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. Tim,

    My prescription for the Hilbert space in no way relies on semi-classical approximations. The notion of a wavefunction over 3-geometries \psi[g_ij] makes no reference to any such approximation, and indeed Thiemann and friends claim that it can be taken as the starting point for an exact nonperturbative quantization of gravity. The same is true for a statement that the wavefunction has support only on connected 3-geometries. On the other hand, talk of spacetimes definitely does make reference to a semi-classical limit. The analogy here is between the space of wavefunctions of a particle \psi(x) and the subspace of wavefunctions which "correspond" to particle trajectories having such-and-such properties. The latter notion is inherently vague. Furthermore, as I understand it, your proposal doesn't even define a vector space: if you have two wavefunctions, each of which approximately describes some semiclassical spacetime, then generically the sum of them will not. So I simply can't use your proposal to make any contact with the CFT, and consequently I lose all interest. The issue here is one of conceptual clarity, not of mathematical rigor.

    And I completely disagree with the rest of your message, about the "whole notion of AdS being problematic" etc. There is every reason to expect that there is well defined and self-contained theory of quantum gravity in AdS, and indeed AdS/CFT is such a theory. Whether such a theory can be thought of as a restriction of a larger theory is a perfectly reasonable question, but one shouldn't have any philosophical preconceptions about what the answer is, as you apparently do. And of course AdS/CFT has been checked way beyond perturbation theory: surely you must have heard of the remarkable computations of black hole entropy in the CFT, which reproduce not just the Bekenstein-Hawking result S = A/4G, but an infinite series of quantum gravity corrections to this result, in perfect agreement with the bulk? It is easy to dismiss AdS/CFT .... until you start to learn about what has actually been accomplished.

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  2. I do not know such a proof.

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  3. bhg,

    you wrote to Tim, "Overall, I am getting the feeling that we have quite a different idea of what it means to "solve" the black info. puzzle"

    This was clear from the start.

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  4. Repeating a quote from Penrose, in Fashion, Faith and Fantasy

    Quote:

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

    End quote.

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  5. BHG—

    If you are happy to use wavefunctions defined over classical 3-geometries, then you have no problem with wavefunctions over classical 4-geometries. Now taking a Bohmian approach to the measurement problem, and letting the local beables of the theory be the metric and whatever is distributed in the spacetime (particles, fields), evolving a wavefunction over 3-spaces will induce a probability current which will correspond to a probability over a set of 4-metrics with the particles or fields distributed on it. From this one gets a probability measure over a set of 4-metrics on connected space-times. If one has implemented the theory in a diffeomrphism invariant way (which is the trick, of course), then the probability distribution over the 4-metrics can be expressed as a wavefunction over equivalence classes of 3-metrics, where each equivalence class contains all the states on maximal hypersurfaces that are in the 4-dimensional solution. The set of wavefunctions will, of course, form a vector space. Furthermore, we implement a solution to the measurement problem, which I gather you don't have.

    Now the question is: will any of the 4-geometries have disconnected maximal hypersurfaces? That's a perfectly well-defined question, and the answer depends on what happened when a black hole evaporates. What does the AdS/CFT duality have to say about that question? As far as I can tell, nothing at all.

    Suppose my solution is correct. Are there a huge number of zero-energy states in the folium of physical states according to my solution? Nope. There are huge number of states on disconnected hypersurfaces, and the disconnected part contributes nothing to the ADM mass of the part with the AdS boundary. But the disconnected parts are not, themselves, in the folium. So there need be no mismatch between the AsD and CFT spectra. And of course no disconnected state appears in the spectrum, since no eigenstate will split.

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  6. Maybe this long discussion is over. RIP. I want to say thanks to Tim and bhg for writing so much and Bee for hosting (and writing too). It was useful to see arguments in such length.

    I think almost all physicists working on BH information paradox believe AdS/CFT is not consistent with remnants or baby universes. It was good to see so clearly why it is so and see attempts to find loopholes fail.

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  7. I don't assume it is over, assume that BHG is busy and needs some time to reflect. BHG has not actually produced any contradiction between the solution I propose and AdS/CFT at all, as far as I can see. Once you try to get clear on what "AdS" even refers to, which is absolutely essential for evaluating any claims here and is completely neglected in the literature to my knowledge, we see that there is no a priori restriction of "AdS" states precluding states on disconnected hypersurfaces, one of which is asymptotically AdS. BHG was under the impression that allowing such surfaces would automatically result in a massive degeneracy of the spectrum of the Hamiltonian on physical AdS states, but no such argument has actually been forthcoming. Without an explicit dictionary translating the CFT states into AdS states in the bulk, nothing on the CFT side can rule this out either. So I have no idea what was "clearly" shown by BHG. One thing we did establish, by agreement, is a calculations based off of perturbation theory applied to the vacuum are not relevant, since those will turn out the same whether on not evaporating black holes yield baby universes.

    I should also note that if there is any salvageable argument at all from AdS/CFT, it cannot possibly carry over to asymptotically flat space-times. Because it is the AdS structure that is supposed to imply the non-degeneracy of the spectrum on the AdS side. So the argument, if there is one, has no application to realistic scenarios.

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  8. It's never over till it's over...

    However, I would be more optimistic about this discussion going anywhere if I could discern in Tim even a hint that he appreciates that there is a real tension between AdS/CFT and scenarios involving info loss in the external region. Appeal to authority is always a last resort, but does he really think that the great figures of contemporary theoretical physics (Polchinski, Maldacena, Strominger, Witten,..) who have thought deeply about this problem over many years are that confused?

    And apparently this hopeless confusion is there in the philosophy community as well! I am curious to know what Tim thinks of this paper (https://arxiv.org/pdf/1710.03783.pdf) by the philosopher David Wallace, which as far as I can tell is perfectly aligned with the conventional wisdom in the physics community (as espoused by the above names) about the incompatibility of AdS/CFT with information loss. I don't see any daylight between his claims and mine.

    A couple of other points: the fact that the CFT reproduces gravity in perturbation theory is highly relevant in the sense that it establishes AdS/CFT as a viable theory of quantum gravity. It is a non-perturbatively well defined theory that reproduces Einstein gravity in the regimes where we have actually tested it. Hence, whatever answer it provides for the outcome of black hole evaporation is of distinct interest. Every other approach to the problem is more or less handwaving.

    Finally, I don't understand Tim's last paragraph. Why does he throw shade on AdS/CFT by saying that asymptotically flat spacetime is more "realistic"? We do not live in asymptotically flat space time, as far as we can tell! We live in an expanding, accelerating universe, best modeled by de Sitter spacetime. But does anybody (perhaps besides Tim?) think that this fact is at all relevant to whether the information in black hole evaporation comes out in the Hawking radiation or not? -- I doubt it. If physics respects even the most basic notion of locality, then a black will behave the same whether it is placed in asymptotically flat, AdS, or De-Sitter spacetimes, as long as the size of the black hole is tiny compared to the curvature scale. To drive this home, recall that around 1998 astronomical observations shifted the conventional wisdom as to the asymptotic structure of spacetime , but did that change anyone's approach to black hole physics? Of course not! In fact, we have no idea what the "realistic" asymptotic structure of our universe is, since we can only observe out to a finite distance. So if it can be shown that for a black hole in AdS, with the black hole size tiny compared to the AdS radius, that the information comes out in the Hawking radiation, it would be utterly radical (and frankly insane) to claim that a different outcome occurs if the asymptotic structure changes. This would be a complete breakdown of locality.

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  9. I am told AdS is gravity in a box. How do I withdraw energy from gravity in a box while remaining consistent with gravity? If the AdS has a black hole and hence is not in a ground state, how can it relax to the ground state? It never does, not if it is in a box. From this point of view the uniqueness of the CFT ground state is irrelevant. The only way the AdS can relax to the ground state is if energy escapes past the boundary at infinity - which it never does.

    Mathematical operations on a physical state that annihilate Hawking photons don’t leave the AdS in a physical state. You need the entire space time history of those annihilators, and once you do that , that space time history no longer contains a black hole. In general, none of the operations that we do on a fixed background space time are available in a gravitational theory.

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  10. BHG—

    Appeal to "authority" on a matter like this is worse than a last resort: you are simply not appealing to authorities on foundational matters. As I have pointed out, a physics education contains no exposure to, or discussion of, foundational issues and has not for over half a century. Even thinking about foundational questions would pretty much sink a career in physics for a couple of decades. You should read Adam Becker's "What is Real?" if you want to get a glimmer of what happened to physics, and why, and where it is now. To be an authority on something you ought to at least be familiar with it and have an opinion about it. I can testify from direct knowledge that Ed Witten, by his own testimony, has not thought about foundational issues in quantum mechanics for over 30 years. In his own words: "My job is to calculate wavefunctions".

    To say that the greatest names in physics (aside from Einstein and Bell and Schrödinger) have not covered themselves in glory on foundational matters for the last century is an understatement. And the way that string theory captured almost all the resources of HEP for decades is another scandal, one that seems to be finally coming to an end with the absence of a shred of evidence for supersymmetry.

    As for the rest: do you want to defend the coherence of the AMPS paper? Of EPR = ER? I'm game. Let's throw in Susskind and "black hole complementarity" as well: even Susskind now admits he has no idea what he was trying to say.

    David Wallace's paper is aligned with the physics community because David Wallace was trained in the physics community. He first got a Ph.D. in physics, and then one in philosophy. What do I think of the paper? I disagree with it. Happens all the time in philosophy. But in any case, Wallace's arguments really have no obvious relation to yours: he is talking about entropy in a way that you have never even raised and that I think is not justifiable. But if you want to work through David's paper, I'm game for that too.

    As for the last paragraph, this is the point: your entire argument has turned on the claim that the Hamiltonian in AdS is not degenerate beyond the degeneracies due to symmetry. And your argument for that turns critically on AdS: the space-time structure of AdS, according to you, acts analogously to a potential well. But if that is true (and I am not actually sure it is true) it is certainly only true for AdS, not for either asymptotically flat or dS space-time. And the gluing theorems of classical GR, as I understand it, make it very hard to see how it could be true in any case. So even if your line of argument works in AdS, it does not work in any realistic space-time.

    We have not gone into the relevance of the gluing theorems either, and I'm game for that. So I am offering a smorgasbord of options. Take your pick.

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  11. bhg wrote: If physics respects even the most basic notion of locality, then a black hole will behave the same whether it is placed in asymptotically flat, AdS, or De-Sitter spacetimes, as long as the size of the black hole is tiny compared to the curvature scale.

    You previously outlined a sequence of repeated measurements involving a spherical shell of detectors surrounding a black hole, but this provoked two concerns, first that it would take infinitely many trials (of identically prepared black holes) measuring all possible things in order to infer the actual wave function, and second that the very act of performing these measurements would restrict us to just an effective wave function, a tiny part of the universal wave function, and this effective wave function is not dual to any CFT. The claim was that this undermines the application of AdS/CFT.

    travis wrote: 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.

    Is there reason to doubt it? Is there something about the AdS/CFT duality that depends crucially on the attributes of the full AdS, as distinct from, say, de Sitter spacetime? (I suppose there is, since otherwise we would be talking about the dS/CFT duality.) Assuming the hope can be justified, is the proposed measurement feasible in principle? In order to infer the wave function from these measurements, would we need an infinite number of trials on infinitely many identically prepared black holes? Our would it require infinitely many identically “prepared” AdS universes, with only one measurement per universe?

    travis wrote: 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.

    Agreed, but if we cannot (even in principle) have any empirical contact with the universal wave function (beyond the tiny effective wave function that comprises the world of our experience - dinosaurs roamed the earth, the Roman Empire rose and fell, measurements had specific outcomes, ...), then it isn’t empirical science. In that case I'd say it’s a purely mathematical question, not a physics question.

    tm wrote: 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.

    I think what’s being investigated here is how the idea of purely unitary evolution of a hypothetical “universal wave function” can be reconciled with the formation and evaporation of black holes. I would avoid claiming that strict unitarity is a general feature of quantum mechanics, as ordinarily understood and applied, because quantum mechanics entails non-unitary measurement (von Neumann’s process 1) as well as unitary evolution (process 2). Von Neumann argued that unitary evolution by itself is vacuous (like no hands clapping), and Bohr, et al, insisted (when chastising Everett) that the concept of a universal wave function was completely unscientific. Of course, none of them ever solved the measurement problem. (As an aside, I don’t believe the Bohmian pilot wave interpretation can solve the measurement problem in a Lorentz invariant way.)

    We can sidestep the measurement problem by stipulating that we exclude all measurements from our considerations, and focus purely on unitary evolution of a universal wave function, but then are we really talking about empirical science?

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  12. Tim,

    "So even if your line of argument works in AdS, it does not work in any realistic space-time. "


    By the same logic, since our world is not Lorentz invariant, all results from QFT based on Lorentz invariance "do not work in any realistic space-time". So we can pretty much throw out all the textbooks, since most of the results described therein are based on assuming Lorentz invariance. The folly of this argument should be just as obvious as for the one you are making about AdS.


    Wallace doesn't really add much to what already appears in the literature, but his overview is succinct and accurate. His recurrence argument is in the same spirit as mine, and also shows why your proposal is impossible. Namely, since the energy spectrum is discrete any initial state prepared at some initial time will, after some time evolution, come arbitrarily close to the initial state. So if the initial state is fully accessible to an external observer the same will be true of the time evolved state, and hence no information can be lost. Turning it around, if information is lost it follows logically that the energy spectrum must have been highly degenerate and/or non-discrete.


    As I have already said, nowhere do I see that you have even addressed, much less resolved, what I and others in this field regard as the real puzzle of black hole evaporation. Namely, the CFT obeys the usual rules of QM: at every time t it has a physical Hilbert space; probabilities are defined by inner products among the physical states; time evolution is unitary; the Hamiltonian is discrete and non (or finitely) -degenerate. The question is whether one can give a bulk description that respects these principles and also respects the standard Penrose diagram. The answer appears to be "no", or at least no one has proposed such a scenario.

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  13. BGG et al.,

    I am just about convinced now that Tim's proposal is inconsistent with a universe in which the spacetime is AdS (specifically: AdS^5 x S^5 - or does the discreteness of spectrum also hold for some CFT dual to Ads^4?). This is interesting and, *potentially*, relevant to the question of whether Tim's way of dismissing the info loss paradox might be realizable in our world.

    The question now, for me, is: how can one argue for this relevance? You reject Tim's quick dismissal:
    TM:
    "So even if your line of argument works in AdS, it does not work in any realistic space-time. "
    BHG:
    "By the same logic, since our world is not Lorentz invariant, all results from QFT based on Lorentz invariance "do not work in any realistic space-time". So we can pretty much throw out all the textbooks, since most of the results described therein are based on assuming Lorentz invariance. The folly of this argument should be just as obvious as for the one you are making about AdS."

    The analogy doesn't seem quite fair, at first sight. Do QFT textbook results rely on Lorentz invariance being global and exact? Or merely on it being locally accurate to a very high degree? Furthermore, as far as I know QFT results don't hinge directly on the global features of spacetime in the way that the non-degeneracy of the Hamiltonian in AdS/CFT hinges on global features of the CFT space (or the AdS, depending on your perspective). Odd and "non-local" as it may seem, for all that I've seen/heard so far, the discreteness of spectrum may not carry over to stringy QG theories in spacetimes with different global features. Perhaps you can comment on this. E.g., how solid are (say) dS/CFT correspondences (I have seen references to such), and in the relevant CFTs is the spectrum also non-degenerate?

    CONT.

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  14. Cont:

    Your direct argument for the relevance goes like this (I quote in full from 2 posts ago):
    "We live in an expanding, accelerating universe, best modeled by de Sitter spacetime. But does anybody (perhaps besides Tim?) think that this fact is at all relevant to whether the information in black hole evaporation comes out in the Hawking radiation or not? -- I doubt it. If physics respects even the most basic notion of locality, then a black will behave the same whether it is placed in asymptotically flat, AdS, or De-Sitter spacetimes, as long as the size of the black hole is tiny compared to the curvature scale."

    The issue is not that Tim, or any other skeptical person, thinks that there is a direct connection between the global structure of spacetime and local behaviors such as black hole evaporation. If that were the claim, your incredulity would be understandable! Rather, the issue is: what reasons do we have, so far, to believe in the discreteness (or non-infinite degeneracy) of the energy spectrum of the correct QG theory *in our world*. AdS/CFT gives us reason to believe that there are logically possible worlds (or 'metaphysically possible' worlds, as a philosopher would say) in which this discreteness is built into the physical laws, and hence that Tim's posited scenario for resolving the info loss problem will not work for such worlds. But we also have ample reason to suspect that our world is NOT one of these worlds described by a string theory in AdS. So: what is the reason for supposing that the actual laws of physics, in our world, resemble the law of these AdS worlds *in the relevant respect*?

    I am not expressing strong doubt that they do, just asking for some positive argument that this should be expected. Saying "black holes will behave the same in any global geometry, it's too local a process to care about the boundary structure" seems to me to beg the question. It presupposes that some stringy QG theory with relevantly the same features is already known to be true for our world - but do we really know that?? Despite some interesting results in the string theory camp, my understanding was that, even among theoretical physicists, the question of what's the *true* QG theory for our world is still wide open. And if that's the case, then we can't be sure yet that the way black holes operate in *our* world is, relevantly, the same as the way they operate in a stringy AdS^5 x S^5 world.

    This is actually a sincere request for information: what experimental or theoretical triumphs of the AdS/CFT approach make you confident that the actual, true physical theory of our world must be very similar? I have spent some time today googling "evidence for AdS/CFT", and the situation looks unclear to me. One thing is evidence for the correctness of the *conjecture* (that a certain stringy QG theory set in AdS^5 x S^5 is mathematically equivalent to a CFT in some lower-dimensional space whose topology I can't remember). There may be all sorts of evidence for the truth of this mathematical conjecture, but that's just (if true) a mathematical fact. An entirely different thing would be evidence that some theory *very much* like that stringy AdS theory is in fact true of our world - that's what I'm asking about.

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  15. Arun,

    The basic AdS/CFT dictionary includes a prescription for injecting and removing energy. You can read all about this in Witten's orginal AdS/CFT paper, for example.

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  16. BHG—

    Your supposed analogy to my argument is completely inaccurate. In GR, there is a property of local Lorentz invariance which holds locally in the limit as one restricts oneself to smaller and smaller regions, and there the divergence from exact Lorentz invariance can be quantified in terms of the order of the approximation. I'm certain you know all this. Insofar as one can argue that *local* Lorentz invariance is all one needs for certain purposes one can justify importing results from SR into GR. But the nature of your entire argument from AdS is of an entirely different character. It turns on the discreteness and non-degeneracy of the spectrum of the Hamiltonian, which is something not approximate but exact. The spectrum of a free particle is simply not discrete, full stop, and the spectrum is an asymptotically flat or asymptotically dS space is massively degenerate, full stop. The form of argument relies critically and inalterably on the claim that the spectrum in AdS is discrete and non-degenerate. I have accepted your claim that this is so without questioning it, but all you have ever done to justify it is make the analogy to a bound state in a potential well. But given the form of argument you are now offering I am reevaluating whether I should have trusted your claim.

    By your form of reasoning, in precise analogy, I can make an argument completely refuting AdS/CFT. Step one: we all know that in an asymptotically flat or asymptotically dS space-time there are states of free particles, and states of free particles have a continuous spectrum Step two: In such a case, the gravitational theory cannot possibly be dual to a CFT since the CFT has a discrete spectrum. Step three: Therefore, the CFT cannot be dual to the gravitational theory even in AdS, because restricting the domain of applicability of an argument to just AdS would be like restricting the domain of applicability of Lorentz invariance to SR and not applying the results in GR. So by your reasoning, I have just refuted AdS/CFT.

    Either the formal properties you are appealing to are restricted to AdS and other space-times with the peculiar features of AdS or they are not. If they are, then results you derive are restricted to AdS and relevantly similar space-times and cannot be exported to asymptotically flat or asymptotically dS space times. If they are not, then results from asymptotically flat spacetimes can be imported back into AdS, in which case AdS/CFT is refuted. What you want to do is have your cake and eat it too: appeal to the peculiar characteristic of AdS for your argument and the say that the results do no depend on those very peculiar characteristics. This argument is plainly unsound. Indeed, it is plainly worthless.

    Con't

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  17. And I will repeat one more time: you say that it is a "usual rule" of QM that "probabilities" are given by the inner product of the state of a system with some other "physical" state. That just is not a "usual rule" of QM. Probabilities for the outcomes of "position measurements" can be calculated in non-relativistic QM can be calculated by taking the "inner product" (as it were) of the wavefunction with a delta function, but the delta function is not a physical state. Probabilities for the outcomes of a "momentum measurement" can be calculated by taking the "inner product" with "momentum eigenstates", but these "momentum eigenstates" are not in the physical Hilbert space.

    Furthermore, you have made the unitarity of the time evolution a "rule" of QM, thereby prohibiting collapse theories. OK: so what are the "probabilities" you are calculating probabilities *for*? Not for collapses! And the idea that any post-collapse state (calculated by projection onto eigenstates of an operator) must be in the physical Hilbert space is undercut, since no such physical collapses occur. So give us a clue: how to you propose to solve the measurement problem while respecting your "rules"? It is not a collapse theory. Is it a hidden variables theory? A Many Worlds theory? The next relevant questions depend on which choice you make here, so please inform us.

    Of course, if your "principles" of QM are internally inconsistent then they can't be reconciled with a Penrose diagram since they can't be reconciled just among themselves. I strongly suspect that that is the case. But you can show I am wrong by clearly answering these precise questions.

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  18. Tim,

    In those examples you gave, you can easily patch it up in order to calculate the probabilities using only physical states; given a position measurement with some precision delta (the only kind of position measurements that can actually be made), you can calculate the probability of a particle being at a certain location with uncertainty delta using only physical states. Same with momentum. Are there any examples of needing to use states which aren't physical in order to calculate probabilities, in which the need doesn't arise merely due to requiring arbitrary measurement precision?

    I just read the Wallace paper, and his recurrence argument is very decisive. I was not considering before that even if you have an infinite number of energy eigenstates, you will still get arbitrarily close to your original state if you are in a bounded system as in AdS/CFT. So no matter what, you have to eventually end up with a state that is arbitrarily close to your original state, and hence information cannot be lost to the inside of a black hole. With this consideration, there is no way Tim's proposal can work in AdS/CFT.

