observables unmodified, then faict changes in subsystem entanglement are of that type.

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  197. "I have been asking could you please be more precise on the statement "except for global symmetries". But what's the formal statement? If you are merely referring to global unitary transformations that leave global observables unmodified, then faict changes in subsystem entanglement are of that type."

    They are global part of the conformal transformations. In the bulk they should be some isometries. If you have one black hole that is not at origin of AdS in your coordinates you can rotate it around the origin without changing energy. It will oscillate or orbit and change in phase of that motion preserves energy. Same for any state in AdS, not just one black hole. These are AdS isometries that commute with Hamiltonian, like translations and rotations in Minkowski spacetime.

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  198. physphil,

    Thanks, that's very helpful. So clearly that's not what an entanglement swap would do. I guess you are right then, if Ads/CFT is correct this means there must be a way to prove that in AdS such a transformation necessarily changes the energy. Extrapolating from your previous comments I assume you do not know of any such proof?

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