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  19. Travis—

    BHG claims that something is a "rule" of QM. I point out that this "rule" is consistently violated in every normal QM textbook. You respond that you can "patch up" the examples. But my point is that the supposed rule just is not a rule. You are proposing to make it a rule. Why should I accede to your proposal? And if there were any rule like that, it would appear to be relevant only to a collapse theory anyway, and in a collapse theory the wavefunction does not always evolve unitarily and information is not always preserved. So how could there be an "Information loss paradox"?

    If one wants to be rigorous about the whole thing, then what one needs is not a "rule" but a solution to the measurement problem, if you want to provide a measurement theory. As I have said, my preferred solution is of the Bohmian type, and it is easy to verify that in Bohmian mechanics no appeal is made to inner products of physical states—at all—in stating the theory or solving the measurement problem or determining the post-measurement state of the universe. So basically this entire line of argument relies on making up a bunch of rules that are not rules and then insisting on them with no justification at all.

    As for Wallace's paper, I hope we can agree first that his line of argument does not intersect BHGs. As for the recurrence argument, it is plainly question-begging. Recurrence arguments only work for closed systems. Is an evaporating black hole system in AdS closed? Well, if my solution is correct, then the answer is "no": the AdS boundary may work like an infinite potential well in the part with the AdS boundary, but the boundary of the disconnected part is not AdS. So that argument is circular: you have to rule out my solution first to get the argument to run.

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  20. Tim,

    Whether or not we want to call it a rule, the fact remains that every quantum system that has been studied follows it, so if we need to reject it then that would be a radical departure from standard QM. Although now that I'm thinking about it, can we say with certainty that QFT follows this rule since Haag's theorem tells us that no interacting QFT even has a well-defined Hilbert space? In any case, as long as the CFT follows this rule, that is sufficient because there is a one-to-one correspondence between states in the CFT and states in the bulk. Does Haag's theorem apply to the CFT?

    I agree that Wallace's line of argument is not the same as BHG's, and Wallace's is more compelling to me. It's not question begging though: as long as you assume that there is a one-to-one correspondence between states in the CFT and states in the bulk, you can just apply the recurrence theorem to the CFT and then deduce that the bulk also will recur.

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  21. Tim,

    Position eigenstates in QM: I agree with Travis here. Position eigenstates are not in the Hilbert space because they have infinite norm. This is fine, since no physical measuring device could ever determine position to infinite accuracy. As the QM texts will tell you, what you should compute probabilities for are wavepackets. OF course, a normalizable wavepacket can approximate a delta to arbitrary accuracy, so it's fine to use delta function position eigenstates as long as you realize that these only arise in a limiting sense. The situation is completely different in your setup: in your case there are physically realistic measurements whose probabilities cannot be computed by the standard rules.

    Wallace's argument: you are completely missing the point if you think it is circular. The CFT is a closed system. Hence, in AdS/CFT so too is the bulk theory. His argument then rules out your scenario from arising in AdS/CFT. It's really as simple as that. This is all true, but note that my argument reaches an even stronger conclusion about what can happen.

    Again, the "measurement problem" of QM has no relevance to this discussion. We can just think of the CFT as a set of abstract rules for computing numbers that obey properties allowing them to be interpreted as probabilities. We need not get into any discussion of what these probabilities mean for a "human observer", or any such thing. In this way we can keep the discussion focused and mathematically precise and avoid all the philosophical issues.


    Lorentz invariance in GR: there is no result about local Lorentz invariance in GR of the sort you are imagining. I can elaborate, but there are obvious physical obstructions which I invite you to discover for yourself. In fact, historically the question of how Lorentz invariance emerges in GR is what led to the notion of asymptotic symmetries, which is how I have been framing the discussion.


    It should be physically obvious that if a black hole embedded in an AdS space whose radius of curvature is, say, a google times the black hole radius, releases all its information in the Hawking radiation, the same will be true for a black hole in asymptotically flat space. Yes, as you take the AdS radius to infinity the spectrum of the Hamiltonian becomes continuous, but it's clear why this is: it has to do with the space of particle states arbitrarily far from the black hole. The energy of spectrum of the black hole itself remains discrete. Putting the system in AdS is a convenient way of avoiding having to "subtract out" the spectrum of distant particles. In any case, I prefer to stay focused on the crispest statement, which is that your scenario cannot arise in AdS/CFT.

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  22. Carl3,

    The QFT theorems in the books do rely on exact Lorentz invariance. Of course, their conclusions will survive in approximate form if Lorentz invariance is approximate. You do of course need some physical insight to understand the regime of validity of such approximations.

    As you can read in my response to Tim, I am definitely not claiming that the energy spectrum is discrete in asymptotically flat spacetime -- indeed, quite the opposite. The point is that the states giving rise to this non-discreteness are totally irrelevant to the local physics of black holes, since the former are simply particles far far from the black hole. To make this whole point even more obvious, imagine a black hole in a spacetime which has a gigantic local AdS space that eventually goes over to flat space. The size of the AdS space is so big that all the emitted Hawking quanta do not reach its boundary before the black hole evaporates. Simply by causality, there is no way that the black hole can "know" that its really in asymptotically flat space. Hence one must get the correct result by computing in an asymptotically AdS space.

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  23. bhg,

    You can't on the one hand use the non-local nature of AdS/CFT to assume the bh infloss problem away and on the other hand argue that you don't need the AdS boundary because the space is locally Minkowski. There is no indication whatsoever the limit \Lambda ->(up) 0 is smooth.

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  24. Bee,

    " There is no indication whatsoever the limit \Lambda ->(up) 0 is smooth."

    I completely disagree. Part of what has come out of AdS/CFT (due to the hard work of many people over the years) is a prescription for extracting the flat space scattering amplitudes from the CFT in the limit that the AdS radius goes to infinity. This is a quite technical but ultimately beautiful story. See papers by Penedones and others. One gets S-matrix amplitudes obeying the desired physical properties, and so every indication is that the limit is smooth.

    Also, I have absolutely no idea what you mean when you talk about "assuming the black hole info loss problem away".

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  25. Travis,

    " since Haag's theorem tells us that no interacting QFT even has a well-defined Hilbert space?"

    That is not what Haag's theorem says. There are plenty of interacting CFTs with well defined Hilbert spaces. That is a mathematically rigorous (by the standard of mathematicians) statement.

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  26. To reiterate, if you start with an AdS/CFT in the vacuum state, and then shoot in from the boundary matter so that a black hole is formed, and then the black hole evaporates, there is no physical process that gets this AdS/CFT back to the vacuum state.

    It should be physically obvious that if a black hole embedded in an AdS space whose radius of curvature is, say, a google times the black hole radius, releases all its information in the Hawking radiation, the same will be true for a black hole in asymptotically flat space.

    The problem is that the Hawking radiation never exits the CFT boundary, it doesn't matter what the ratio of radii of black hole to AdS is. What the CFT never again sees is an AdS ground state; nothing escapes the box. So non-degeneracy of the CFT's vacuum is utterly irrelevant to ruling out Maudlin's solution, because the CFT is never going to return to the vacuum, and so it cannot rule out a remnant/disconnected piece/whatever by a vacuum non-degeneracy argument.

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  27. bhg,

    Do or do they not prove that the limit is continuous? Just to avoid that we are talking past each other, I mean that whatever they have calculated there, they have shown the limit \Lambda \to 0 from below is actually the same you would get for having Lambda=0 in the first place. I would settle for a proof that the limit up is the same as the limit down.

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  28. BHG,

    Thanks for the reply. I see why you think it is physically obvious that how a black hole works should be unaffected by the global structure of spacetime that it is embedded in (at least, if the spacetime is hugely bigger than the region in which the whole evaporation process takes place).

    But you are assuming at least this much: that whatever the true QG theory is, it is compatible with spacetimes that are (global) AdS and also spacetimes that are dS or FLRW (so that the actual universe is covered). You need this, because you want to take a feature of black holes in AdS (discrete spectrum) and say that black holes in our universe must have the same feature; this only makes even prima facie sense if you're assuming the same laws of nature in both cases.

    So let me re-phrase my evidence question. What would you point to as evidence that we have, to support the claim that the true physics of our world is compatible with spacetime being (at whatever classical level of description is appropriate) AdS?

    Again, a genuine request for info, not sarcasm or snark. One reason to *doubt* it does occur to me. Suppose we think that, ultimately, Lambda is not a primitive term but somehow is derivable from the full QG theory. Then depending on how this derivation goes, it might be the case that Lambda being negative is physically impossible. In which case, spacetime being AdS would be impossible too, right?

    I am not so much interested in trying to save Tim's scenario from refutation as I am interested in understanding what what reasons we have for thinking AdS/CFT teaches us important things about the QG theory of our actual world.

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  29. BHG,

    I believe you, but what is it that causes the CFT to break free of Haag's theorem? Is it the fact that it is a CFT instead of an ordinary non-conformal QFT, or is it the fact that it doesn't live in Minkowski space?

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  30. BHG—

    I don't think you mean to say that position eigenstates "have infinite norm". Either the norm is not defined or it is 1. You seem to be thinking of momentum or energy eigenstates in the free theory. Of course, it is rather ironic that in a free theory the energy eigenstates are not in the physical Hilbert space since your argument turns on the spectrum of the Hamiltonian, and so by your official criterion the Hamiltonian has no spectrum in the free theory. That appears to make a hash of the idea that anything like this argument could apply in a physically realistic spacetime.

    As I have said, from a physical point of view the focus on the spectrum of the Hamiltonian seems odd since the eigenstates are static and hence have little physical significance. We often use energy eigenstates in the analysis of *subsystems* of the universe, but it does not follow that they are are much use in the analysis of the *universal* wavefuncton.

    And I can only repeat again that the "rules of QM" that you keep adverting to are neither clearly articulated anywhere nor justifiable in this context. It is these vague "rules" that are used to handle the measurement problem in standard QM, and they do not provide any clear coherent answer to that problem. If they did, the standard approach would not have a "measurement problem". No theory I know of that actually solves the measurement problem in a coherent way employs any of these "rules".

    The more I reflect on this appeal to recurrence (which you never even hinted at) the more irrelevant it seems. Even if some recurrence theorem could be proved, what would it show? Nothing. No proposed solution to the paradox invokes recurrence. They all have to do with the informational content of the Hawking radiation. Recurrence has nothing at all to do with that.

    The measurement problem is not a "philosophical issue". It is a physical issue. If readers of this exchange come away with nothing else, I hope they come away with that fact. Physicists have now formulated a perfect defense against facing up to the physical inadequacy of the standard presentations. If you discuss the measurement problem but the resolution is not relevant for some issue they are trying to resolve, then it is called a "philosophical problem". But if you argue that the resolution of that problem *is* relevant to the physical issue they are trying to solve they avoid facing that by calling the measurement problem a "philosophical issue". The measurement problem is physics, and physics that is critically important for the question of information loss. Which is what we are seeing here.

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  31. Bee,

    It depends on what you mean by "prove", but here is the sort of thing one can show. Take a theory corresponding to, say, \phi^4 in AdS, and its dual CFT, and consider, say, 2->2 scattering. We can write down a CFT correlator whose bulk version involves the particles scattering near the center of AdS. We can then take the AdS radius to infinity. We can then ask whether the limiting result equals the 2->2 scattering amplitude for \phi^4 theory in Minkowski space. The answer is yes, and this can be shown very explicitly. So the AdS theory (and its CFT dual) contain within it the physics of flat space scattering amplitudes. The key here is to focus on some localized process and then take the limit of that.

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  32. Carl3,

    I agree with you general sentiment, although I might phrase it a bit differently. An important thing to keep in mind is that there is not necessarily a single Theory of Quantum Gravity. There could be many such consistent theories, and our world is only described by one of them. So even if AdS/CFT describes one such consistent theory, and even if one could prove rigorously that black holes in that theory don't lose information, there is always the possibility that physics in our world is simply different. It could even be that the QG theory governing our world is incompatible with AdS -- who knows. My point all along is that there is precisely one theory of QG which is presently sufficiently well defined that one can even address this problem without furious hand waving, and so it is certainly of interest to know what that theory says. And maybe this will turn out to be an accurate description of how black holes work in our world.

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  33. Travis,

    Haag's theorem is a statement about the (non)existence of the interaction representation. CFTs can be formulated without reference to such a construction (instead directly in terms of the operator algebra and so on). And even if one did use the interaction representation things would still be fine, since one can use conformal invariance to map the theory to a spatial sphere where Haag's theorem doesn't apply. Also, you may know this, but Haag's theorem is not really viewed as a significant issue among practitioners of QFT, partly because you can always avoid it by simple maneuvers like placing your theory in a very large box with periodic boundary conditions. It is usually viewed as a technical nuisance more than anything else. Most QFT books don't even mention it.

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  34. BHG,

    "I agree with your general sentiment, although I might phrase it a bit differently. An important thing to keep in mind is that there is not necessarily a single Theory of Quantum Gravity. There could be many such consistent theories, and our world is only described by one of them. So even if AdS/CFT describes one such consistent theory, and even if one could prove rigorously that black holes in that theory don't lose information, there is always the possibility that physics in our world is simply different. It could even be that the QG theory governing our world is incompatible with AdS -- who knows. My point all along is that there is precisely one theory of QG which is presently sufficiently well defined that one can even address this problem without furious hand waving, and so it is certainly of interest to know what that theory says. And maybe this will turn out to be an accurate description of how black holes work in our world."

    Great - this makes perfect sense to me. Thanks.

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  35. Tim,

    Yes, when I say infinite norm that is the same as saying the norm doesn't exist. The rest of your comment about norms is wrong because in AdS energy eigenstates of the free theory are normalizable, as I have stated many times.

    Wallace's recurrence argument does rigorously rule out the possibility of information being forever lost, which is what I gather you have been proposing. On the other hand, the recurrence time is much longer than the evaporation time. To argue that the info comes out at the evaporation time you need an argument like mine.

    Wallace is a philosopher, but note that in his review of the topic he never (as I recall) mentions anything to do with the measurement problem of QM. I assume this is because he, like I, doesn't see any relevance of the measurement problem to the black hole information puzzle. The kinds of measurement we are talking about (collecting photons etc) are not subtle or exotic, and the two outcomes (info loss or not) make dramatically different predictions: the von Neumann entropy of the radiation is either zero or macroscopically large. So there is absolutely no reason to expect that we need to get into the subtleties of measurement anymore than what an astronomer does when they analyze the spectrum of a star -- which is to say, not at all.

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  36. BHG—

    It would help there to be progress if you are just straightforward in what you say, and also if you pay closer attention to what I am arguing. It is simply false, as far as I can tell, to say that the norm of the position eigenstates is "infinite". Insofar as there are "position eigenstates" at all, the are delta "functions", and the issue with delta "functions" isn't that they have infinite norms but that they aren't functions at all: they are distributions. On the other hand, *in non-relativistic quantum mechanics* the problem with energy and momentum eigenstates in the free theory not that they aren't functions at all—they sure are—but they have infinite norms and so cannot be normalized. This is a plain fact about non-relativistic quantum mechanics as it is usually presented: nothing to do with AdS, nor did I claim that it has anything to do with AdS.

    Why did I bring it up? Because you keep making claims about what the "rules of QM" require that are conditions that plain vanilla non-relativistic QM does not satisfy. I still have no clear notion what you mean by the "physical Hilbert space" of a theory in general and the "physical Hilbert space of AdS" in particular. And it is not at all clear that the "rules of quantum mechanics" that you are proposing (in order to say that my proposal violates them) are satisfied by plain vanilla non-Relativistic QM, from which it follows that they are not "rules of QM" at all. Let's be quite precise.

    What is the "spectrum of the Hamiltonian" of a theory? Well, it could be the set of eigenvalues of all eigenstates of the Hamiltonian operator *whether the eigenstates are in the physical Hilbert space or not*, or it could be the set of all eigenvalues of the eigenstates *in the physical Hilbert space*. I take it as obvious that you must mean the latter. Because you claim that the duality between AdS and the CFT implies that their respective Hamiltonians have the same spectra, or at least isomorphic spectra. But surely the spectrum of the quantum gravity Hamiltonian is not discrete relative to the class of all possible states. The quantum gravity theory had better be able to handle free particles, where the spectrum is continuous, for example. So you mean to restrict the states relative to which the spectrum is defined to "physical states in AdS". But we as yet have no clear definition of what this restriction amounts to. I do not think you have offered a clear definition, and if you have I missed it. At one point you seemed to suggest that the "physical states" could include states on disconnected surfaces, but then, as far as I can tell, you took it back. So how about offering a clear definition of what you mean?

    Here is a problem you will bump up against, illustrated in the non-Relativistic theory. If you restrict the spectrum of the Hamiltonian to the set of eigenstates in the physical space, then in the free non-Relativistic theory there is no spectrum because there are no eigenstates in the physical Hilbert space. And here the problem really is that the norm is infinite. But what is usually said is that in the free theory the spectrum is continuous, not that it does not exist. The "usual way" these concepts are deployed even in non-Relativistic QM is at best problematic and at worst inconsistent. And I am really not very concerned about whether my solution accords with some inconsistent set of "rules".

    Con't

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  37. The whole reason we have gotten into the discussion of the measurement problem is not because I brought it up, but because you and physphil did. It was you and physphil who repeatedly claim that I am unable to provide a "measurement theory" that is in accord with the "rules of QM". And what I have said over and over is that there are no coherent and precise "rules of QM" of the sort you keep trying to refer to. The entire "Information Loss Paradox" is predicated on the assumption that a fundamental feature of quantum theory is that the wavefunction evolves always unitarily, and therefore never loses information. This is a feature that is *preserved* in my solution. It is therefore consistent with the AdS/CFT duality, as far as I can tell. Put the other way round, AdS/CFT does not rule out my solution on those grounds. We got into the whole issue of measurements due to you and physphil trying to argue that my solution is inconsistent with AdS/CFT. And in order to make such an argument, it was you, not me, who brought up all this stuff about measurement theory and the "rules" of QM. And as far as I can tell you have not been able to formulate an argument that my solution is inconsistent with AdS/CFT or with the "rules of QM" that is not also an argument that AdS/CFT, all by itself, is inconsistent or violates those same "rules".

    So if you want to avoid bringing up the measurement problem, fine. Start again. State in as exact and precise terms as you can why my solution to the problem is inconsistent with AdS/CFT. If you insist on a term like "physical Hilbert space", please be sure to define, as best you can, what it is. Let's see if you can articulate your criticism without adverting to measurements and hence to the physical problem of accounting for measurement outcomes. If you can, fine. If you can't, don't blame me.

    By the way, you obviously cannot "collect photons" in any way so as to determine the von Neumann entropy of the incoming photon state. I take that that is obvious. If not we should have a discussion. You keep acting as if there were physical procedures that allow you to calculate certain theoretical quantities when they plainly can't.

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  38. My report of the death of this thread was an exaggeration.

    Still, I see badly misinformed comments like

    Tim: "No proposed solution to the paradox invokes recurrence. They all have to do with the informational content of the Hawking radiation. Recurrence has nothing at all to do with that."

    Wrong. Recurrence is a consequence of unitarity of boundary CFT. It has everything to do with that.

    Arun: "To reiterate, if you start with an AdS/CFT in the vacuum state, and then shoot in from the boundary matter so that a black hole is formed, and then the black hole evaporates, there is no physical process that gets this AdS/CFT back to the vacuum state."

    Not true, as has been repeated many times. AdS/CFT does not require the box is sealed. I suggested a simple way to see conflict. Form a black hole. Make the boundary leaky so all or most Hawking particles escape. After a long time you are in a near vacuum state for exterior. There are very few states near CFT vacuum. If Tim were correct there must be many such states because black hole interior can be in many states. So, Tim's idea conflicts with AdS/CFT.

    There are many arguments like that. As bhg says again and again it is really not hard to understand. A disconnected internal region with many states obviously means large degeneracy for boundary description. CFT does not have that.

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  39. Ok, I'm no longer convinced by Wallace's argument. If the spacetime splits, and then stays split for a very long time until some very unlikely conditions occur which causes the process to reverse, that would definitely still count as Tim's solution being correct, and the recurrence theorem doesn't rule this out.

    I think BHG's argument does rule out explicitly disconnected surfaces, that is, it rules out having amplitudes assigned to {g_ij} in which the {g_ij} are disconnected, but nothing so far has ruled out the possibility that the surfaces remain very approximately disconnected for a long time after the evaporation event. I guess we already decided that's just a remnant solution, but there's no reason why the effectively disconnected surface can't contain all kinds of interesting structure, so I see no reason not to call that a baby universe solution.

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  40. bhg,

    I don't understand how it's interesting to compare local physics in AdS with that of Minkowski space. You want to solve the bh infloss problem, it's the non-local or even global behavior that you need to carry over in the \Lambda->0 limit. As I said, thinking you can do this by arguing that the local physics works the same doesn't help you.

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  41. physphil

    And it won't end as long as there are still errors like yours to correct. You are right about one thing: there are many arguments like the ones you gave, namely invalid arguments.

    No proposed solution of the problem has ever invoked recurrence. If it did, it would argue that the system has to return to the original state, the state without any Hawking radiation. And AdS isn't even globally hyperbolic! There is no way in the world that recurrence can come into the discussion. None.

    The "argument" you offer to Arun has so many holes I don't know how to count them. Reflect on whether you really want to try to defend it or prefer to just retract it. That could save us some time.

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  42. Bee,

    "You want to solve the bh infloss problem, it's the non-local or even global behavior that you need to carry over in the \Lambda->0 limit."

    That's wrong: BH evaporation is a local process. Proof: the black hole forms and evaporates in some time T. So surround the system by a sphere of size, say, 10T. By causality, what happens outside this sphere (i.e. whether the space is asymptotically flat, AdS, or whatever) has no influence on the evaporation process.

    If what you claim were true then the outcome of BH evaporation for some solar mass, say, black hole could depend on what the universe looks like arbitrarily far from the BH. This would make the entire problem hopeless, since we know nothing about the latter.

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  43. Tim,

    "The quantum gravity theory had better be able to handle free particles, where the spectrum is continuous, for example."

    You lost me here: the spectrum of free particles in AdS is of course discrete. What are you claiming is continuous?

    "The whole reason we have gotten into the discussion of the measurement problem is not because I brought it up, but because you and physphil did. It was you and physphil who repeatedly claim that I am unable to provide a "measurement theory" that is in accord with the "rules of QM". And what I have said over and over is that there are no coherent and precise "rules of QM" of the sort you keep trying to refer to."

    Please go back and read what I actually wrote, which was a request for the rules for computing probabilities. This is precisely what is spelled out in the QM textbooks, e.g. Weinberg. You manifestly violate these rules.

    " At one point you seemed to suggest that the "physical states" could include states on disconnected surfaces, but then, as far as I can tell, you took it back. So how about offering a clear definition of what you mean?"


    No, I did not take it back, and indeed explained precisely what I means here several times. Scroll up and you will find your question answered.

    Also, I have explained very clearly on multiple occasions what defines the physical Hilbert space: it's the space of wavefunctions that obey the constraints, and have finite norm, and have finite matrix elements with the Hamiltonian. In other word, I mean what every other practitioner of canonical quantum gravity means.

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  44. bhg,

    You write

    "Proof: the black hole forms and evaporates in some time T. So surround the system by a sphere of size, say, 10T. By causality, what happens outside this sphere (i.e. whether the space is asymptotically flat, AdS, or whatever) has no influence on the evaporation process."

    That's the most ridiculous "proof" I have ever seen. I'll try and help you listing the assumptions you started from:

    A) Assume it is correct that a gravitational theory in asymptotic AdS space can equivalently be expressed by a CFT on the boundary of that space.

    B) Assume that all fields in that AdS space can be expanded around the boundary (and converge suitably towards the boundary).


    It is then widely believed that from A and B it follows

    C) Black hole evaporation is unitary and obeys the BH entropy.

    Your claim is now that if you drop assumptions A and B, C still follows. Now please explain what you think it follows from.

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  45. Bee,

    Since neither the assumptions (A) and (B) nor the conclusion (C) bear any relation to what I wrote, I have no response to your comment -- you will have to argue with yourself about this. I will simply repeat what I did say, explaining the sense in which black hole evaporation is a local process.

    If under assumptions X (whose nature need not concern us at the moment) it can be established that a black hole which evaporates in a time T does or not does release its information in Hawking radiation, then that conclusion is independent of what the spacetime looks like outside a sphere of radius, say 10T, surrounding the black hole. That follows from causality. So contrary to your statement, it is indeed very interesting to compare local physics in AdS with local physics in Minkowski, because a) in a region of fixed size the two geometries approach each other to arbitrary accuracy as L_AdS -> \infty, b) BH evaporation is a local process.

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  46. BHG—

    A free particle in an asymptotically flat or dS spacetime, of course. Or are you claiming that you have a theory of quantum gravity that cannot handle those possibilities?

    If a physical state has to obey all the constraints, does that include the Hamiltonian constraint? Then, at least in the bulk, it is the space of solutions, as I said, and your whole argument collapses.

    If all you want are rules for computing probabilities, I can use whatever mathematical methods I like. They need have no physical significance. That's exactly what standard QM does, and I can use the same standards. If the rules have to somehow flow from a physical analysis of the situation, then I can do that too, and without taking an inner product. That's called giving a principled solution to the measurement problem. And you can't do that at all. So what you are saying is that I have to put a physical significance on a completely non-physical method for computing probabilities that has no foundational justification because that's what you do, and nothing else is any good?

    By the bye, with respect to what you said to Sabine, your entire argument has hinged absolutely crucially on the precise asymptotic structure of the space-time. The only structure for which you have any reason to think the Hamiltonian has a discrete spectrum is AdS. But now you are arguing that what happens when a black hole evaporates is a local matter that has nothing to do with the asymptotics. The contradiction in the way you argue from one moment to the next could not be more stark.

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  47. bhg,

    "If under assumptions X (whose nature need not concern us at the moment) it can be established that a black hole which evaporates in a time T does or not does release its information in Hawking radiation, then that conclusion is independent of what the spacetime looks like outside a sphere of radius, say 10T,..."

    Well I think you should concern yourself with the assumptions X. If they depend on what the spacetime looks outside of the sphere, then the conclusion is is not independent on what the spacetime looks like outside of the sphere.

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  48. BHG—

    Suppose the spacetime is asymptotically flat, so the spectrum of the Hamiltonian is not discrete and the Hamiltonian is massively degenerate. Then you have no objection to my solution, and concede it can handle black hole evaporation without the information coming out in Hawking radiation. Now if the black hole evaporates in time T surround it with a sphere of radius 10 T. Clearly, the adequacy of the solution cannot depend on what happens outside the sphere. So even if outside is AdS you have no objection.

    See how that works?

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  49. Tim,

    You're missing the point of the argument. The argument is that in AdS/CFT, the Hamiltonian is discrete and we can conclude from this that, whatever physics is going on, it causes the information to escape from the black hole. We then assume that, if information escapes from a black hole in AdS, it makes sense that it would also escape in an asymptotically flat space since we can make the AdS radius as large as we like and it would be weird if dramatically different physics took place merely because the curvature far from where the black hole is is slightly different. You can reject this assumption, but that's the assumption we have to make if we're going to allow AdS/CFT to inform us at all about our universe. BHG's argument this whole time has merely been that *within AdS/CFT*, information escapes from the black hole. If he demonstrates this, he has succeeded; the implications for our universe is a separate discussion.

    You give this as an example of his supposedly flawed reasoning:

    "Suppose the spacetime is asymptotically flat, so the spectrum of the Hamiltonian is not discrete and the Hamiltonian is massively degenerate. Then you have no objection to my solution, and concede it can handle black hole evaporation without the information coming out in Hawking radiation. Now if the black hole evaporates in time T surround it with a sphere of radius 10 T. Clearly, the adequacy of the solution cannot depend on what happens outside the sphere. So even if outside is AdS you have no objection."

    But BHG is deriving a contradiction between black hole information loss and AdS/CFT. If dropping the assumption of AdS no longer gives a contradiction *using the same argument*, that has no bearing on whether the full quantum theory would or would not contradict your solution via a more complicated argument. He's making inferences about what the full quantum theory must be like based on certain features that show up when we look at AdS. So saying he would have "no objection" to your solution is not true; the objection is that physics should work approximately the same in asymptotically flat space as in a space with a large AdS radius, and we have a contradiction in the case of a large AdS radius, and therefore we have good reason to think there would be a contradiction in the asymptotically flat case *even though we no longer have easy arguments for the contradiction*. It's exactly the same as if we deduced that a scattering process of two particles in a box must be unitary no matter how large the box (using the box to make calculations easier), and then concluded from that that the scattering process of the two particles would still be unitary if they were not in a box.

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  50. travis,

    I'll tell you the same as bhg: Go and write down your assumptions and you will (hopefully) see that you are merely producing logical mush. You write

    "We then assume that, if information escapes from a black hole in AdS, it makes sense that it would also escape in an asymptotically flat space since we can make the AdS radius as large as we like and it would be weird if dramatically different physics took place merely because the curvature far from where the black hole is is slightly different. You can reject this assumption, but that's the assumption we have to make if we're going to allow AdS/CFT to inform us at all about our universe..."

    You are assuming what you want to show, namely that what you conclude in AdS bears relevance for asymptotically flat space, even though you have zero reason to think that's the case. If you believe you have a reason, please state it. An assumption is not a reason.

    As I said above, the benefits from AdS/CFT come from the global structure. An AdS space in the Lambda\to zero limit does not reproduce Minkowski space any more than a sphere of infinite radius is a plane. They're locally equivalent, all right - no one doubts that. But for your (or bhg's) argument to work you need (as Tim points out correctly) the global structure.

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  51. I think BHG and Travis' arguments lately have been flawed, and would refer you to my exchange with BHG of a few days ago, and particularly his last reply.

    Basically, they are making tacit assumptions that need justification, and not providing the justifications. Let me see if I can articulate some of them.

    When BHG makes his 10T radius argument, he overtly assumes "causality", which personally I harbor some doubts about, but let that pass. He tacitly assumes that, in the full QG theory that his favored stringy theory set in AdS_5 x S^5 is meant to be a perturbative approximation to, there are solutions that are compatible with both AdS structure in the bulk and other structures, e.g. dS or asymptotically flat, or whatever. [This is a weird assumption to make, it seems to me, in light of the AdS/CFT conjecture itself. If the conjecture is correct, the bulk QG theory can really ONLY be set in an asymptotically AdS space; a bulk theory set in, say, asymptotically flat space would need to have a degenerate spectrum and hence could not be dual to that very same CFT...]

    Alternatively, he and Travis may be tacitly assuming that the "way BH evaporation works" in this hypothetical QG theory will reveal to us how BH evaporation works in any reasonable QG theory, including those compatible with non-AdS spacetime structures (locally and/or asymptotically). But just writing this assumption down is enough to make it obvious that it is highly questionable, for reasons that have nothing to do with causality/locality. There may be many consistent QG theories out there waiting for us to discover them, and some may only be compatible with certain types of (approximate, classical-level) spacetime structures. Indeed, if there is a consistent stringy QG theory lurking underneath the "AdS" bulk theory we have been discussing here, it seems entirely plausible to me that it may not have any models corresponding to dS or \Lambda > 0 FLRW spacetime. Which is why I asked BHG to tell me what evidence there is out there, for the claim that that whatever QG theory our actual world instantiates, it should be the same in relevant respects as "AdS". But so far he has not answered this request.

    Just to nail down the point: the way that BH evaporation works may be "local" in some important sense, but that does not mean that the way it works in "AdS" is identical to the way it works in any QG theory with any global spacetime structure. Some further argument is needed.

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  52. Sabine and Tim,

    I agree that if you want more than plausibility arguments, you need to use the global structure. I should have been more careful and said it this way: even if we assume that BHG is right that results in AdS tell us about non-AdS and vice versa, Tim's parody of BHG's argument doesn't work because the argument

    lack of argument against viability of solution in non-Ads implies lack of argument against viability in AdS

    is a completely different form of argument compared to BHG's

    nonviability of solution in AdS implies nonviability of solution in non-AdS.

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  53. Carl,

    I agree with your remarks. The question of applicability of AdS results to non-AdS spacetimes is not guaranteed and is just based on plausibility. I think though that the fact which BHG has mentioned that AdS/CFT has been able to calculate the entropy of a black hole which is in perfect agreement with the entropy that was predicted without assuming AdS is pretty good evidence that AdS/CFT contains at least something deep about the way quantum gravity works in general.

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  54. Travis,

    Thanks for reminding me of the evidence coming from the successful calculations performed under AdS/CFT, such as the calculation of BH entropy. I accept that BHG and many other physicists see this as important and even compelling evidence. I think it's also fair to say that many philosophers of physics would find it less compelling (and perhaps also some physicists?). This strikes me as the kind of issue on which perhaps a useful conversation between communities (phil phys and theoretical phys) could take place, if both sides would take time to sketch out their reasons in detail. But it would be important to start with an open mind and a charitable interpretive attitude, on both sides.

    In any case, that would be a new topic, not a continuation of the battle over Tim's paper on info loss.
    C3

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  55. Tim: "No proposed solution of the problem has ever invoked recurrence. If it did, it would argue that the system has to return to the original state, the state without any Hawking radiation. And AdS isn't even globally hyperbolic! There is no way in the world that recurrence can come into the discussion. None."

    Stephen Hawking: "The question of whether information is lost in black holes is investigated using Euclidean path integrals. The formation and evaporation of black holes is regarded as a scattering problem with all measurements being made at infinity. This seems to be well formulated only in asymptotically AdS spacetimes. The path integral over metrics with trivial topology is unitary and information
    preserving. On the other hand, the path integral over metrics with non-trivial topologies leads to correlation functions that decay to zero. Thus at late times only the unitary information preserving path integrals over trivial topologies will contribute. Elementary quantum gravity interactions do not lose information or quantum coherence."

    Juan Maldacena: "We propose a dual non-perturbative description for maximally extended Schwarzschild Anti-de-Sitter spacetimes. The description involves two copies of the conformal field theory associated to the AdS spacetime and an initial entangled state. In this context we also discuss a version of the information loss paradox and its resolution.
    ...
    But since the evolution is just adding
    phases to the state, by the quantum version of the Poincare recurrence theorem after a
    very long time will get arbitrarily close to the initial state and therefore we would recover the geometric interpretation, but with slightly different initial conditions..."

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  56. Tim,

    "Suppose the spacetime is asymptotically flat, so the spectrum of the Hamiltonian is not discrete and the Hamiltonian is massively degenerate. Then you have no objection to my solution, and concede it can handle black hole evaporation without the information coming out in Hawking radiation. Now if the black hole evaporates in time T surround it with a sphere of radius 10 T. Clearly, the adequacy of the solution cannot depend on what happens outside the sphere. So even if outside is AdS you have no objection."

    That is flawed logic, to put it mildly. It is logically equivalent to the following argument. Let's consider the hypothesis that Mary's eyes are blue. If I am in a dark room with Mary I indeed can't come up with any argument as to why they are not instead, say, green. However, when we walk out into the sunlight I can look and see that they are (perhaps) green. Just as there is some environment (a dark room) in which I can't refute the blue eye hypothesis, so too is there an environment (asymptotically flat space) in which I can't rule out your scenario. However, just as I can turn on the light and see the color of Mary's eyes, so too can I put the black hole in AdS and answer the question there. And crucially, I can do this for an AdS radius of arbitrarily large size, such that the local environment of the black hole is indistinguishable in the two cases. So unless you are going to defend the rather bizarre claim that local physics in AdS of arbitrarily large curvature is measurably different than local physics in flat space, the conclusion follows. And if you do want to defend this bizarre claim you are essentially abandoning everything we know about physics.

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  57. Bee,

    Your comments don't make much sense to me, so perhaps it's simplest for me to point out that what you are claiming conflicts with explicit computations. Namely, one can do CFT computations corresponding to colliding particles in a local region of AdS. They then take the AdS radius to infinity to get an expression for an S-matrix, which can include quantum gravity effects. They can then do a completely separate scattering computation in asymptotically flat spacetime. The two S-matrices agree. So your statement that there is "zero reason" to conclude that "AdS bears relevance for asymptotically flat space" is both odd on its face and explicitly disproved. There was even an example of such a paper on the arxiv yesterday! The formation and evaporation of a black hole is just a type of scattering process, and there is every expectation to believe that the same story holds there.

    Important point: I have mentioned this many times, but I am emphatically not claiming that this "proves" that black holes in flat space release their information in Hawking radiation, for the simple reason that there might not be a unique answer to this question. But this is totally separate from the misguided belief that local physics in AdS with a length scale of 46 googleplex light years bears no relation to local physics in flat space.

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  58. Tim,

    "If a physical state has to obey all the constraints, does that include the Hamiltonian constraint? Then, at least in the bulk, it is the space of solutions, as I said, and your whole argument collapses. "

    Of course it includes the Hamiltonian constraint. The rest of your message makes no sense.

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  59. bhg,

    As I have told you several times now, no one doubts that the local processes can be mapped onto each other. It would be odd if that wasn't so. What I am telling you is that you need the global or at least non-local properties to carry over to Minkowski-space to get any benefit out of the whole game, and you have zero reason to expect that to be the case *exactly because* the benefits come from AdS having a conformal boundary. An infinitely large box is not no box - the former has a boundary, the latter doesn't. If you think they're both the same, you are assuming what you want to prove. Again, please make an effort and write down your assumptions and conclusions, maybe it'll help.

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  60. Bee,

    If you really believe this I don't know how you think any physics can ever be done. The only assumption being made here is the crudest form of locality. Most computations in QFT are based on Lorentz invariance whereas we live in a Universe which is not asymptotically Minkowskian -- it might be De Sitter, but truthfully we have no idea. Nonethless, people compute S-matrix elements for the Standard Model and use them to make predictions at colliders, even though the entire foundation of S-matrix theory (based on LSZ etc) heavily uses exact aymptotic Lorentz invariance. According to your argument this is all based on unwarranted assumptions about global properties, and we should distrust the whole process, there are no "benefits" to it etc. Conceptually, using the CFT to compute local processes is absolutely no different than using S-matrix theory to compute the same. IF you want to fret about this, be my guest. Fortunately, other people have done actual computations showing that things work as expected: both the CFT and S-matrix theory correctly describe local physics in spacetimes that are completely different globally. Do you think they just got miraculously lucky? If not, I don't see your point at all.

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  61. BHG—

    Of course it is flawed logic! That's the whole point of it. It is an analog of your argument, constructed to show how flawed it is. Maybe you need to slow down so you reflect on what is being said and the points that are being made. And maybe you can respond to an argument with something more helpful that "That makes no sense." My arguments have premises and argue to conclusions. If there is something wrong, then it is in a premise or an inference. If there is a sentence you can't follow then it is a particular sentence. Make a specific objection or ask a specific clarificatory question, or else you don't have a rational objection and you are not entitled to make any complaint.

    So in the physical Hilbert space the bulk has to obey the Hamiltonian constraint. Excellent! The Hamiltonian constraint plays in the WdW patch the same role as the boundary Hamiltonian plays for the part of the wavefunction out at the boundary. In the bulk, the right thing to do (indeed the only thing to do) is to think of the wavefunction as characterizing the entire solution in the WdW patch. That is, the elements of the physical Hilbert space correspond to full solutions, not to the states on spacelike hypersurfaces. In fact, this is absolutely necessary for the argument you want to give: take a pair of maximal hypersurfaces that coincide from some point onward out to the AdS boundary but diverge inside. If these count as *different* elements of the physical Hilbert space then your claim that the spectrum of the Hamiltonian is not massively degenerate is clearly false. So they must count as *the same* element of the Hilbert space. Which is to say, as I have said over and over, that every element of the physical Hilbert space corresponds to a whole space-time, not to a unique state on a hypersurface. If the Cauchy surfaces of a Wheeler deWitt patch become disconnected, then the spectrum of the Hamiltonian must be the same as the spectrum for the connected hypersurfaces. And if your arguments show that the spectrum of the latter is discrete, then so is the spectrum of physical states on the former. So, as I said, your whole argument collapses.

    I concur also with Sabine: try to actually write down your arguments in explicit detail. Then we can note, for example, that this sentence: "So unless you are going to defend the rather bizarre claim that local physics in AdS of arbitrarily large curvature is measurably different than local physics in flat space, the conclusion follows. And if you do want to defend this bizarre claim you are essentially abandoning everything we know about physics" can be used against you. Your argument—which I don't accept anyway—that my solution cannot be right hinges critically on the discreteness and non-degeneracy of the spectrum. But that in turn depends on what the boundary is like, even if the boundary is unimaginably far away. But, as you say, the evaporation is a local process. So you are claiming that the fundamental nature of a local process depends critically on the structure of a boundary that can be arbitrarily far away. But then "you are essentially abandoning everything we know about physics". Once again, you want to have your cake and eat it too: insist on the critical importance of the AdS boundary in one breath and then deny that it can have any relevance to the evaporation process in the next. If it takes you longer to work things out and get your arguments straight, just take the time. And if you can't come up with a specific objection or request for clarification then don't make any objection at all.

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  62. Travis,

    Your characterization of the situation is interesting because it misses the mark with respect to what BHG and I have been arguing about all this time, so if the characterization seems right to him then we have hit upon one reason this argument never ends.

    There is no issue between us about whether information is lost. I have never been trying to advocate that information is lost. Indeed, I have consistently been arguing that information is not lost (well, at least in a globally hyperbolic setting) *in the only sense of "not being lost" that there was ever any rational ground to be interested in*. For example, suppose I give you the state on a maximal hypersurface (so in a globally hyperbolic setting on a Cauchy surface). Then I grab that away from you and replace it with the state on a proper part of that maximal hypersurface (and ergo, in a globally hyperbolic setting, on a non-Cauchy surface). Have you lost information in that exchange? Of course you have! So what? That is trivial, and has no particular bearing on any interesting question of physics. And you will lose information similarly in the transition from the state on a maximal hypersurface to the state on a non-maximal hypersurface at a later time. That is again uninteresting and trivial and cannot violate any fundamental principle of QM or anything else. BHG and I agree that no information is lost in the interesting sense and of course that information is lost in the boring a trivial sense. So how do we disagree?

    BHG thinks that it follows from the absence of interesting information loss that the information about what fell in to form the black hole must come out in the Hawking radiation, so the radiation cannot really be thermal. I have been arguing that the information never comes out in the Hawking radiation—indeed, it would violate QM if it did come out that way!—because the information is still in the interior of the black hole even after the black hole horizon has evaporated. We do not disagree about whether it is lost, we disagree about how it is preserved. BHG thinks it is preserved on the region connected to the AdS boundary and I do not.

    How does AdS/CFT come into this dispute? The connection is actually very roundabout and honestly still obscure. You write: "The argument is that in AdS/CFT, the Hamiltonian is discrete and we can conclude from this that, whatever physics is going on, it causes the information to escape from the black hole." Well what does the discreteness of the spectrum of the Hamiltonian have to do with anything here? And indeed, what does the "discreteness of the spectrum of the Hamiltonian" even *mean*? This is what we have been arguing about. Note that BHG says that he has an argument that the spectrum of the AdS gravity theory must be discrete that is based on the structure of AdS, making no mention of the CFT at all. Anyway, as far as I can tell, BHG has been trying to argue that no maximal hypersurface can be disconnected because that would imply that the spectrum of the Hamiltonian is continuous and massively degenerate rather than discrete and non-degenerate (up to symmetries). I have yet to be able to make out what that argument is supposed to be. That has gotten us bogged down in the definition of the "physical Hilbert space", etc., etc.

    And now that I come to think on it, I can't even see the *possibility* of any argument that the spectrum is not discrete if there are disconnected parts to a maximal hypersurface. If the AdS structure of the asymptotic geometry comes from the value of lambda, then each disconnected piece will be AdS. And if being AdS entails discreteness, then the spectrum of the Hamiltonian will be discrete for each taken by itself and therefore discrete with both taken together. So I see no hope of any argument that there is anything wrong with my solution.

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  63. Physphil,

    I say that no solution to the information loss paradox makes mention of recurrence. You then post a quote from Harking that make no mention of recurrence. And then a quote from Maldacena that mentions recurrence, but not in the context of the information loss problem.I can't see what relevance either of the quotes have to my claim. Since AdS is not even globally hyperbolic you have zero chance of proving an interesting recurrence theorem for it.

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  64. bhg,

    As I have said several times, no one doubts that AdS (or dS or really any other sufficiently smooth manifold) is locally flat and hence you can transfer local results from one to the other provided they're 'local enough' in quantifiable ways.

    What I am, evidently pretty unsuccessfully, telling you is that the whole reason you have a benefit from AdS is that it's not *globally* the same as Minkowski (or dS for that matter). If it was, then why don't you go and do your unitarity argument in Minkowski space to begin with. Oh, it doesn't have a conformal boundary, oopsi. You are *assuming* that matching the local processes is sufficient to carry over the global benefits, not showing that this is so.

    I am telling you that it's incoherent: You can't on the one hand use the global structure of asymp AdS to argue you solve the bh infloss problem and then claim because the physics is locally the same as in asym Minkowski this should work in our universe too.

    Take your example with blue-eyed Mary, but forget about the color and ask what's the size of her pupils. Maybe not such a catchy example, but it'll do. The moment you take her out of the room the size will change. If you want to make an inference from the outside to the inside, you have to do more than just looking at her. I don't want to make this analogy more contrived than it is already, but maybe it helps you see the missing step.

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  65. Tim,

    By information loss, I just meant information loss from the exterior; it stays in the interior and does not come out in the Hawking radiation. So we all agree here on what we are talking about.

    The discreteness of the spectrum comes into play because you need it for the recurrence theorem to work (recurrence is only guaranteed for quantum systems with discrete spectra). For concreteness, we can say that the spectrum refers to the spectrum of the CFT, about which there is no ambiguity. Then, under the assumption that there is a one-to-one correspondence between states of the CFT and states in the bulk, we then deduce recurrence in the bulk. So it's definitely the case that, if AdS/CFT is true, you have to eventually end up arbitrarily close to your original state. You don't need to assume anything about global hyperbolicity or anything. It follows entirely from the discreteness of the CFT spectrum.

    Like you said, the degeneracy argument is the main argument BHG has been making. This argument does not depend on discreteness in any way, at least as far as I can tell. We just start with the known fact that the CFT spectrum is nondegenerate (up to symmetries). The argument is that in your model, at late enough boundary times, there would have to be states which only contain amplitudes assigned to disconnected surfaces. Since any state can be written as a superposition of energy eigenstates, the state at late enough boundary time would have to be a superposition of energy eigenstates which themselves contain only disconnected surfaces. Something something something degenerate energy spectrum, contradiction. I'm honestly a little confused how this part of the argument is supposed to go.


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  66. Tim,

    you write "And then a quote from Maldacena that mentions recurrence, but not in the context of the information loss problem." Earlier you wrote "No proposed solution of the problem has ever invoked recurrence...There is no way in the world that recurrence can come into the discussion. None."

    The first part of the Maldacena quote is the abstract of a famous paper on black holes in anti de sitter spacetime from 2001, where he writes "...we also discuss a version of the information loss paradox and its resolution." The second part of quote is from the conclusion of same paper, where he explains how recurrences are related to resolving the info paradox. It would be nice to see you admit you are wrong in such a clear case. But, I do not think you can do that.

    The Hawking quote is from a paper from 2004 where he says he resolves info paradox, using Maldacena's idea from the 2001 paper (he does not say that they are Maldacena's ideas, but they are). Hawking does not call it recurrence but it is the same physics as in Maldacena.

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  67. Bee,

    imagine we make black hole in AdS by firing particles inside from a big spherical device. Black hole is very small and AdS radius very big. Hole evaporates long before earliest Hawking particle travels one AdS radius distance or even reaches back to device. Spherical device is also a detector and is also much smaller than one AdS radius (but much bigger than hole). It catches all Hawking particles and measures them. After many repetitions we can be sure particles are in pure state (or not).

    This is experiment in AdS, but AdS curvature is not important and we can make it less and less important by making AdS bigger and bigger. Entire experiment can be described in AdS/CFT if that is true duality. For reasons bhg says there cannot be any remnant or baby universe left after evaporation according to CFT. So, if AdS/CFT is true it shows that there is one theory of quantum gravity where black holes form and evaporate without leaving remnants or baby universes behind in a space that is arbitrarily close to flat. I think that is very interesting and powerful statement even if it does not prove anything about strict Minkowski limit. You do not agree?

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  68. physphil,

    I agree that it's interesting. Gravity in 2 dimensions is also interesting. Unfortunately it doesn't describe our universe.

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  69. physphill—

    I am going to surprise you here. I freely admit I was wrong about no one being silly enough to advert to recurrence in this context. And I freely admit to not having read much of this literature. And I freely admit to having formed the opinion—on the basis of how it is spoken of—that Maldacena's work which is supposed to be so important is somewhat serious, despite the fact that I know that EPR = ER is ridiculous. So I made the error of being charitable where I ought not to have been. My bad. I thought I could not underestimate how poor the work by esteemed physicists in this field could be, but I was apparently wrong about that. What you have opened my eyes to seems to be that physicists have been taking seriously claims that are even more nonsensical than I had imagined. I am grateful to you for that.

    Since I somehow imagine that you will not actually appreciate my openly and freely admitting my error here, given the accompanying commentary, let me explain. Do you have any conception of the recurrence times for a classical box of gas? Or that the recurrence times go up as the box gets bigger? So if you have a proof that makes any reference at all to the recurrence time for the matter that collapsed to form an actual black hole, where you are treating the dimensions of the entire AdS spacetime as the "box", you are making reference to a time period so unimaginably enormously longer than the projected time for the universe to come to thermal equilibrium that the numbers involved—without even calculating them—or off the scale of anything that could possibly be relevant to any actual phenomenon? Or did you—or Maldacena or Hawking—ever consider that recurrences are recurrences, i.e. returns to something close to the original state of the system, i.e. in this case the state of a collapsing star? So you are tacitly making reference to the following scenario: a large star, having fused all of its hydrogen and helium into heavier elements collapses to a large black hole, which then slowly evaporates by putting out Hawking radiation, most of which is in the form of thermal gamma rays. And then you are going to wait long enough for that thermal photon gas, because it is confined by an AdS boundary, to somehow evolve back again into a collapsing ball of heavy elements. And in all of this the AdS boundary, which you are likening to the walls of an infinite potential well, do not even form a globally hyperbolic space-time. And you are presenting an evolution on this unimaginable time-scale as somehow relevant to understanding the nature of the Hakwing radiation?

    In short, the idea of referring to recurrence in this context is so completely absurd that I could not imagine anyone bringing it up. But you have proven me 100% wrong.

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  70. Bee,


    I sincerely hope I can get you to see the light on this point -- here comes my best shot. Keep in mind that what I am specifically arguing against is your statement that there is "no reason to believe" that we can carry over results about BH evap from AdS to Minkowski in the way I am claiming. My position is *not* that I have some "rigorous proof", but rather there is very strong reason to believe we can do this.


    First a question: consider two experiments in which I collide matter to form a black hole that then evaporates. The first experiment occurs in a bubble of Minkowski space of very large radius R embedded in asymptotically AdS space. The second occurs in Minkowksi. If you were shown the results of the first experiment, are you really arguing that there is "no reason to believe" that the results of the second experiment will be the same up to corrections that are suppressed by the small parameter R_{BH}/R ? If yes, then I have to question your judgement. If no, then no matter what theoretical method I use to compute the outcome of experiment of expt 1, the computation will apply with high accuracy to expt 2


    Next, you ask why don't I run the arguments in Minkowski to begin with? As anyone who has studied GR beyond the intro level knows, asymptotically flat spacetimes are far more subtle than AdS. But this is due to IR issues that (apparently) have nothing to do with local physics. For example, the asymptotic symmetries of asymp flat space continue to be studied and argued about to this day, while the AdS case was easily handled long ago. By putting the BH in AdS we can remove all the complications of what is going on arbitrarily far away and focus efficiently on the physics at hand. We do this kind of maneuver all the time, and I will give you two examples.


    First, we live in something like de Sitter space, yet we do QFT computations relevant to colliders by assuming Lorentz invariance. Your stated objection could be carried over word for word to this case, just by substituting AdS -> Minkowski, Minkowski -> de Sitter. Trying to formulate an S-matrix in de Sitter is fiendishly subtle, so we instead do the computation in Minkowski, even though the "entire benefit" is coming from the different global structure. The situation is exactly parallel: we do this because of the reasonable assumption that local physics doesn't jump discontinuously between Minkowski and large radius de Sitter. But given your position, I don't see why you aren't complaining that we have "no reason to believe" in the validity of S-matrix computations as applied to our world, since no one has proven that there is no discontinuity.

    cont

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  71. cont

    Second, as undergrads we learn about the technical simplification of putting a system in a box, computing, and then taking the box size to infinity. This makes life easier in many ways, and we are told that as long as you are studying some local process you should get the same answer as if you did the computation in infinite space. Why doesn't this also bother you? How do we know that if we do this for some strongly interacting QFT (where we can't establish the validity of the procedure explicitly) we won't get the wrong answer? Again, the situation in our case is the same: The simplifying feature of AdS is that it functions as a (very symmetric) box, and extracting Minkowski space by taking L_AdS->\infty is no different than taking our box size to infinity.

    Finally, you said that "you can transfer local results from one to the other provided they're 'local enough' in quantifiable ways."." Good, so you at least agree that this is procedure is sometime sensible. Is it sensible for colliding electrons at 1 TeV, say? How about at 100 Tev? How about at 100 M_planck? Do you have some reason to believe that there is some sudden transition where these collision processes, all of which are occurring in a small bounded region of spacetime, become sensitive to the global structure of spacetime? It's logically possible, but surely not the expected outcome given everything we know about physics. The "quantifiable" measure that you request is clear: this procedure should work if the distance scale R of the measurement is small compared to the AdS radius. Can I prove this? No. But saying that "we have no reason to believe" its validity is a very odd claim to make.

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  72. Tim,

    I do think it's important that you make some legitimate attempt to address the recurrence argument. As physphil says, this argument has been discussed in around 100 or so papers since Maldacena's original one -- it is not some very recent idea.

    Wallace's argument rigorously establishes that the wavefunction returns arbitrarily close to itself at some finite time t. So if the information about the state is present in the AdS region at some original time, then it will also be at some later time. This is in conflict with your proposal, since you are (as far as I can tell) claiming that information about the state is irretrievably lost to a disconnected region of spacetime.

    This conclusion of this argument is much weaker than what one ultimately wants to show, which is why I didn't mention it until recently. But your proposal is so strong that even this weak conclusion rules it out.

    Global hyperbolicity is a non-issue: AdS spacetime with standard boundary condition is every bit as predictive as a global hyperbolic spacetime. Again, please consult the textbooks on this. Even the simplest computations reveal this fact, which is known to everyone in the field.

    Anyway, if you don't even try to respond to this argument it's hard to take you seriously.

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  73. bhg,

    "First a question: consider two experiments in which I collide matter to form a black hole that then evaporates. The first experiment occurs in a bubble of Minkowski space of very large radius R embedded in asymptotically AdS space. The second occurs in Minkowksi. If you were shown the results of the first experiment, are you really arguing that there is "no reason to believe" that the results of the second experiment will be the same up to corrections that are suppressed by the small parameter R_{BH}/R ? If yes, then I have to question your judgement. "

    AdS has a second length scale which is the AdS radius. If you make your bubble large enough you'll see corrections from that.

    "Your stated objection could be carried over word for word to this case, just by substituting AdS -> Minkowski, Minkowski -> de Sitter."

    Neither Minkowski nor de Sitter have a conformal boundary, so they are much more similar than AdS is to either. Having said that, I actually do think there are various QFT theorems for Minkowski space that do not apply for dS and we hence shouldn't worry about, but that's a different thing altogether.

    As to your QFT-box, this box is merely an intuitive way of doing an epsilon \to zero limit. The usual boxes of course do not restrict what you call 'physical states'. In AdS/CFT in contrast, the CFT on the boundary is the whole reason you can claim that unitarity is conserved and you do this by virtue of removing states in the bulk that would conflict with that (see earlier discussion about states not reaching the boundary and/or Tim's disconnected hypersurfaces). You can take the AdS radius all the way out to infinity, but once you remove the boundary, you throw away the extra constraints that you argued came with it. Hence, you have no reason to think the results carry over.

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  74. Come to think of it, if you'd take this limit correctly you would probably get gravity in asymptotic Minkowski-space with a CFT lumped onto it that doesn't live anywhere in particular but that (in some sense) keeps track of what's going on in the Minkowski-space. What I am saying is that I don't see a reason to think that's what's actually going on with gravity in our universe.

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  75. What is weird:

    1. The global structure - topological and geometric - of spacetime seems to be deeply implicated in holography, unitarity and information-loss in quantum mechanics. It would be nice if we could reduce each of these properties to something that can be checked with a local computation, e.g., the change of an amount of stuff in a volume is accounted for by the inflows and outflows of stuff.

    2. If you could show, eg., there is some kind of boundary theory, like not conformal, on any boundary of a spacetime volume with gravity, that perhaps exceptionally simplifies in the case of AdS, then the handwaving would not be needed.

    ---

    Q1. The matching of counts of degrees of freedom in the bulk and in the CFT on the boundary of a spacetime at least in the original AdS/CFT required a large-N Yang Mills, N --> infinity; it is not clear what happened to this requirement in the current discussion of AdS/CFT?

    Q2. What can we say about the CFT ground state if there are N to some positive power of lowest energy excitations of mass N to some negative power, as N --> infinity?

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  76. BHG—

    Quite by coincidence I was talking to someone today who was aware of a distinction between the Hilbert space and the "physical" Hilbert space. At least a distinction between those is explicitly made by Shankar. Here is the quote:

    "In restricting ourselves to real k we have restricted ourselves to what we will call the physical Hilbert space, which is of interest in quantum mechanics. This space is defined as the space of functions that can be either normalized to unity or to the Dirac delta function, and plays a central role in quantum mechanics. (We use the qualifier "physical" to distinguish it from the Hilbert space as defined by mathematicians, which contain only proper vectors, i.e. vectors normalizable to unity. The role of improper vectors in quantum theory will be clear later)" p. 72.

    Is this what you have had in mind by the "physical Hilbert space"? If so, there has been a lot of time wasted by not making that clear. The terminology was not familiar to me, and is very misleading, since the "physical Hilbert space" is not a Hilbert space and the improper vectors are not vectors and also do not seem to correspond to physical states. I have been using the mathematician's usage, as defined above. So can you say whether this is what you having been meaning ?

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  77. Bee,

    "AdS has a second length scale which is the AdS radius. If you make your bubble large enough you'll see corrections from that. "


    That's not really answering the question. Let me try again. Take R_AdS >> R_{bubble} >> R_{black hole} with the ratios as large as you like. Am I correct that you think "there is no reason to believe" that the outcome of black hole evaporation in such a case will be the same as in Minkowski space, up to corrections suppressed by the small ratios?


    I am not going to argue with you anymore about this, but if you answer "yes" I will say that if I shared your opinion I would probably quit physics. By the same token, any QFT computations based on Lorentz invariance are totally inapplicable to our universe, which is not Lorentz invariant (sorry, but de Sitter is clearly very different from Minkowski at the global level, more so than AdS differs from Minkowski). I suppose you must regard it as an amazing coincidence that all of the AdS computations people have done in this regime (from black hole entropy, including quantum gravity corrections, to scattering amplitudes) have agreed with those performed in Minkowski spacetime.

    Anyway, you are entitled to your opinion, and I will now stop griping.

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  78. bhg,

    No, I don't think it's a coincidence. As I said earlier, you have constructed the theory so that these results agree.

    Yes, that's exactly right. You have no reason to believe the outcome is the same because you have used a global constraint which you do not have for \Lambda = 0, hence no reason to think the limit \Lambda \to 0 is continuous.

    "if I shared your opinion I would probably quit physics"

    It is clear to me that you don't want to consider what I am telling you may be correct, but I hope you'll at least think about it. The whole miracle of AdS/CFT comes from the global structure. Solving the bh infloss problem while maintaining the BH entropy requires non-local corrections to the Hawking-process (see eg the firewall issue). The idea that demonstrating that local processes in AdS are the same as those in Minkowski space means the solution carries over doesn't make any sense whatsoever.

    I think Tim used the right idiom there when he said you can't have your cake and eat it too.

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  79. Sabine,

    I wrote up a post on recurrence that seems to have gone astray. Can you check?

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  80. Tim, can you clarify how your post on recurrence answers the point that Travis and BHG have been making? Travis said, several posts ago, that if a quantum system has a discrete spectrum of the Hamiltonian, then a quantum recurrence theorem applies. The argument then seems pretty simple/compelling: If the recurrence theorem applies to black holes in AdS, then your scenario is ruled out because the information gets stuck for the rest of time inside the event horizon (or so everyone seems to be assuming). Therefore even if a new star forms after all the Hawking radiation bounces off the boundary and returns, and it then collapses and forms a black hole, it shouldn't be the same sort of state as before, in fine detail. But the quantum recurrence theorem (I guess) does guarantee the same sort of state as before, in fine detali.

    Which part of this argument do you reject?

    From your post addressing Physphill, all I got was that the recurrence time would be unimaginably long. The answer to that, it seems to me, is "so what"? The argument is just supposed to establish that the information must get out of the black hole if we assume AdS/CFT, and I don't see how the length of the recurrence time matter for that question.

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  81. Bee,

    "Yes, that's exactly right. You have no reason to believe the outcome is the same because you have used a global constraint which you do not have for \Lambda = 0, hence no reason to think the limit \Lambda \to 0 is continuous. "


    Just one followup on this, since I can't tell if you are actually answering my question due to the appearance of the mysterious phrase "you have used a global constraint..." To be totally clear, my question has nothing to do with AdS/CFT, so please put that out of your mind for a moment. I am asking you if, having witnessed an experiment (in particular BH formation/evap) in the Minkowski bubble, you say there is "no reason to believe" the result will carry over to Minkowski itself, even if the bubble and AdS radius are arbitrarily large. I am not "imposing any constraint" here beyond specifying the experiment s I have just described. If you would just make clear that you are answering *this* question, I will drop the issue.

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  82. Carl,

    To be fair to Tim, the recurrence theorem also "rules out" the second law of thermodynamics. If Tim is going to claim that the information is always irretrievably lost to the interior forever, even on ridiculously long time scales, then he would be in conflict with AdS/CFT. But if he merely wants his solution to be as correct as the second law of thermodynamics, that is, he wants a solution that follows from any non-hyperfinetuned initial condition and remains a good description of the physics for an extremely long time, then the recurrence theorem won't hurt him.

    Tim, can you verify that you agree at least that on ridiculously long time scales, recurrence of the bulk state must occur in AdS/CFT?

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  83. By the same token, any QFT computations based on Lorentz invariance are totally inapplicable to our universe, which is not Lorentz invariant (sorry, but de Sitter is clearly very different from Minkowski at the global level, more so than AdS differs from Minkowski).

    Ouch. Lorentz invariance relates the observations of observers with different 4-velocities at a point in space-time; i.e., Lorentz invariance applies in the tangent space. To relate observations at different points, one must use the connection given by the space-time metric.

    Think of the two-metric on the surface of a balloon. It is Euclidean everywhere.

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  84. Carl,

    The point is that recurrence times are so mind-boggling long that they can't have anything to do with a question about what happens in the process under consideration. I mean, this is much more obvious than BHG's attempt to appeal to the size of the boundaries in his example, which he mistakenly thought he could appeal to. The difference in the two cases is that his entire argument up to that point had appealed to the effect of the AdS boundary in a critical way: it way central to the argument. But you can't do that and then, another context, go on to argue that the nature of the boundary is irrelevant to the process, so it will also happen in Minkowski. That's the "have your cake and eat it too". Now in this case, none of my argument had to do in any essential way with the very-long-term future. And as such, it is just obvious that the situation can't depend on what happens in such an absurdly, unphysically long term-future. I mean that could only to at all relevant if you thought that the theory you were analyzing were the perfectly and precisely true ultimate theory of physics, and also if none of you arguments ever made an approximation of any kind, threw away any higher-order terms, etc, etc. That's why Boltzmann's response to the recurrence argument was "You should live so long". He was not concerned with Poincare, and was 100% right. All he had to do was concede that his argument was not a 100% rigorous derivation, which physics arguments never are. (Recall the amount of trouble Russell and Whitehead had to do through to make "1 + 1 = 2" a logically rigorous theorem, and there your are dealing with mathematical claims that are perfectly precise. Physics is just never meant to be that precise: all the results worth taking seriously must be robust against some epsilonics. So anyone who thought that Maxwell's account of the Second Law was flawed because of recurrence just did not understand the sort of certainty and precision that pervades such a physical argument. As Aristotle said, only expect as much precision in a science as the subject matter allows. None of the arguments in this part of physics are proof against what happens in such a long term future. (Boltzmann had to worry more about the time-reversal argument than the recurrence argument, and even that he could immediately see how to answer. Recurrence he just rightly dismissed.)

    Now if you want my response to be more precise, how's this: recurrence requires that a state eventually evolve to a state *abritrarily close* to the initial state, not that it recur to the initial state. Which it typically never will. Never. So recurrence always loses information.It can't be relevant to the question of information loss.

    It is just a matter of keeping track of what principles are actually used in an argument. BHG's arguments, for example, have turned critically on the spectrum of the Hamiltonian being discrete. Critically. That being so, to argue that you would have to get the same result in Minkowski because, well, if the discrete eigenvalues are plenteous enough the spectrum "looks" continuous, is the height of absurdity: you switch from saying that the fine structure is critical to the argument to saying it can't really be important. Having cake and eating it.

    The point about recurrence being an approximate concept and the sense of "information loss" we are worried about being precise is pointing out a conceptual mismatch of two contexts in which the claims are made. I can't be much more precise about what I am reacting to...if you are following the arguments in a certain way it is just manifestly clear that recurrence just cannot be relevant.Similarly, if someone pointed out that my solution to a paradox requires a violation of the Second Law I would be worried. But if they the said that to observe the violation you had to wait recurrence time, then I would just laugh. You should live so long.

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  85. Tim,

    On the recurrence argument: since you offered no counter argument (just childish insults and a rambling word salad), I take it you concede that this kills your scenario. One can show rigorously that the CFT has a *finite* recurrence time, whereas in your scenario, as you have explained it, there are no recurrences. Case closed. Wallace clearly understands this point.

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  86. bhg,

    From your question it is clear that you are still avoiding even thinking about what I am telling you.

    You are basing your argument on some 'physics intuition' which you don't have (noone has ever done an experiment in AdS) and that, moreover, isn't logically consistent. You are avoiding to address that AdS/CFT is not just physics in AdS. We have talked about this several times before. For AdS/CFT to work you must have restrictions on the fields that propagate in the bulk because you need them to be expandable around the boundary. Without this requirement the duality will not work. What I am telling you is that this is a global constraint (that results in non-local correlations) which are responsible for the AdS/CFT magic. If you put a bubble in AdS, this selection will still be there - but it's not encoded in the bubble. It will vanish the moment you remove the boundary. That's what I am telling you: The local processes in the bubbles may be the same, but that doesn't prove what you want to prove.

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  87. Bee,

    Is it too much to ask that you answer the question? Is it not clearly stated? I don't understand the evasiveness (actually, I do). Once again, with emphasis, my question has NOTHING TO DO WITH ADS/CFT, it is a question about locality. Just ignore AdS/CFT for the moment.

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  88. Tim,

    BHG is not trying to have his cake and eat it too. We can plausibly expect that a black hole will behave in approximately the same way in an AdS with a very large radius as it will in dS, while at the same time utilizing the ease of calculation in AdS to extract information about what that behavior must be. The claim that information does not come out in the Hawking radiation in dS but does in AdS would be a much more radical and implausible claim than the claim that BHG is making.

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  89. Travis,

    No, I would not concede that at all. This idea that the Poincaré recurrence theorem has any validity at all here has no plausibility, and even if it were rigorously accurate, it would have no bearing on the information loss scenario as I have already explained.

    The recurrence theorem can be proven for a system with a finite phase space whose dynamics obeys the Liouville theorem. That's what Poincaré proved. In classical theory, to get a finite phase space you put up an infinite potential. Then it is easy to see that the theorem follows, but bear in mind that "recurrence" does not mean "return to the starting state", it means "return to arbitrarily close to the starting state".

    Now: in AdS/CFT do we actually have a system with a finite phase space (or in the relevant sense space of possible physical states) and a dynamics that obeys Liouville's theorem (or an appropriate analog to Liouville's theorem? This is not even a well-posed question, because it assumes that we have a perfectly well-defined theory under consideration. When it suits him, BHG will insist that of course we have no idea what goes on at Planck scale, or as you approach a singularity, etc. And without all those precise details you don't have a precise enough theory to ask any questions of (especially when you realize the time scale on which you demand that the conditions rigorously hold). As I have said over and over, AdS is not even globally hyperbolic! And the theory that is supposed to be dual to the CFT is some sort of limit as N goes to infinity. If you think that there is a well-defined theory here that can even be analyzed you are kidding yourself.

    If you want to understand the process of Hawking radiation and black hole shrinkage then focus on giving a physical account of that. Even if the account is only approximately correct, if it can get you to an understandable post-evaporation state then you have at least sketch an approach to this problem. As soon as you say that the issue is not getting a post-evaporation state but to the state that would obtain after the recurrence time, you have lost touch with any coherent physical or mathematical problem.

    Anyway, my paper certainly does not offer any precise theory of quantum gravity to be analyzed for recurrence. What I offered was an argument that 1) the standard presentation of the so-called paradox, if analyzed by straightforward accepted principles drawn from GR, yields a completely non-paradoxical solution and 2) Hawking—and many others—missed that solution because of a possibility that they were conceptually blind to: that the "missing information" can exist in the interior of the black hole after the black hole has evaporated. Hawking's explicit question—Where did the information go?—is answered cleanly and completely. I do not claim to have worked out all the details of such a solution, and indeed there are various important choice points in implementing the solution such as whether to put in the EE and break the manifold structure, that would have to be addressed. But my point is that there is no objection at all to this obvious solution once you get the conceptual situation clear. The fact that several people have mentioned the paper of Banks, Susskind and Peskin as a convincing objection to this view, when that paper is incorrect and has been known to be for decades, indicates that this solution has not been seriously considered. What we seem to have is something parallel to the use of the von Neumann "no hidden variables" theorem: the theorem itself is, as Bell said, "foolish", and was seen immediately to be foolish by Grete Hermann and Einstein. But that did not stop generations of physicists, most of whom had never even read it, from citing it as definitive. All of this AdS/CFT literature and Information Loss literature appears to be no better. Or at least BHG, who professes to be an expert, cannot cite anything better.

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  90. BHG

    I concede no such thing. Three observations.

    1) I actually have given arguments. If you think they are "word salad", then cite the specific sentences you have trouble understanding. If you can't, then you are just bluffing.

    2) Your response to Sabine at one point was if Sabine was correct then you would probably quit physics. That is not, in any way shape or form, an actual response to Sabine's argument, which I completely endorse. The fact that you are reduced to such a display is rather telling.

    3) Now you have jumped on the idea that some recurrence theorem really holds the key to refuting my scenario. We are approaching 1,100 posts, and this has been going on nearly a year. If you can cite a single place anywhere where you so much as mentioned recurrence before you cited David Wallace's paper, please do so. I have no memory of such a thing. So if recurrence is somehow really the key, and if Wallace is expositing the true argument that all the cognoscenti understand, then it appears that you yourself did not understand it until you learned it from David. If so, please just concede that you don't know what you are talking about and stop wasting all our time.

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  91. bhg,

    No need to get upset. I already answered that question way above, so I simply don't understand why you keep repeating it. If you have a locally flat patch in either space, the physics is plausibly the same. Why do you want me to repeat my answer? I also told you several times already that this isn't relevant. Do I need to boldface that you can't use local identifications to carry over global conclusions for you to get it?

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  92. Travis,

    Yes he is. The argument that there will be recurrence at all depends on the discreteness and non-degeneracy of the spectrum. The argument for the discreteness and non-degeneracy of the spectrum depends on the AdS boundary. So if you eliminate the boundary, you undercut your whole argument. Whatever results there might be in AdS that depend on this argument do not export to dS or flat space-time.

    Of course, there may well be physical phenomena *that do not depend on the boundary* that are more easily calculated in AdS, so you use instances in AdS to learn about them. That's fine. It is not that every use of AdS calculations to draw non-AdS conclusions is obviously unreliable. But BHG's is, as Sabine keeps pointing out.

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  93. travis,

    "We can plausibly expect that..."

    No, it's not at all plausible. If you think it is, write it down. Come, do it. Black hole evaporation in AdS is unitary because it's dual to a CFT on the boundary for which we know evolution is unitary, therefore if you remove the boundary then .... what?

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  94. Tim,

    thanks for responding to my question about where you think the recurrence objection goes wrong. I think you raise many good points in your answer, points which forcefully address the following question:
    1. If AdS/CFT, when interpreted as literally and rigorously true, is incompatible with the Maudlin solution to the info loss paradox, then how damaging is that fact to the interestingness/physical relevance of the latter?

    However, I don't see most of your remarks about precision in physics as being relevant to this distinct question:
    2. If AdS/CFT is taken as literally and rigorously true, is it compatible with the Maudlin solution to the info loss paradox?

    Regarding this question we have two arguments on the table for the view that the answer is 'No'. One is the non-degenerate energy spectrum argument, and the other is the recurrence argument. [Incidentally, I don't think you should criticize BHG for bringing up the latter now, I think he has just given up on trying to persuade you with the former argument and is having a go with the Wallace argument.] Let's set aside the non-degenerate spectrum argument for now and focus on the recurrence argument.

    I said just above that most of your remarks in your last reply to me are not relevant to question 2. The exception is where you say "Now if you want my response to be more precise, how's this: recurrence requires that a state eventually evolve to a state *abritrarily close* to the initial state, not that it recur to the initial state."

    This strikes me as an interesting point, one which raises questions about the logic of the recurrence argument as an argument against your solution. For example: suppose that before that finite time T at which the recurrence theorem guarantees that the universal state will return to within epsilon of its pre-BH state, the universe goes through a gazillion BH evaporation/re-formation stage (none of which came within epsilon of the initial state). Finally, at T, we get back that close to the initial pre-BH state. Did that happen because no information ever stayed behind inside a horizon, or because a gazillion monkeys typing on typewriters are bound eventually to write a page of Shakespeare? Does the recurrence theorem have any resources to answer that question?

    If we recall that "info loss" is really just shorthand for failure of retro-determinism, then we see that there might be a way to use perfect recurrence to argue against your solution. Assuming that we start with an initial pre-BH formation that evolves unitarily, retro-evolving from that state just gets you a star, and whatever earlier states brought about that star. But if unitarity is not violated and some later state of the world recurs precisely to the same pre-BH state, after passing through one or a gazillion BH evaporation phases that left disconnected pieces of spacetime hidden behind event horizons, then we do have a contradiction. Retro-evolving from the late copy of the state gives us reverse black hole evaporations, then reverse black hole formations, etc., while - by hypothesis - the actual initial pre-BH state did not arise from that sort of pre-history.

    This argument looks tight to me. But it only works for perfect recurrence, not for recurrence to within epsilon.

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  95. Bee,

    Thank you -- that is the clear answer I was looking for. I kept asking because it was not clear to me that that was really what you were saying, and I wanted to make sure I wasn't misunderstanding your position.

    So we agree that if you have a local region of Minkowski space (call it R_M) of size R, then the result of experiments inside R_M and at scales much smaller than R will give universal results, regardless of what spacetime looks like outside R_M. So in particular, if I form a black hole that evaporates, and if the evaporation time T obeys T << R then the outcome of this expt is independent of physics outside R_M (since no causal signal can go between the black hole and the region outside R_M). In particular, the question of whether the Hawking radiation is in a (exact or approximate) pure state is independent of this. So if, by whatever method I have at my disposal, I can establish that this is the case when the spacetime outside R_M happens to be AdS, then this will also be true when the spactime outside R_M is Minkowski. Please notice that I am not invoking AdS/CFT here, and indeed imagining for the moment that AdS/CFT has not been discovered. And even if you believe in a firewall (for whatever reason), the nonlocality here is occurring at the scale of R_{BH} << R, (or perhaps at scale T), so this changes nothing in the above. Causality ensures that the details of the spacetime outside R_M can have no effect on the localized (in space and time) expt. occurring in the center of R_M.

    I assume you agree with all this? As you requested, I am being completely explicit in my assumptions, and the only assumption I am making here is the one that you just confirmed yourself, "you have a locally flat patch in either space, the physics is plausibly the same. ".

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  96. bhg,

    It follows by way of your logic that black hole evaporation in AdS is not unitary and the BH-entropy does not count microstates as per the argument Tim already mentioned earlier: You can play this game either which direction. I was, maybe incorrectly, assuming that that's not what you want. But if you wish, we can settle on this conclusion.

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  97. Tim and Sabine,

    If you remove the boundary, we don't know what happens because we don't know how to calculate anything. So yes of course the whole argument goes away if you remove the boundary, but that doesn't mean that the boundary causes the black hole to do something radically different from what it would have done if there was no boundary. Just to be clear: are y'all saying that it seems *more* plausible to you that information will escape in AdS but not in dS than that whichever scenario happens in one happens in the other?

    Re: the recurrence theorem, I don't know the math of AdS/CFT myself but it's my understanding that the CFT is a perfectly well-defined theory and that we know with absolute confidence that it has a discrete and nondegenerate spectrum. That's all you need for the quantum recurrence theorem (you actually don't even need nondegeneracy). Wallace has a paper on recurrence that proves this (https://arxiv.org/pdf/1306.3925.pdf). I agree of course that ideally we should focus on what happens at times near the evaporation, but if your solution is in conflict with this rigorously proven theorem then there's no need to focus on earlier times to establish the conflict. Again though, I endorse the idea which you've mentioned that there's no need to suppose that your solution is so exact that it will hold even on recurrence time scales; if information is lost to the interior for a googol years that's long enough to say that your solution is basically correct. So can we agree that your solution, if taken to hold exactly for all times, is in conflict with AdS/CFT?

    Also, I don't think it's right to say that Hawking didn't notice that the information was in the interior of the black hole. I'm pretty sure he knew that, it's just that he was thinking that when doing path integrals over all possible trajectories for calculating scattering cross sections, you would have to include a trajectory for forming a black hole which evaporates, and therefore scattering amplitudes would have to be nonunitary, hence the dollar matrix.

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  98. Carl,

    Let me try this in some bullet points. Reasons why no mention of recurrence is or can be relevant to our problem.

    1) Recurrence is an approximate notion. It refers to a return near to the initial state, not to the initial state. In the context of Poincaré's point, that is OK, since the Boltzmann entropy has the same approximative character: two nearby states in phase space have the same or nearby Boltzmann entropies. Unitarity and information preservation are not approximative in this way: they are exact. Further, to even evaluate how the approximative character comes in, we need an explicit characterization of what "nearby" even means here. By what measure? If it is by inner product, then I do not believe that any analog to Poincaré's result exists. Take the entire universe. Then, as you know, I can take any given state, change the physical state of only a single electron, and get a state as far as physically possible from the original, i.e. an orthogonal state. So we need to start with a clear account of what the analogy to the Poincaré argument even is supposed to be. I do not think there is any analog.

    2) Given the delicacy mentioned in 1), every single claim upon which the argument depends must be exact and exactly proven. No approximations, no ignoring higher-order terms, etc. Because if the AdS/CFT duality is off by as much as the spin of a single electron in the whole universe, the argument breaks down (if there is an argument in the first place) completely. There is not a shred of proof that AdS/CFT is so exact, or even can be formulated to be so exact. Certainly, if I ask for the dictionary translation of a claim about the spin of a particular election in the AdS into the language of the CFT, no one can provide it. And given the approximative character of the calculations used to test AdS/CT they provide no evidence for this sort of exact correspondence.

    3) What we are seeing is that AdS/CFT is being used in these discussions as a sort of magic talisman you invoke to arrive at whatever conclusion you desire. It is not a serious physical or mathematical claim. I spent the early days with BHG trying to get a clean statement of AdS/CFT and he never gave one and has not to this day. Recently I was informed that the conjecture asserts the isomorphism of the Hilbert spaces and the operator algebras of the two theories. I immediately responded that it is trivial that the Hilbert spaces are isomorphic, so that means nothing, and would BHG please identify the operators in the relevant operator algebras. Radio silence. Recall that on the AdS side we are suppose to have a quantum gravity theory, which is a type of theory that we don't have yet, and on the CFT side there is a limit taken as N goes to infinity. I do not believe for a second that the precise operators and hence operator algebras on either side are well enough defined to even pose an exact question here. It is hand-waving, not physics or mathematics.

    4) In the right setting, slightly imprecise arguments are fine. Boltzmann's explanation of the Second Law was perfect as an explanation. It was not perfectly precisely formulated or argued. Poincaré's criticism of it as not being a precise physical theorem that would hold forever, while accurate, was also (as Boltzmann immediately saw) also completely irrelevant to what he was doing. Poincaré's argument was not fundamentally serious: it was pointless heckling. Referring to recurrence here, where we are not even dealing with exactly defined theories, is even worse than heckling. It is idle fantasizing about the precise mathematical structure of non-existent theories. The combination of arrogant pronouncement and absence of any serious theory or even clear statement of the hypothesis is the sort of thing physicists have been getting away with for too long.

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  99. Bee,

    "It follows by way of your logic that black hole evaporation in AdS is not unitary and the BH-entropy does not count microstates as per the argument Tim already mentioned earlier "

    Sorry, I don't follow. Can you explain how you get from what just agreed on, as stated in my last message, to your conclusion above? So far we just agreed that physics is local in the crudest sense possible: that the results of experiments in R_M do not depend on what spacetime looks like in regions that are causally disconnected from the spacetime region enclosing the experiment in R_M. This says nothing about whether BH evaporation is unitary or not, or whether BH entropy counts microstates. Hence, I have no idea what you are talking about here.

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  100. Tim,

    It became clear long ago that you need to massively violate the rules of QM for your scenario to evade my argument. At this point I consider the issue to be dead and buried, but at the same time I realize you will never admit to this, so I will not waste my breath.

    As far as "word salad" goes, here is a random example:


    " As I have said over and over, AdS is not even globally hyperbolic! And the theory that is supposed to be dual to the CFT is some sort of limit as N goes to infinity. If you think that there is a well-defined theory here that can even be analyzed you are kidding yourself. ".


    This is babble. I keep explaining to you, to no apparent effect, that AdS with standard boundary conditions is every bit as predictive as a globally hyperbolic space. This is a rigorous mathematical fact about initial value problems that you are apparently incapable of grasping. The trouble here, I think, is that you never make the effort to dirty your hands with any calculations, and you think you can make progress with intuition alone. Why don't you go and play around the the field equations in AdS with standard boundary conditions. Right before your eyes you can see that there is no issue. And then you further go on to suggest that large N gauge theories are ill defined, and that people who say otherwise are kidding themselves. Wow. This will be a surprise to to the people who compute exact results in such theories, and to those of us who know about the rigorous lattice constructions of these theories. You are spreading disinformation.


    I brought up the recurrence argument (which is really due to Maldacena, but is pedagogically reviewed by Wallace), because it is an even simpler way to kill your proposal in the way you have stated it. Let's review it now. First of all, yes of course the recurrence time is very long for a large black hole, but so what? The evaporation time for a solar mass black hole is already far bigger than age of the universe, so we are just talking about thought experiments here. If you like, we can restrict attention to small black holes, with mass say, 10 M_Pl. Then the evaporation and recurrence times are brought down to a reasonable scale.


    Here is what kills your proposal. Consider the evaporation of a black hole formed from a pure state of diffuse matter in AdS, and let's in particular focus on the von-Neumann entropy of the state (or rather density matrix) in the AdS region. In your scenario, at late time this von Neumann entropy is large: S_vN \approx S_rad, where S_rad is the entropy of the density matrix describing thermal radiation of total energy equal to the black hole mass. The AdS/CFT claim is the S_vN =0 at late times, since the radiation is in a pure state. Now, by assumption, at early time we have S_vN = 0. The recurrence theorem says that there exists some time in the future for which S_vN < \epsilon, where \epsilon is arbitrarily small. The killer is that you are claiming that S_vN >> 1 for all time, since the quantum state is a highly entangled state, with the entanglement coming from correlations with the disconnected region. That contradicts the recurrence theorem, so it cannot happen. QED.


    Now, if you were serious you would acknowledge this and try to move forward. I can actually imagine conceivable modifications of your proposal that might evade the theorem. You would need the disconnected regions to reconnect in some way. This would be interesting to discuss, but I doubt we will get to this, since I expect a giant cloud of fog to now be produced, obscuring the plain fact of the recurrence theorem.

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  101. Carl

    Maybe this analogy of the situation will also help. Consider the case of chaos. In classical physics, it is provable that certain systems (e.g. the double pendulum) are chaotic. And apart from such a mathematical result, a physical double pendulum is, we can say, observably chaotic, in a sense obvious to anyone who watches one. So that's just a physical fact.

    Now, many physicists have believed that there is something puzzling about this because the double pendulum is, in fact, governed not by classical physics but by quantum physics, and quantum physics is unitary. Unitarity means that the inner product between states is preserved by the evolution. So in the sense of the inner product, the time evolution of quantum states does not result in their "getting further apart". But the whole idea of chaos is that under a chaotic dynamics, initially "nearby" states become "very far apart" in phase space. This result is sometimes reported as "quantum chaos in impossible". That raise both an empirical question and a logical one. The logical one is: if classical behavior is supposed to "emerge" from quantum theory at macro scale, as the correspondence principle asserts, how is that possible here? The empirical one is: Since there is observational or empirical chaos as defined above, if there is no quantum chaos due to unitarity, doesn't that refute quantum theory or at least unitarity?

    Now the argument just given, which has puzzled many physicists, is specious. It somehow sounds rigorous, and appeals to some formal properties, but it proves nothing, and there is not even a prima facie tension between the unitarity of quantum theory and chaos. Why?

    Take the classical analysis, and the actual observed behavior, of the double pendulum. Classically, you compare the trajectory of the system through phase space for a given initial push with the trajectory of a "nearby" slightly greater or slightly smaller push, and find that the trajectories diverge in the natural phase space metric. Both analysis and observation agree about this. But what does this translate into in QM? In QM, the state of the system with a given initial push and the state of the system with some other initial push are—no matter how small the difference—orthogonal in Hilbert space. So all unitarity tells you is that the two systems' states will always be orthogonal: as far apart as possible in inner product. Whether *in some other sense of "nearby" the initial states are "nearby" and the later state no longer "nearby" is untouched by this observation. To even make sense of such a question some notion of "nearness" other than the inner product must be specified. Without that, you can't even ask the right question in the QM setting.

    As far as I can tell, this appeal to AdS/CFT and recurrence is completely parallel. There just is no there there. The relevant notions are not even well defined. And the fact that there is appeal to recurrence time is a huge red flashing warning sign.

    Page made a calculation of the recurrence time for the visible universe under the supposition that it were put in a proper box. He got 10^10^10^10^10^1.1, which is said to be the longest finite time ever calculated on physical grounds. If you think that there is anything even vaguely compelling about BHG's argument that the "boundary can't matter" because it is too far away (which we know is a bad argument), the argument that what happens after recurrence time is irrelevant is unimaginably more compelling.

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  102. bhg,

    I have been jumping ahead in the hope to shortcut this but it seems not a good idea, so please continue with your argument. I wish to object, however, that you have replaced the vague expression 'physics' with 'outcome of experiments'. For all I can tell you seem to be speaking of S-matrices, and it would better if you could stick to something concrete. Best,

    B.

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  103. Travis,

    First, see my response to Carl about recurrence. That may help some. With regard to Wallace's paper, please go take a closer look: his theorems do not run on just discreteness of the spectrum (with or without degeneracy) nor could they. He needs in addition that the spectrum be 1) bounded from below and 2) have only a finite number of values in any bounded interval. Now it is this second condition that is not at all obvious for the CFT used in AdS/CFT. As I understand it, there are various forms of the conjecture (none of which have been proven) and some of which involve the CFT as the limit where N—>infinity, with N being the number of branes. I see no reason in such a setting that the CFT should satisfy Wallace's conditions. Once again, BHG and physphil constantly write as if there is even some perfectly clear account of what the AdS/CFT conjecture is, which was the very first question I asked here months and months ago. I never got a sharp answer because there isn't one. In his last gambit, BHG said that the conjecture is that the Hilbert spaces of the theories are isomorphic and the operator algebras are isomorphic. I replied that it is trivial that the Hilbert spaces are isomorphic—because all infinite dimensional Hilbert spaces are—so it would all hinge on the operator algebras, and would he specify what those are. Radio silence.

    I understand that there is a great rhetorical advantage to acting as if there are a bunch of rigorously proven theorems, just like it was rhetorically advantageous to say the von Neumann had proven that completions of quantum theory are impossible. But I do not myself believe it, and the level of clarity and rigor with which BHG and physphil answer questions do not inspire a drop of confidence here. So I am not willing to say that my solution is in conflict with AdS/CFT. I don't think the conjecture is even precisely enough formulated to say what it would imply about evolutions on the time scale at issue.

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  104. Tim,

    Wallace says that the spectrum of CFT being bounded below, discrete, and at most finitely degenerate (which gets us most of the way to every bounded interval having a finite number of eigenvalues) is a standard result about CFT's on compact spaces. Is he wrong?

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  105. Bee,

    Good -- I am interested to know if we actually end up disagreeing about anything.

    So, I think we have concurred that if you are a physicist doing experiments in R_M which are sufficiently local in the sense I defined, then you need not concern yourself in any way with what goes on outside R_M. Your theoretical computations and expectations are the same regardless, just like we make calculations for experiments here on earth without worrying about what spacetime looks like far beyond our cosmological horizon.

    We can first of all consider processes in R_M that are well described by perturbation theory, basically local scattering experiments. We can compute cross sections for these using standard physical principles and verify these experimentally. On both the theoretical and experimental sides, we proceed identically regardless of what goes on outside R_M.

    Now AdS/CFT enters the story. One thing we can do is to compute CFT observables that yield the cross sections etc. for the scattering experiments just mentioned. We do this by computing a CFT observable corresponding to sending in particles from the boundary of AdS that are aimed to interact purely inside R_M. A priori, you might worry that the fact that the particles have to go from the AdS boundary to deep in the middle will lead to different predictions than what standard physics inside R_M would yield. But here we can just do the computations and verify this is not the case. The accurate statement is that at present it has *almost* been proven that such CFT computations are in agreement with computations based on local Lagrangians inside R_M. There are of course technical issues that we can get into here if you wish, but I think I can fairly state that there is very strong evidence that this works out. By transitivity, it follows that the CFT correctly such perturbative local phenomena in Minkowksi space. In other words, for such local phenomena we don't expect any discontinuity upon taking the AdS radius to infinity.

    We could next discuss non-perturbative effects not including gravity, such as low energy QCD processes. Here it is harder to back up statements with explicit computations, but I don't see any reason why the situation should be any different here than above.

    Finally, we come to black holes. We can first talk about an experiment involving black hole formation and evaporation, such that the entire spacetime region describing the experiment is deep inside R_M, and hence protected by causality from influences outside R_M. This is not yet the situation relevant to AdS/CFT (which involves injecting particles from outside R_M). But I am going to pause here before getting into that.

    If you disagree with any of the above I am sure I will hear about it! I am being very methodical here, since you requested several times that I spell out all my assumptions.

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  106. Travis,

    No idea. But "most of the way" to the conditions of a theorem is worth...nothing.

    Especially the case of the limit as N —> infinity, I can see no reason at all for the actual condition needed for recurrence to be satisfied. And the case of the limit as N —> infinity seems like the most plausible case for the Duality to hold: in essence, taking N —> infinity is akin to adding a new dimension so far as the state space goes, so the mismatch of dimensionality between the AdS and the CFT seems less threatening.

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  107. bhg,

    First let me note that you state

    "I think we have concurred that if you are a physicist doing experiments in R_M which are sufficiently local in the sense I defined, then you need not concern yourself in any way with what goes on outside R_M"

    despite my explicit objection on the matter of doing experiments. No, I do not concur on anything that regards 'physicists doing experiments' and so on, because you never stated what a physicist is in your vocabulary and what you mean by experiment, which presumably includes a measurement, and so on. As I already said above, I would appreciate if you could stick with precise statements.

    Second, your previous statement that I agreed upon came with the provision "I am not invoking AdS/CFT here, and indeed imagining for the moment that AdS/CFT has not been discovered"

    now you are clearly invoking AdS/CFT.

    To put this differently, the statement that I agreed upon is: time evolution in asympt AdS (without CFT) is locally equivalent to that in asympt Mink. You now want to restrict this to asympt AdS states (of fields and metric) that have a CFT dual. That is, as I have said several times before, not a local criterion.

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  108. Some intriguing (to me) statements in this, which in my muddled mind are relevant to the year-long argument raging here.
    https://arxiv.org/abs/1405.6394
    Finite N and the failure of bulk locality: Black holes in AdS/CFT
    Daniel Kabat, Gilad Lifschytz

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  109. Tim and Travis,


    In the AdS/CFT correspondence N is equal to the ratio of the AdS radius to the Planck length raised to a positive power. The statement about N->\infinity is the statement that in this limit the CFT yields *classical gravity* in AdS. The full correspondence holds for finite N. Also, note that 1/N perturbation theory is equivalent to bulk perturbation theory in graviton loops. Now, at finite N the CFT on a compact space certainly has a discrete spectrum with finite degeneracies, so the conditions for recurrence hold.

    Travis: good luck getting Tim to see the light on the recurrence argument. If you get anywhere I will applaud.

    Tim complains that I haven't explained in detail how the operator algebras match between AdS and CFT. But I did say that bulk field operators are dual to single trace primary operators smeared by the HKLL kernel, and that's all that's needed for present purposes. Now I realize that this answer is probably incomprehensible to TIm because he has not studied AdS nor CFT. At this point I think he should make the effort to consult one of the (many) pedagogical introductions to AdS/CFT. Previously, I tried teaching Tim a bit about how gauge invariance works in QFT, but I got back an insult laden tirade about how I don't know calculus etc, all due to the fact that Tim did not understand what the words "Gauss' law" meant. I am not going to subject myself to that again. Tim should expend a little effort to learn the basics of AdS and CFT before making grand pronouncements, and isn't it a bit odd that he apparently won't bring himself to do so? Is this how things work in philosophy -- you just argue, and whether you know the basic facts is irrelevant?

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  110. Bee,


    ". No, I do not concur on anything that regards 'physicists doing experiments' and so on, because you never stated what a physicist is in your vocabulary and what you mean by experiment, which presumably includes a measurement, and so on."

    OK, I don't understand your objection here, but I will comply with your request for more precision. The "physicist" is Sabine Hossenfelder, and she is equipped with rulers, clocks, particle accelerators, etc, and she can use these to carry out experiments that are bounded in space and time, as measured by her clocks and rulers. Is that OK? Now, do you agree that under these conditions the calculations and experimental findings of SH will be independent of what spacetime looks like outside R_M?

    If I invoke AdS/CFT at any point I will explicitly say so. Now, extending the previous paragraph, and *not* invoking AdS/CFT, suppose our experiments consist of injecting particles into R_M from outside, such that no two particles get close to each other until they are well inside R_M. The particles are prescribed to enter R_M with specified energies and position (localized wavepackets). E.g. suppose we fire in an electron and positron and ask for the total probability to produce hadrons. Do you agree that our computations for this probability and our experimental expectations are independent of what spacetime looks like outside R_M. Again: No AdS/CFT in sight here.

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  111. BHG,

    Do you know of any proof that the CFT satisfies the condition necessary for the recurrence theorem, that there are finitely many eigenvalues in every bounded interval of the spectrum?

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  112. bhg,

    No, sorry, I can't agree on this. The physicist who sits in AdS knows that if she measures precisely enough she'll always be able to find traces of the AdS curvature. She also knows that quantum measurements are at least in the standard formulation inherently non-local, which makes her very suspicious of the whole box-business. I think it's irrelevant to your point, but, well, you asked.

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  113. Bee,

    Actually I had in mind that inside R_M the spacetime is precisely flat, so there is no curvature to discover. I am imagining something like a scalar field potential with a local extremum at zero cosmological constant, and the field inside R_M is placed at this value. Also, even if you were to say that quantum measurements are inherently "non-local", I presume you still believe in causality, so if the boundary of R_M is farther away than can be reached by any light signal during the course of the experiment, I don't see how details outside R_M could influence the outcome of any experiment deep inside R)M. So I will ask again for your opinion on this setup.

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  114. bhg,

    I don't believe in causality, but not sure that's a good place to get into this.

    So your scalar field potential, is it discontinuous at the box-wall?

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  115. Travis,

    There is no single CFT, rather there are various specific CFTs relevant to AdS/CFT. Now, on general grounds, we expect almost every QFT to have such a discrete and finite degenerate spectrum when it is defined on a compact spatial surface. Imagine if this were not the case: it would say that the thermal ensemble is ill-defined, since the Boltzmann sum diverges at any nonzero temperature. So there is no thermodynamics in such systems. Now, for the most famous CFT appearing in AdS/CFT, namely supersymmetric Yang-Mills theory, one can explicitly verify the discreteness/degeneracy claim at zero gauge coupling, since this is just free field theory. It's then clear that to every order in perturbation theory in the gauge coupling the claim will continue to hold. At strictly finite coupling I don't know that there is really a proof of this statement in the rigorous mathematical sense, but for it to be otherwise is highly implausible for the reason just noted, and it would go against everything we expect of a QFT.

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  116. Bee,

    Ok fair enough. Let me replace the statement that the region inside R_M is precisely flat with the statement that I can make it as close to flat near the center as I want (by making the bubble larger and larger for example). So my modified question is: for fixed experimental resolution, do you agree that I can arrange a setup in which it is impossible to distinguish between the two cases for the types of experiments I described (the cases being AdS or Minkwoski outside R_M)?

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  117. BHG,

    Ok thanks. Discrete and finite degeneracy doesn't quite get us finitely many eigenvalues in every bounded interval. It's possible that there could be an infinite sequence of eigenvalues which approaches a finite limit. This doesn't seem likely to me, but do we know for sure that that doesn't happen?

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  118. Travis,

    If you look at Wallace's paper, is seems clear that he was aware of no such proof. Note what he says about whether QFT obeys the condition:

    "When will this somewhat abstract condition be realised? On heuristic grounds we would expect it to hold in nonrelativistic QM whenever the system is spatially bounded and has a potential bounded below: in this situation, a solution to the Schr ̈odinger equation can only have discretely many nodes, and any increase in nodes increases the average curvature of the solution and so the kinetic energy. In fact this can be formally proved in functional analysis (see [3, p.1330]). More generally, [1] has advanced arguments to the effect that even in quantum field theory, there are only finitely many energy eigenstates of any spatially bounded theory below a given total energy."

    I have not looked at the paper, but "X has advanced arguments to the effect" indicates that there is no proof.

    I point out once more: in all this time BHG never so much as mentioned recurrence. The idea that this is the golden key to whatever it is he is trying to show it simply not credible, else he would have brought it up.

    I also note for the record that Wallace's theorems appear to be about Hilbert spaces, and as we have seen the "physical Hilbert space" that BHG goes on about appears not to be a Hilbert space. (He refuses to affirm or deny that Shankar's definition is what he had in mind, so who knows?) If so, there may well be a mismatch between the conditions needed for the proofs that Wallace cites and the physical conditions assumed here.

    I also commend that you reflect on how completely incredible all of this reference to recurrence here is. As I said, the period of time needed for recurrence to so incomprehensibly longer than any physically relevant time period that you would have to assume that you have a perfectly defined theory that has been perfectly analyzed with zero tolerance for approximation in order to trust a prediction over that time scale. And we just are not dealing with any such thing.

    Con't.

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  119. It is telling that in place of physical considerations, BHG makes himself feel better by citing the complexity of the mathematics: "Tim complains that I haven't explained in detail how the operator algebras match between AdS and CFT. But I did say that bulk field operators are dual to single trace primary operators smeared by the HKLL kernel, and that's all that's needed for present purposes. Now I realize that this answer is probably incomprehensible to TIm because he has not studied AdS nor CFT." I'm sure that BHG felt smugly superior composing that. But recall what the actual situation was: I just wanted to know what the *content* of the AdS/CFT conjecture is (as it turns out, there are several such conjectures, of different strengths. This I could find out in a few minutes of research, but BHG has not mentioned it in a year.) In particular, I wanted to know the precise mathematical content of the claim that the two theories are "dual". I was told that they are *the same theory*, and when I inquired what exactly that meant, I was told that it meant that both the Hilbert spaces and operator algebras are isomorphic. And when I pointed out that the first condition is trivial for infinite-dimensional Hilbert spaces, and so we ought to focus on the second, so could the precise operator algebras please be specified, BHG went silent. Now he says that this one duality "is all I need for present purposes", but citing just that one does not begin to answer the question of what the relevant operator algebras are, and what the evidence for the duality is. (All of this stuff about "We get the same answers where we can do calculations" is evidently pointless: you get the same answers calculating in GR and in Newtonian gravity in the low-energy limit, but they are hardly "the same theory".)

    On purely physical grounds, recurrence is just a red herring here. Over the period of time mentioned, all sorts of things could happen the would strike you as physically impossible, as the classical recurrence theorem proves. Completely decoherent systems, for example, will recohere, making rather a hash of any Many Worlds approach to understanding the physics. And it is perfectly possible that the "splitting off" of worlds envisaged in my solution will be undone by the "merging" of worlds. So this entire line of inquiry is just a distraction.

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  120. Sabine,

    On the "have your cake and eat it too" front, note that BHG appears to be making use of the classical GR "gluing theorems", which I mentioned a little while ago. That is what lies behind the claim that there can even *be* a strictly Minkowski region in a space-time that is asymptotically AdS. And classical GR agrees.

    But if you accept that this is possible, then BHG's whole argument goes right out the window. It means that no measurements made at the boundary can determine even as much about the bulk as whether it contains a Minkowski patch. And that means that there are infinitely many different bulk states consistent with all measurements at the boundary. And that means that the boundary Hamiltonian is massively, probably infinitely, degenerate. And the means all of the things he has been arguing are wrong. Indeed, it seems to mean that AdS/CFT is itself wrong. (I have for a while noted that BHG's attempts to prove a contradiction between my solution and AdS/CFT appear to hinge on arguments that make no reference to my solution, i.e. to arguments that show AdS/CFT all by itself is inconsistent. This is one of those.)

    This is typical of his style of argument. Where it is convenient, he cites some conditions as critically important and then later gives arguments that the very same conditions are completely irrelevant. The mother of all such examples is being an asymptotically AdS space-time. When convenient, it is a condition that limits the bulk physics in a fundamental way, and then at other times it is "obviously" supposed to be irrelevant to the bulk physics, so the results will export to flat and dS space-times.

    I know you know this....this is more a comment for the lurkers.

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  121. BHG:

    " it is impossible to distinguish between the two cases for the types of experiments I described (the cases being AdS or Minkwoski outside R_M)?"

    These terms do not have a chance in the world of being given a rigorous content, such that anything at all can be demonstrated from them. They are, in the language of John Bell, "unprofessionally vague". The question at issue—can the fundamental evolution of the quantum state be unitary through black hole evaporation, and the slightly more sketchy "if information is preserved through the evaporation process, where does it end up"—are precise enough. You just can't possibly reach such precise conclusions from such vague premises. This is what a lifetime evaluating arguments teaches you, and a lifetime doing calculations does not.

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  122. Tim,

    I don't think he wants to actually have a discontinuous transition, exactly because it would go bye-bye if you try to encode it on the boundary, but even so it's not going to help because it doesn't matter if the local processes are the same if the states are not.

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  123. bhg,

    Now you have brought in a 'fixed experimental resolution' which you haven't defined. Note that since you are rescaling dimensionful quantities, this is not a trivial assumption. Again I am not sure how it matters, but I can't agree on the the statement.

    In the hope for moving this forward (sorry, patience is not my strong side), is it possible you are simply trying to rephrase what you said earlier, some S-matrix calculation that in the \Lambda \to zero limit reproduces the Minkowski result? If so, why not say it?

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  124. Sabine,

    I'm not sure if this is relevant, but the classical gluing theorems do not involve in discontinuities, I believe. Indeed, the data and solutions are everywhere smooth, and they agree outside of the gluing region to within arbitrary epsilon, I think. That is, by any standard, there are distinct solutions that cannot be be distinguished by any actual experimental means, i.e. an experiment with finite accuracy, away from the actual gluing spots.

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  125. Arun,

    Yes, here is an interesting claim:

    "At finite N in the CFT, or equivalently at finite Planck length in the bulk, it seems clear that any attempt to construct a local bulk quantum field must fail. Holographic theories have an entropy bound [16], and as a result the CFT has far fewer degrees of freedom than would be necessary to define a local field in the bulk [17]. Local bulk effective field theory is only an approximation, albeit an excellent approximation under ordinary circumstances."

    But note footnote 1. The duality appears to be being stipulated into existence rather than proven.

    Also note that perturbative results in the low-energy limit of the AdS theory are really of no significance for our question, since the formation and evaporation of a black hole are not within that regime.

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  126. When bhg writes "we expect almost every QFT to have such a discrete and finite degenerate spectrum when it is defined on a compact spatial surface. Imagine if this were not the case: it would say that the thermal ensemble is ill-defined, since the Boltzmann sum diverges at any nonzero temperature." he means also that there are finite number of energy states in any interval. That is necessary condition for finiteness of Boltzmann sum.

    I also do not know of a proof of this for general interacting QFTs. However I would not be surprised if there is one. If it is not true, much of foundation of QFT would erode. It would mean those theories cannot describe real world, because there would be no thermodynamics. Heat capacity in finite volume would be infinite. Of course this is not true for QFTs that describe real world. Also numerical simulations would all be wrong.

    Another point. If we are accepting AdS/CFT as true, we can check if CFT has this property using AdS side. On AdS side, if we believe Hawking and Bekenstein then we know it does. On bulk side, thermal ensemble is an eternal black hole that has finite entropy. Therefore there are finite number of energy states in any interval. (If we do not believe Hawking and Bekenstein this debate is without any meaning.)

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  127. Black hole evaporation doesn't appear to be local. Perhaps some QFT scattering processes are really effectively at some tiny neighborhood of a spacetime event where spacetime curvature can be neglected. But blackhole evaporation isn't one of these processes, is it? The modes involved are at least the size of the diameter of the horizon? The Page time paradox as told in David Wallace arXiv:1710.03783 suggests that the story of evaporation says that time periods comparable to the black hole lifetime are relevant.

    Lastly we don't seem to know how to define unitarity, information-preservation or loss, etc., in quantum mechanics in anything but the context of global spacetime. If we could calculate information leaking out of the blackhole like Hans able to stick his fingers into holes in the dike, black hole information loss wouldn't be so mysterious. What one ideally would have is a set of local assertions, that can be calculated and verified anywhere in a dynamic spacetime, that if true at each spacetime event, provide certain physics guarantees. But such may not be the nature of the physical world.

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  128. Bee,


    Is this a physicist's version of good cop/bad cop, where first you tell me to slow down and spell out every minor assumption, and then you complain and tell me to speed up? Fine, I will get to the point, but please do your part as well and answer the question succinctly.


    I take it as "obvious" that we can engineer two spherical regions R_M, one contained in asymptotically AdS and the other in asympt Minkowski, such that the size of R_M is as large as I want, and R_M is locally as close as I want to Minkowski space in the sense that the Riemann tensor and values of matter fields can be taken to be arbitrarily small within R_M. Then, from the outside I inject two particle entering R_M at antipodal poles. The locations, energies, spins, etc, of the particles are arbitrarily close in the two setups. The particles are aimed to collide in the center of R_M. SH is near the center, armed with pith balls, calorimeters, tracking chambers, etc. We repeat the experiment many times, slowly increasing the center of mass energy from, say, 1 Mev to 10^5 M_Planck. Question: do you expect that SH will observe any difference in the experimental outcomes, as viewed from inside R_M, that cannot be made arbitrarily small by increasing the size of R_M, preparing the incoming state more carefully, etc.? If yes, at what energy does this occur?


    No AdS/CFT anywhere in sight!

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  129. bhg,

    I didn't ask you to slow down, I asked to to be more precise. In any case, about your recent statement, the size of R_M cannot, of course, be "as large as you want" because I guess it should be below the AdS radius. I am assuming you take that to be the case. I am also further assuming you still have \Lambda be generated by a scalar field potential that goes to zero towards the AdS center. I also assume you want the scalar field potential to be analytic in the asymptotic AdS space. I am also guessing that you further want to keep flexible the ratio of R_M to L (where by 'L' I mean the AdS radius) because I assume you don't want me to be able to actually measure the slope of the potential (or its boundary if its a pit).

    Please let me know in case you want to object on any of my interpretations of what you probably mean.

    I am now not sure what you mean by 'particle'. What wave-function in which Hilbertspace and how is the Hilbertspace and the function defined? Presumably you want the particles to be localized?

    Best,

    B.

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  130. Tim,

    A discontinuity in any derivative of the field is sufficient to prevent you from making an expansion around the boundary from which you can fully reconstruct the field, though the higher the order the less relevant it'll be for observables. Keep in mind though that bhg explained earlier (I didn't check put presumably true) that it's sufficient if the field is expandable around one point on the boundary.

    The case in which you have entirely disconnected pieces of space-time is a particularly obvious example for which this is not the case. But you don't need to disconnect the space-time to have functions that don't fulfill the assumption. None of these functions can be captured by the CFT side of AdS, so they're not in the full space of solutions. Anything that's entirely hidden (and remains hidden) behind a horizon, for example, is of this type. So just by stating that one uses the duality one has assumed these cases away. What I am saying is merely that the whole AdS/CFT miracle relies on that, and it's clearly not a local assumption, hence no reason it should survive if you remove the boundary.

    You could of course simply assume that the "real physics" in asymp Mink space is whatever you get from AdS/CFT in the limit \Lambda -> 0, but just why one should believe that I don't know.

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  131. Bee,

    " In any case, about your recent statement, the size of R_M cannot, of course, be "as large as you want" because I guess it should be below the AdS radius."

    No, I can take it to be as large as I want. By "AdS radius" I mean the curvature length scale of the AdS space (this is standard usage). AdS has infinite spatial volume, so there is no problem placing an arbitrarily large bubble of Minkowski space inside an AdS space of fixed AdS radius. So, regardless of the details of my scalar potential, by taking the size of R_M to be as large as I want I can always engineer a region of spacetime of arbitrary size that is arbitrarily close to Minkowski space (locally). I hope that answers your questions about these issues. I am fine with your other statements as far as I can tell, although I don't think they are all necessary.

    Yes, the particles should be well localized compared to the size of R_M. Now you ask me what i mean by "particle". Hmm, I am not sure what you are getting at here. In fact, all I care about is that we are firing in some localized blobs of energy and momentum, and that what is injected in the two cases is the same to desired accuracy. I hope that is sufficient detail, although I worry you are now going to ask me "what do you mean by energy". Since I don't see how this point is going to be of any particular relevance to anything in my argument, I am unsure how to respond.

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  132. I take it as "obvious" that we can engineer two spherical regions R_M, one contained in asymptotically AdS and the other in asympt Minkowski, such that the size of R_M is as large as I want, and R_M is locally as close as I want to Minkowski space in the sense that the Riemann tensor and values of matter fields can be taken to be arbitrarily small within R_M. Then, from the outside I inject two particle entering R_M at antipodal poles. The locations, energies, spins, etc, of the particles are arbitrarily close in the two setups. The particles are aimed to collide in the center of R_M. SH is near the center, armed with pith balls, calorimeters, tracking chambers, etc. We repeat the experiment many times, slowly increasing the center of mass energy from, say, 1 Mev to 10^5 M_Planck. Question: do you expect that SH will observe any difference in the experimental outcomes, as viewed from inside R_M, that cannot be made arbitrarily small by increasing the size of R_M, preparing the incoming state more carefully, etc.? If yes, at what energy does this occur?

    Since it isn't obvious that we humans can engineer two such spacetimes with the desired asymptotic properties, this is a purely computational exercise.

    So what you are asking is - is the asymptotic nature of the spacetime a relevant input into any physics computation we do within R_M? The answer is - obviously, yes. In an asymptotically AdS spacetime, I can presumably do computations that are beyond our capabilities in asymptotically Minkowski spacetime.

    If quantum field theory in curved spacetimes was identically tractable on asymptotically AdS and asymptotically Minkowski spacetimes, then I could do the computations and tell you at which energy scale of collision there might be a difference or maybe no difference at all.

    A counter question is - can the computed effects of a loss of quantum mechanical unitarity be made arbitrarily small in the computational scenario that you have defined?

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  133. I have a more definite answer to the question posed here:
    I take it as "obvious" that we can engineer two spherical regions R_M, one contained in asymptotically AdS and the other in asympt Minkowski, such that the size of R_M is as large as I want, and R_M is locally as close as I want to Minkowski space in the sense that the Riemann tensor and values of matter fields can be taken to be arbitrarily small within R_M. Then, from the outside I inject two particle entering R_M at antipodal poles. The locations, energies, spins, etc, of the particles are arbitrarily close in the two setups. The particles are aimed to collide in the center of R_M. SH is near the center, armed with pith balls, calorimeters, tracking chambers, etc. We repeat the experiment many times, slowly increasing the center of mass energy from, say, 1 Mev to 10^5 M_Planck. Question: do you expect that SH will observe any difference in the experimental outcomes, as viewed from inside R_M, that cannot be made arbitrarily small by increasing the size of R_M, preparing the incoming state more carefully, etc.? If yes, at what energy does this occur?


    Yes, assuming that the experimental set up can indeed be made, SH will observe a difference in experimental outcomes. This is because if I understand the words of Gary Horowitz http://www.ctc.cam.ac.uk/stephen70/talks/swh70_horowitz.pdf:

    "For Minkowski or de Sitter, it has been shown that small but finite perturbations remain small (Christodoulou, Klainerman; Friedrich).

    This has never been shown for AdS.

    WHY NOT?

    It is just not true. "


    "Anti-de Sitter spacetime is nonlinearly unstable: generic small perturbations become large and (probably) form black holes. "

    "Generic small (but finite) perturbations of AdS become large and eventually form black holes. The energy cascades from low frequency to high frequency modes in a manner reminiscent of the onset of turbulence."

    "Conclusion of spherically symmetric scalar field evolution in AdS:
    No matter how small you make the initial amplitude, the curvature at the origin grows and you eventually form a small black hole. "

    ---
    IMO this intuition that in AdS one can have a lone black hole that forms and then evaporates and all the Hawking radiation escapes past the boundary at infinity may be mistaken; maybe we just get a cascading series of black holes if the generic small perturbation can collapse into a black hole.

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  134. We repeat the experiment many times, slowly increasing the center of mass energy from, say, 1 Mev to 10^5 M_Planck.

    I just sit around repeating the collision experiment ad infinitum at 1 MeV. If I'm in asymptotically anti-deSitter my apparatus will eventually detect a tiny black hole. Some perturbation somewhere will produce a black hole in my instrumented volume R_M. If I'm in asymptotically Minkowski, that won't happen. It is just a matter of twiddling my thumbs long enough. :)

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  135. Travis,

    "It's possible that there could be an infinite sequence of eigenvalues which approaches a finite limit. This doesn't seem likely to me, but do we know for sure that that doesn't happen?"


    I don't know of any proof, but nothing of this nature has ever shown up in numerical simulations of QFT, as in lattice gauge theory. A continuous spectrum, or some limit points would imply nonexistence of the Boltzmann distribution, and this would have an obvious and dramatic effect on the numerics. So sure, absent a proof it's fine to assign some \epsilon probability that something like this could happen, but there is absolutely no argument, handwaving or otherwise, that it actually does, and it goes against all intuition gained from many decades of study of QFT.


    That's really the only loophole in the recurrence argument. As soon as you assume a minimum spacing of the energy levels, the recurrence theorem is elementary and takes just a few lines to prove. This obviously kills Tim's scenario of information being forever lost to disconnected regions, which is why he offers no counterargument but instead turns on the fog machine. But I do want to stress that the recurrence argument only establishes something much weaker than what we really want to show, namely that the info comes out after the much shorter evaporation time. For that, you need something along the lines of the argument I was giving.

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  136. bhg,

    I was thinking of a different radial coordinate. In any case, I am asking what you mean by particle just to figure out whether you are referring to energy eigenstates and whether you are assuming you have a complete basis of the space, but apparently not. So, fine, then please go on.

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  137. Tim,

    I didn't notice your message about Shankar's book, but when I keep saying that states in the physical Hilbert have finite norm, I indeed mean that their norm is finite. Not a delta function -- finite. Shankar's definition is nonstandard, and I have not heard it before. Anyway, in AdS there is no reason to introduce the notion of delta function normalizability since energy eigenstates have finite norm.

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  138. Bee,

    So you have the question before you. If we consider the two realizations of R_M, both arbitrarily large and arbitrarily close to Minkowksi space in the sense I described, and if we have two "particles" entering the region at distant separation that are aimed to collide in the middle, do you expect to see any difference in the experimental outcomes, as observed in R_M, that cannot be made arbitrarily small by taking R_M larger and larger and preparing the states more and more accurately? Again, I ask this question in the case that the collision energy ranges from say 1 MeV to appreciably above the Plank energy.

    As I see it, although I cannot "prove" anything here, every expectation is that the experimental outcomes will be essentially the same in the two cases. The particles don't have any "memory" of whether they came from an external AdS or Minkowski space, and once they enter R_M they encounter the same local physics. I just don't see any basis for believing that the outcome will be appreciably different under these circumstances. You would have to believe in some memory effect, or action-at-a-distance, it seems to me.

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  139. BHG

    Maybe things would not seem so foggy to you if you paid attention. I nowhere said that information is "forever lost" to disconnected regions. Forever is a long time. Indeed, 10^10^10^10^10.1 is a long time. I don't recall making any claims at all about that. I can't see that it has any relevance to anything. That is why I found any mention of recurrence so off-the-wall as not to even be worthy of serious discussion. And apparently neither did you until you read Wallace. Can you just acknowledge for once, that this is red herring and not waste more time with it?

    Well, since you have never adequately defined the "physical Hilbert space", I had a brief hope that you might have meant what Shankar does, since he explains what he means. If the elements of the physical Hilbert space must satisfy all the constraints, including the Hamiltonian constraint, then I can't see that it means anything else than the solution space. But you have consistently denied that. Just as you denied that there was any bulk Hamiltonian at all, just the boundary one. But we have seen that the bulk evolution is governed by the Hamiltonian constraint, which is a form of a Hamiltonian.

    Since Shankar's definition, you say, is non-standard (and that is a very well-regarded book) how about you cite a definition that is standard? That should be simple enough.

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  140. bhg,

    I have already said a few times that I agree on this, so I am not really sure what we learned from this. What I asked you to be more precise about is the reason why you think that the insights you have gained by using the AdS/CFT duality still hold in asymptotic Minkowski space. Let us take two concrete examples: Unitarity of black hole evaporation and the microcanonical interpretation of the BH entropy, because this is what we've been talking about. I am saying I cannot see any reason why these would still apply in asymptotic Minkowski space. Your elaboration so far doesn't address why I should.

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  141. BHG

    "The particles don't have any "memory" of whether they came from an external AdS or Minkowski space, and once they enter R_M they encounter the same local physics."

    QM is not local. "Energy" in QM is not a locally definable quantity. The question of whether or not the spectrum of the Hamiltonian is discrete is not locally determinable. Since you have no position eigenstates there is no such thing as the "position" of a particle: "particles" are not localized. In Bell's terminology, the wavefunction of a system is not a "local beable", so unless you are using something like Bohmian mechanics, you have zero local beables.

    So even if you won't define a "physical Hilbert space" will you please do me the favor of defining "local physics"?

    I realize that you are likely to characterize the question I just asked as "foggy" and "word salad". This is yet another "have your cake and eat it too" moments. When it suits you, you make free use of all sorts of notions that cannot be locally defined. Or to be even more precise, you are using a set of concepts *none* of which can be locally defined since you have no local ontology in terms of which to define anything. You cite as critical to your argument properties of theories that are global properties.The mother of all these is *being AdS*. But then when asked to prove your assertions, you don't even pretend to offer proofs: you appeal to your intuitions about "memory effects' or "action-at-a-distance". That is, you pretend that the theory you are dealing with *is not QM*, that it is instead some local theory or physics whose predictions can be approximated, within epsilon, by some local theory. QM is not local and cannot be approximated by any local theory. Violations of Bell's inequality are quite noticeable and cannot be recovered by any local physics.

    You can't define "where the particles entered" or "aimed at each other" or "local physics" or "memory effect" or "action at a distance". You can't even properly define "unitary evolution" because you want to use the phrase in a collapse theory..

    All Sabine is asking is that you be precise. And you can't. What Sabine has noticed is what everybody has noticed: for months you have been arguing exclusively about AdS/CFT and in that context you have constantly been appealing to global characteristics of the physics. On the QM side, in fact, *every* characteristic is global since all you have is a wavefunction and that is a global (non-local) entity. And even on the GR side, being AdS is a global characteristic that cannot be defined (and hence determined) locally. But when it suits you, all of a sudden none of the very properties and characteristics you have been appealing to for months can possibly be relevant to anything because of intuitions about the "local" physics that the "particles" will "encounter".

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  142. Sabine

    When I said that in the classical GR context the gluing is smooth I meant infinitely differentiable. That is my understanding of what the theorems say. But I am no expert.

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  143. If you are a physicist doing experiments in R_M which are sufficiently local in the sense I defined, then you need not concern yourself in any way with what goes on outside R_M.

    I think what matters for locality isn’t just a limited spatial region but the entire past light cone of that region, which extends to arbitrarily distant locations. Assuming the universe has always been AdS, can we really claim that any local region is causally isolated from the AdS character of the universe?

    I just don't see any basis for believing that the outcome will be appreciably different under these circumstances. You would have to believe in some memory effect, or action-at-a-distance, it seems to me… I ask this question in the case that the collision energy ranges from say 1 MeV to appreciably above the Plank energy.

    Isn’t is plausible that the local structure of spacetime is - and needs to be - consistent with the global structure? For example, suppose each event in 3+1 dimensional spacetime contains (at the Planck scale) several curled-up dimensions comprising a Calabi-Yau space. There are many different possible such spaces, and it may be that some physical phenomena – especially at the Planck scale – depend on which of these Calabi-Yau structures actually exists. Likewise there are a variety of possible global structures of the 3+1 dimensions. It doesn’t seem implausible (to me) that there is some relationship between these structures. In other words, an AdS spacetime might be compatible only with a certain class of Calabi-Yau spaces (or none at all), and a dS spacetime might be compatible only with another class. I don’t think such a consistency constraint would imply action-at-a-distance. (Classical general relativity already provides an example of a non-trivial consistency constraint – the Bianchi identities – being satisfied locally over the whole manifold.) This may not be relevant to the question being discussed here, because maybe all these structures have the same black hole behavior. I just don't think we can assume that all local physics is logically independent of the global structure of the universe.

    More fundamentally, I don’t even see how a theory of quantum gravity along with the idea of a unitarily evolving wave function can be represented by any single spacetime manifold. I think that any given initial prepared state (with a certain information content), prior to formation of a black hole, could sometimes lead to a black hole and other times not. So, the wave function evolving unitarily from the initial state must be a superposition of “branches” with different numbers of black holes forming and evaporating, and the supposedly conserved information is spread throughout all these branches - nearly all of which are empirically inaccessible to us. (If we make measurements of the evaporation radiation along the way, we are aligning ourselves with just a tiny part of the hypothetical overall wavefunction, and the rest is inaccessible.) What is the Penrose diagram of a superposition of “black hole” and “no black hole” and “two black holes colliding”, etc.? How can we represent this entire unitarily-evolving wavefunction with just a single spacetime manifold?

    As an aside, I wonder if the “multiverse” idea is supposed to extend to global structure. In other words, is the multiverse supposed to contain some AdS universes and some dS universes and some asymptotically flat universes, etc., with different local physics and vacuum structures in each? Would we expect black holes (along with the rest of physics) to “work” the same way in each of these universes?

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  144. Bee,

    Well, I would have thought that the rest of the argument writes itself, but anyway here it is. So we have agreed that if you perform this type of scattering experiment then we plausibly expect the outcome to be the same regardless of what spacetime looks like outside R_M. In particular I can perform the experiment at high enough collision energy such that I form a black hole that then evaporates, and I can use my measuring instruments in R_M to see whether the state of the Hawking radiation is pure (this requires repeated experiments of course). So we have agreed that if the emitted radiation is pure when R_M is inside AdS then the same will be true when it is inside Minkowski. The point, of course, is that the AdS experiment just described is precisely of the sort I know how to translate into CFT language. I inject particles in from the boundary such that they collide inside R_M, and then examine what comes out, in particular whether the radiation is in a pure state or not. Putting 2 and 2 together, it follows that I can use the CFT to answer the question in Minkowski as well. And I again remark that this sort of thing has been done explicitly for scattering at sub-Planckian energies, and indeed one recovers the expected flat space results.

    As I already noted, I am not claiming this "proves" that this procedure gives the "right" flat space result, if for no other reason than that there might not be a unique "right result" -- there could be many consistent theories of QG. But what makes no sense to me is your claim that one has "absolutely no reason to believe" that the results carry over, given that this gives a valid description of QG in some Minkowskian world, and so far is the only such description available.

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  145. Tim,

    I don't see any relevance for the Page number you quote: what's at issue here is the existence of a finite recurrence time, not its particular value. But if this bothers you, just consider starting in empty AdS and creating a black hole of mass, say, 10 M_Pl. That brings the recurrence time down to a far more reasonable number, and the same argument applies. Of course it is still very long, but any timescale involving black hole evaporation is going to be very long, so this is hardly a compelling argument


    I also don't understand your point about Shankar. I am using the standard definition of physical state that you will find in any reference on canonical quantum gauge theory or gravity. The state should have finite norm and be annihilated by all the constraints. Where we differ is your additional stipulation (which I don't even know how to make sense of) that the state should satisfy "all dynamical equations" including apparently the Schrodinger equation (which is not a constraint equation!) H \psi = i d\psi /dt, where t is boundary time and H is the total Hamiltonian including boundary term. I have never heard of such a thing: the condition for a state to physical is a condition at fixed boundary time, not a condition about its evolution in boundary time. Taken at face value, it also appears to add nothing, since every physical state at a given time defines initial data for some solution of the Schrodinger eqtn.

    If you have presented an actual counterargument (as opposed to side-commentary) to the recurrence argument I definitely missed it. Here is what you have to confront

    1) Assume a minimal energy spacing for the CFT (I will admit that this has not been proven as far as I know, but if this is the only loophole you can find then that's good enough for me)

    2) Start with a state \psi(0) at t=0 that describes a pure state in AdS that is collapsing to a black hole

    3) By the recurrence theorem, for any epsilon there exists a time T such that \psi(T) is arbitrarily close to \psi(0), in the sense that ||\psi(T)-\psi(0)|| < \epsilon

    4) According to you, the AdS part of the state at late time is strongly entangled with a disconnected region. Hence the von-Neumann entropy of the density matrix for the AdS region is large S_vN >> 1 (this is what we actually mean by info loss to the AdS region)

    So, the state \psi(T) on the one hand is arbitrarily close to \psi(0) in the above sense, yet its von Neumann entropy is very far from that of \psi(0), since the latter state has no entanglement with a disconnected region and hence has S_vN =0.

    The challenge for you is to come up with a state \psi(T) that can accomplish this, and I predict with high confidence that you will be unable to do so. So how about addressing this directly?

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  146. bhg,

    "So we have agreed that if the emitted radiation is pure when R_M is inside AdS then the same will be true when it is inside Minkowski."

    Yes, but you haven't shown it's pure in AdS.

    "The point, of course, is that the AdS experiment just described is precisely of the sort I know how to translate into CFT language. I inject particles in from the boundary such that..."

    No, that's where it goes wrong. You have now assumed you work in the theory that's on the AdS-side of the CFT. That theory has additional constraints, as we discussed earlier. These constraints are global.

    How come you do not notice you make an additional assumption here? That's why I asked you to write it down. The moment you speak of the CFT you restrict the AdS fields to states that are expandable around the boundary.

    The easiest way to note what you say is wrong is what Tim and I mentioned a few times earlier. You could equally well turn the argument around and note that Hawking's original calculation applies to good precision in asymptotic AdS, hence the microcanonical interpretation of the BH must be wrong also in AdS. And that's correct, of course. Just that the Hawking-evaporation scenario doesn't have a CFT dual. Best,

    B.

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  147. Bee,

    You're not understanding the argument. The easiest way that you can see this, if you are truly interested, is to examine cases where the procedure I described is carried out explicitly, correctly yielding quantum gravity effects on scattering amplitudes that agree with independent computations in Minkowski space. So there is simply no question that AdS/CFT correctly describes some legitimate quantum gravity effects in Minkowski space, and does so in a totally explicit manner that anyone can unpack for themselves. I can't tell if you don't believe me here or if you think there will be some sudden failure of this approach once black holes are formed.

    Here is one example of something you write that is missing the point: " The moment you speak of the CFT you restrict the AdS fields to states that are expandable around the boundary. ". This ls equivalent to statement that in Minkowksi space "the moment you speak of an S-matrix you restrict the Minkowksi fields to states that are expandable at infinity". Yes, an S-matrix formulation does refer to in and out states, but that doesn't make it wrong, and indeed to the contrary we know that it is right in the very real sense of making correct experimental predictions. You are confusing this with the different question of whether such a formulation is complete in the sense of being able to answer all possible questions of interest. Same goes for AdS.


    Anyway, this discussion has clearly run its course, so I will end here and let you have the last word if you so desire, unless you have some specific question you want to ask.

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  148. bhg,

    ""The moment you speak of the CFT you restrict the AdS fields to states that are expandable around the boundary.". This ls equivalent to statement that in Minkowksi space "the moment you speak of an S-matrix you restrict the Minkowksi fields to states that are expandable at infinity".

    You just conjectured an equivalence. Now prove it.

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  149. BHG

    I have no idea what you mean by a "far more reasonable number". Remember, we are calculating the recurrence time, not the evaporation time. For a light black hole the evaporation time may be made reasonable, but I can't see any reason to think that the recurrence time we be appreciably different. Can you supply such a reason?

    My point about Shankar is simple: we have been going back and forth about what you mean by the "Physical Hilbert space" for some time. And as to what I pointed out, Shankar gives a clear and explicit definition of the "physical Hilbert space" as opposed to the Hilbert space. I understand what Shankar has defined, and why. And what he has defined has direct relevance to other points I have brought up. So it seemed possible that the distinction you were drawing is just the distinction that Shankar made.

    Your response is to say that Shankar's usage is "non-standard". So I asked for something very simple: give a reference to a "standard" definition in a "standard" textbook that gives the definition as clearly and cleanly as Shankar. This is a simple request. Since it can be found in "any reference on canonical quantum gauge theory", just pull one out and provide the page where a definition as clear and precise as Shankar's is given. That ought to be child's play. I don't understand why you prefer to argue rather than just provide something so easy to come by.

    As for your argument, of course I answered it. Neither I nor anyone else has any vague idea what could happen to a system over the periods of time you have to reference. I am sure you are not working with any quantum gravity theory that is precisely enough defined to calculate with, since no approximations or idealizations could possibly be allowed. How about your write down the exact quantum gravity theory whose time evolution you think is so exact.

    In any case, as I mentioned, in any time-reversible theory whatever can be done can be undone. So if Cauchy surfaces can split they can also recombine, If I were to take your challenge as a serious problem. Then it is self-answering: eventually the split surfaces will revert to unsplit states. So there is nothing at all contrary to my argument, which is concerned about realistic periods of time.

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  150. Bee,

    I expect you use compute S-matrices in asymptotically flat spacetime and use them to predict results in our observable universe, even though our universe is non asymptotically flat (more like De Sitter, which is totally different). Please prove that what you are doing is correct, being careful to state all of your assumptions, and defining what you mean by concepts like "physicist" and "particle". You will find that you have absolutely no reason to believe that what you are doing is correct, because you are making global assumptions that do not apply to the problem of interest.

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  151. Tim,

    Poincare recurrence time is exponential of entropy. The entropy of black holes is small when the mass is small. So, the recurrence time is not so long if the black hole is small.

    It is remarkable that you engage in this debate and do not know that.

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  152. bhg,

    I don't have to prove it works to convert asympt Minkowski calculations to asympt dS, I can just go make a measurement and see that it works. Besides, I'd be willing to accept your argument to transition from Minkowski to dS because in that case at least we know it's not wrong. In your case, we do know it's wrong. Proof: If it was correct, black hole microstates in AdS/CFT wouldn't respect the BH entropy.

    Look, I have said this a few times before, but there is no qft with a unitary time-evolution at asymptotic infinity of Minkowski space. The global assumption that I am speaking of has nothing to do with the curvature as you seem to think. The curvature is a local property, no problem with that. It's the extra constraint that comes from the duality itself that is the problem. The whole point of doing the calculations in AdS/CFT is that the result for black hole evap is *not* the same as in asympt Minkowski.

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  153. Bee,

    I read what you write but I also am not really understanding your objection.

    The question is whether black holes form and evaporate according to unitary (external) operator. In Minkowski spacetime that is whether S-matrix for black hole formation/evaporation is unitary. bhg is saying that Minkowski S-matrix is arbitrarily well-approximated by CFT correlators (in correct limit). That has been demonstrated for scattering below BH formation threshold where we know what Minkowski result is. If this same approach still works above threshold for BH formation, it means flat space BH formation/evaporation is arbitrarily well-described by a unitary S-matrix. No?

    Maybe it stops working as soon as you go above BH formation threshold. But why? BH can be much smaller than AdS, and evaporate much faster than AdS time. How would BH know it is in AdS and should misbehave to fool too-clever physicist and vindicate one very confident philosopher?

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  154. Bee,

    "In your case, we do know it's wrong. Proof: If it was correct, black hole microstates in AdS/CFT wouldn't respect the BH entropy. "

    I do not understand you. Can you explain?

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  155. physphil,

    Same thing I've said before. Take Hawking's calculation in asymptotic Minkowski space. Radiation doesn't carry information, hence the entropy of what remains doesn't respect the BH-entropy, evaporation isn't unitary because some rays end at a boundary that isn't at infinity (assuming you know the story but I can go into great detail if you wish). You don't need quantum gravity to see that something is at odds here, that's the whole reason people discuss the matter to begin with. It is of course the very process that bhg is concerned with, some transition from in-states to out-states and so on.

    Now he goes and conjectures that the process in the AdS-dual of the CFT is the same as that in asymptotic Minkowski-space. He does that in the hope of convincing someone (not sure whom) that the AdS-calculation is the one that should apply to our universe. I am saying (as Tim did before) that if that was correct you could use the same logic to proclaim that the Minkowski-calculation should apply in the AdS-dual. Which it does not according to bhg's own argument. It's not adding up is what I'm saying. It's an internally inconsistent set of assumptions.

    "bhg is saying that Minkowski S-matrix is arbitrarily well-approximated by CFT correlators (in correct limit)."

    What he said is that the Minkowski S-matrix is arbitrarily well-approximated by the AdS S-matrix. Fine. You can then *for the cases in which you are in the AdS-dual of the boundary CFT* express this by the CFT correlators. But with this qualifier you have thrown out your justification for carrying over the time-evolution to Minkowski-space because its not a local requirement. Please, I ask you the same thing as bhg, write it down. Take a sheet of paper, list your assumptions. Best,

    B.

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  156. Tim,

    Shankar is not (I presume) talking about gauge systems, so this appears to have no relevance. If want a ref you can go back to the source, which is Dirac's lectures. It's all there. Now, how about you find *any* reference which says that to define the physical Hilbert space you need to solve not just the constraints but also the time dependent Schrodinger equation.

    You can make the recurrence time as small as you want by taking the energy of your system as small as you want. In any case, it seems you agree that on long time scales the Penrose diagram must break down. This takes the legs out from under your argument. Information loss proponents claim that as far as the AdS region is concerned there is an evolution from pure states in the far past to mixed states in the far future. Recurrence rules this out, because the state becomes pure at an infinite number of times in the future, again in gross violation of the standard Penrose diagram. My previous argument has also established that the information comes out much sooner, unless you are willing to violate the axioms of QM, which is rather self-defeating.

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  157. Bee,

    thank you for your response. But, your argument "you could use the same logic to proclaim that the Minkowski-calculation should apply in the AdS-dual..." has force only with equivalence between the calculation of Hawking and calculation in AdS/CFT. But they are very different. Hawking used QFT on a fixed background, or on pretend background that ignores large quantum gravity corrections near singularity/evaporation point. That is an approximation. It might not include necessary ingredients to see if the evolution is unitary (or not). On other hand, CFT is (claimed) to be exactly dual to full quantum gravity.

    So, there is no reason to believe the result of Hawking for subtle questions like unitarity. But there is a reason to believe the CFT result (reason is all the evidence for AdS/CFT. The two arguments are not equivalent.

    "You can then *for the cases in which you are in the AdS-dual of the boundary CFT* express this by the CFT correlators. But with this qualifier you have thrown out your justification for carrying over the time-evolution to Minkowski-space because its not a local requirement. "

    Maybe it is not a local requirement, I am not sure. But it still describes a process where BH forms and evaporates, and is unitary, in space that is arbitrarily close to flat. I do not see the difference with the cases below BH formation threshold. The same AdS/CFT procedure correctly computes (some) flat space S-matrix elements. You can check that it works. Now we use it in the same way to compute some BH formation/evaporation S-matrix elements, and they are unitary.

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  158. Physphil,

    I have to congratulate you. You have produced something like the Platonic Form of a comment that encapsulates perfectly the situation in foundation of physics today. It is a miracle of compression and conciseness. Here it is in full:

    "Tim,

    Poincare recurrence time is exponential of entropy. The entropy of black holes is small when the mass is small. So, the recurrence time is not so long if the black hole is small.

    It is remarkable that you engage in this debate and do not know that."

    Under 50 words. Amazing.

    Since you have your sheet of paper and pencil out for Sabine, please try the following calculations. The system in question is a black hole of mass 1 kilogram, in an asymptotically flat space-time (Minkowski).

    Allow the black hole to evaporate. Now,

    1) Calculate the recurrence time.

    2) Using the answer from 1), use your formula to calculate the entropy of the black hole.

    3) Using your answer from 2), use Bekenstein's formula to calculate the area of the event horizon of the black hole.

    4) Report any anomaly or puzzle that the result of 3) creates. Try to resolve that puzzle.

    I eagerly await your results.

    One thing I will admit: I simply came across Page's number and assumed that he was doing something sensible. I may well have been being too charitable about that.

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  159. physphil,


    "So, there is no reason to believe the result of Hawking for subtle questions like unitarity."

    Of course! I entirely agree on that! And since there is no reason to worry about the loss of unitarity can you remind me why we need AdS/CFT to begin with?

    "Hawking used QFT on a fixed background, or on pretend background that ignores large quantum gravity corrections near singularity/evaporation point. That is an approximation. It might not include necessary ingredients to see if the evolution is unitary (or not)."

    My response to bhg referred to the BH entropy and not to unitarity. That was to avoid this distraction, but apparently my attempt to get straight to the point was rather futile. I believe you can make the argument also for unitarity, but it's more complicated and why bother. One contradiction is sufficient.

    "Maybe it is not a local requirement, I am not sure. But it still describes a process where BH forms and evaporates, and is unitary, in space that is arbitrarily close to flat. I do not see the difference with the cases below BH formation threshold. The same AdS/CFT procedure correctly computes (some) flat space S-matrix elements. You can check that it works. Now we use it in the same way to compute some BH formation/evaporation S-matrix elements, and they are unitary."

    All of this is fine with me. What I am asking is: Why should I believe that the limit \Lambda->0 describes what actually happens in our universe? Bhg's argument is local equivalence. But that, as I point out doesn't hold water because the requirement that what's going on in your AdS space has a CFT dual isn't local.

    As I said before, you can of course simply postulate that the \Lambda->0 limit of the AdS-side of the duality is what we would observe could we observe a black hole evaporate. I merely say I don't know why one should believe this describes physical reality. Best,

    B.

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  160. Maybe, crudely put, if the "black hole wave function" - whatever that means - goes to zero much faster than how one approaches the boundary of spacetime, then the black hole doesn't care if it is in AdS or Minkowski space. On the other hand, its dynamics are then perhaps invisible to the boundary, and AdS/CFT can't say anything about it.

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  161. Tim,

    "The system in question is a black hole of mass 1 kilogram, in an asymptotically flat space-time (Minkowski)."

    That is not what we are discussing. We are discussing recurrences for black holes in AdS. Also, 1 kg is heavy. I prefer a mg.

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  162. Bee,

    "All of this is fine with me. What I am asking is: Why should I believe that the limit \Lambda->0 describes what actually happens in our universe?"

    You do not have to. Maybe there is more than one theory of quantum gravity, and answer is different in different theories. I do not find that very likely, but it is only opinion.

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  163. physphill,

    Yes, of course. But that's not the way the argument is usually presented, and bhg demonstrated just exactly the confused logic for why we should believe it that I have had to endure countless times. Hence the above exchange. For what I am concerned it would be progress if we could agree that there is no reason other than personal conviction that \Lambda -> 0 actually describes the universe we inhabit. I'll still grumble that that conviction is nonsense, but at least it's not as obviously nonsense as the reference to locality.

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  164. Sabine,

    Whether it describes the universe we inhabit or not, the only relevant question here is whether there are good reasons to believe that the same general sort of thing should happen with a black hole that is placed in an AdS that is dual to a CFT vs. one that is placed in dS. A physical process whereby a black hole remains on a connected spacetime is a very different physical process from one where the spacetime splits. It would be odd if the small nonlocalities due to restriction to a CFT dual were responsible for this drastic difference, don't you agree?

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  165. travis,

    I don't think they are 'small non-localities', or maybe I don't know what you mean by that.

    ReplyDelete
  166. Arun,

    You write:

    "Maybe, crudely put, if the "black hole wave function" - whatever that means - goes to zero much faster than how one approaches the boundary of spacetime, then the black hole doesn't care if it is in AdS or Minkowski space. On the other hand, its dynamics are then perhaps invisible to the boundary, and AdS/CFT can't say anything about it."

    Yes, roughly speaking that's the issue. A Gaussian wave-packet, eg, can't be expanded around infinity. States of that kind therefore cannot appear in the AdS-side of the CFT. They can perfectly well appear in AdS space, per se, but once you assume it has a CFT dual, you limit the space of allowed states in the bulk. This is (imo) clearly a global constraint and is a restriction that will go away discontinuously once you remove the boundary.

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  167. physphil

    Adjust the numbers as you like. Put it in whatever spacetime you like. Just do the problem and get the point, OK?

    ReplyDelete
  168. Tim,

    I do not think you have any point. But, maybe this will help you. OK.

    "Allow the black hole to evaporate. Now,

    1) Calculate the recurrence time.

    2) Using the answer from 1), use your formula to calculate the entropy of the black hole.

    3) Using your answer from 2), use Bekenstein's formula to calculate the area of the event horizon of the black hole.

    4) Report any anomaly or puzzle that the result of 3) creates. Try to resolve that puzzle. "

    I choose M = .65 mg.

    1) t=Rexp(S)=10^7 s=three months (factor R is arbitrary, depends on definition of t)

    2) S=ln(t/R)=115

    3) A=4GS=1.1E-67m^2

    4) There is no anomaly or puzzle.

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  169. Bee,

    " A Gaussian wave-packet, eg, can't be expanded around infinity. States of that kind therefore cannot appear in the AdS-side of the CFT. "


    I vowed to step away from this exchange as it seemed fruitless. But when I read the above comment I feel compelled to respond out of a fear that some impressionable reader might actually believe this nonsense.


    To put it bluntly, the statement above is Wrong. Not wrong is some subtle interpretational way, just false in the same way that 1+1=3 is false. I can easily write down the CFT state, completely explicitly, that accomplishes what you say can't be done. This is a standard exercise. I am pretty sure I know the elementary error you are making here, but I won't put words in your mouth. If you wish to stand by the above claim please say so, and I will be happy to set you straight in more detail.


    Everything else makes sense now. If your claim above were true, then indeed it would be a fantasy to recover Minkowski space physics from the infinite radius limit of AdS. But once you take away this error you will (if you are honest) understand that my argument makes sense.

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  170. bhg,

    The "states of that kind" were not referring to Gaussians in AdS but were an analogy to states that can't be expanded around the AdS boundary. I believe we talked earlier about how tails from massive fields reach the boundary or did we? In any case, sorry about the sloppy phrasing; I was hoping it would be an illustrative example :/

    The only way to resurrect your claim is to prove that there aren't any states that can't be expanded around the boundary.

    ReplyDelete
  171. PS: Trust me, if I thought it was that easy, I'd have said it earlier.

    ReplyDelete
  172. Bee,

    Well, at least you agree about being wrong/sloppy about that wavepacket statement. To be clear, there exists a CFT state for any possible wavefunction of a free particle/field in AdS.

    You claims still make absolutely no sense to me, but it's possible some of this is due to similarly bad phrasing -- who knows? Your argument appears to be a perfect example of circular reasoning. You say that Hawking's computation showed that info is lost, therefore any computation, e.g. involving a CFT, that disagrees with it must be wrong. If there is any more to it than that, I missed it. I will just make a few comments, although these are mainly aimed at any other interested readers, as you are clearly locked into your position.

    First, the exact same logic that would lead one trust Hawking's computation in Minkowski applies to the AdS case as well. However, in that case we have an alternative *nonperturbative* formulation of QG that is: a) known to reproduce standard Einstein gravity at low energies b) has the property that the info *does* come out in the radiation, contra Hawking (Tim disagrees here, but he needs to violate the axioms of QM to get this, which is self-defeating). Therefore, in the AdS case, there is apparently a flaw in Hawking's approximations, and it stands to reason that the same flaw is present in the Minkowski case as well, but one needs a non-perturbative formulation to expose it.

    Now, we know that the flat space limit of AdS/CFT does correctly yield the S-matrices for, say, QED coupled to gravity, in the perturbative regime. Now, if we crank up the energy past the Planck scale there is no particular reason to expect that things suddenly break down. The BH info puzzle can be cast as a question about S-matrix (non)unitarity for scattering above the Planck scale, and this is precisely what we can access by this approach. So this procedure is the one and only way to getting at black hole evaporation in Minkowski via a non-perturbative theory of QG, although it is admittedly rather circuitous and surely not the final word. But if some one comes to you saying that they have a non-perturbatively formulated theory that agrees with standard perturbative QFT + gravity S-matrices at low energy, and can make predictions about super Planckian scattering that creates black holes, you reaction would really be "nah, not interested: this came from AdS so it won't apply to our world". Sorry, but it doesn't matter where it came from, if it's a formulation that possesses the above traits, it should be taken very seriously.

    Your comments about the CFT not describing states that are not "expandable around the boundary" (whatever that means) is totally irrelevant. If there are some states that cannot be produced by throwing stuff in from the boundary, they by definition have no effect on S-matrix computations. In flat space, you can imagine there are some kind of states that are not in the in/out scattering state Hilbert space, but if they are never produced this has no bearing on the correctness of the S-matrix computed in their absence.

    Finally, I again point out that it is plainly illogical for you to believe in the applicability of Minkowski S-matrices to de Sitter physics, but not in the setup I am describing, based on saying that the former agrees with experiment. I have tried explaining in vain that we *know* that the CFT will produce the correct perturbative Minkowski S-matrices in the flat limit, so you can equally well apply your experiment based criterion here.

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  173. Tim,


    Shankar is not (I presume) talking about gauge systems, so this appears to have no relevance. If want a ref you can go back to the source, which is Dirac's lectures. It's all there. Now, how about you find *any* reference which says that to define the physical Hilbert space you need to solve not just the constraints but also the time dependent Schrodinger equation. Or one that says that probabilities are computed from inner products of states not in the Hilbert space. I am sure you will not find one, since the rules you are espousing are not those of QM.


    You can make the recurrence time as small as you want by taking the energy of your system as small as you want. In any case, it seems you agree that on long time scales the Penrose diagram must break down. This takes the legs out from under your argument. Information loss proponents claim that as far as the AdS region is concerned there is an evolution from pure states in the far past to mixed states in the far future. Recurrence rules this out, because the state becomes pure at an infinite number of times in the future, again in gross violation of the standard Penrose diagram. My previous argument has also established that the information comes out much sooner, unless you are willing to violate the axioms of QM, which is rather self-defeating.

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  174. Tim,

    I sent response yesterday but must be lost. Here are answers

    "1) Calculate the recurrence time." t=R exp S (prefactor of R depends on exact definition)

    "2) Using the answer from 1), use your formula to calculate the entropy of the black hole." S=ln t/R

    "3) Using your answer from 2), use Bekenstein's formula to calculate the area of the event horizon of the black hole." A=4GS

    "4) Report any anomaly or puzzle that the result of 3) creates. Try to resolve that puzzle."

    There is no puzzle.

    If I use M~.65mg, I can get t~3 months and S~100.

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  175. bhg,

    "Your argument appears to be a perfect example of circular reasoning. You say that Hawking's computation showed that info is lost, therefore any computation, e.g. involving a CFT, that disagrees with it must be wrong. [...] the exact same logic that would lead one trust Hawking's computation in Minkowski applies to the AdS case as well."

    Ugh. Both Tim and I have used this argument to show that your claim is circular. I suppose I should be happy that you finally agree, but I'd appreciate if you didn't claim that I actually intended to use it to show the AdS/CFT calculation is wrong. I merely said your logic is flawed.

    "the exact same logic that would lead one trust Hawking's computation in Minkowski applies to the AdS case as well. However, in that case we have an alternative *nonperturbative* formulation of QG that is..."

    Note that once again you have skipped from AdS space to the AdS dual of the boundary CFT without mentioning it.

    "Your comments about the CFT not describing states that are not "expandable around the boundary" (whatever that means) is totally irrelevant. If there are some states that cannot be produced by throwing stuff in from the boundary, they by definition have no effect on S-matrix computations"

    Your argument happened in a box and the stuff didn't come in from the AdS boundary but from the box boundary.

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  176. BHG

    This is getting rather frustrating.


    If by "Dirac's Lectures" you mean the book "Lectures on Quantum Mechanics", I cannot find any mention of a "physical Hilbert space", or indeed a "Hilbert space" in the book. If you mean his book "The Principles of Quantum Mechanics", then I have run a search and I find zero mentions of that phrase or, indeed, of the single word "Hilbert". So I cannot find this defined anywhere in these. You direct me to go to "Dirac's lectures", and I take the time to dine these up. Nothing. Since you claim that yours is a standard definition to by found anywhere, please actually find a definition and post it.

    This is a particularly odd case, hardly ever discussed, I imagine. Using the Wheeler DeWitt equation for a globally hyperbolic case, the entire dynamics is given in the Hamiltonian constraint. This is due to diffeomorphism invariance. In the case if AdS, because it is not globally hyperbolic, the complete dynamics uses both the Hamiltonian constraint and also a Schrödinger equation. I image this mixed case needs some special discussion. But before even attempting that, I would like to nail down what you have in mind, and I still cannot find a thing. The phrase "Definition of physical Hilbert space" returns only 6 hits, one of which is from this blog! So this seem not to be a very common notion. Just find an actual presentation that define what this means and give the exact citation.

    I cannot for the life of me figure out what you mean by saying "You can make the recurrence time as small as you want by taking the energy of your system as small as you want." Let me ask you to consider the same questions I asked physphill, who has still not got the point:

    Please try the following calculations. The system in question is a black hole of mass 1 kilogram, in an asymptotically flat space-time (Minkowski).

    Allow the black hole to evaporate. Now,

    1) Calculate the recurrence time.

    2) Using the answer from 1), use your formula to calculate the entropy of the black hole.

    3) Using your answer from 2), use Bekenstein's formula to calculate the area of the event horizon of the black hole.

    4) Report any anomaly or puzzle that the result of 3) creates. Try to resolve that puzzle.

    I eagerly await your results.

    ReplyDelete
  177. Physphil

    I rather thought it would be obvious that I did not intend you to use the trivialization of questions 1 and 2 that you did, so you have completely missed the point. In fact, the point in the Minkowski space-time I started with is immediate, but I tried to play along with you. I can't tell if you understand the point and are not admitting it or still don't get it. In case it is the latter, let's try this:

    OK, you just did the calculation in some AdS space of radius alpha. (Query: do you find it odd that you insisted on doing this in AdS but alpha does not seem to appear in any of your calculations? Hint, hint). OK, now double alpha and redo the calculations. Highlight how the outcome depends on the value of alpha.

    If you are tempted to just send back identically the same calculation, please do not. Stop and think. How *should* the calculation depend on the value of the radius of AdS?

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  178. Bee,

    Again, you are just assuming what you want to establish, so it's circular. I am challenging Hawking's original conclusion, so you can't use as a counterargument the fact that I get an answer that disagrees with Hawking -- there's your circle.

    The argument is so straightfoward that I really don't understand why it is confusing you

    1) fact: the CFT describes a nonperturbative theory of quantum gravity in AdS that reproduces standard perturbative gravity/field theory, i.e. Einstein's equations coupled to matter

    2) fact: the flat space limit of perturbative AdS correlators (no CFT here) yield the correct perturbative Minkowski S-matrices. Ie. we reproduce flat space S-matrices computed via standard Feynman diagrams when we take the limit.

    3) Points (2) and (3) imply, by pure logic, that CFT correlators, upon taking the flat space limit will correctly reproduce perturbative S-matrices in flat space. This has indeed been verified.

    No assumptions so far, just verified facts and logical reasoning.

    Now, one way of viewing the problem of QG is that we want a mathematical formalism that computes S-matrices that reproduce standard perturbation theory at low energies but are defined non-perturbatively past the Planck scale. There may or may not be a unique solution to this problem -- we don't know. However, the procedure described above provides one such formalism, and is in fact the only option on the table at the present time. So clearly it is of great interest, especially given how much effort has been put into this problem from various other approaches, without success.

    As far as I can tell, your counterargument is the following: this procedure produces an answer that I don't like, therefore there is no reason to believe it. The thing is, people who study quantum gravity in a mathematically serious way, rather than by wild speculation, appreciate how incredibly hard it is to satisfy the conditions in the last paragraph, so the fact that we have an actual concrete theory that does this is of immense interest.

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  179. bhg,

    "Again, you are just assuming what you want to establish, so it's circular. I am challenging Hawking's original conclusion, so you can't use as a counterargument the fact that I get an answer that disagrees with Hawking -- there's your circle. "

    As I just told you, I know it's circular. I used this to make you see the circularity of your own argument. You are assuming the AdS/CFT result is the correct one for Minkowski space so you can't use it to argue that the Minkowski result is wrong. You get out what you put in.

    Use your three facts and explain why the BH entropy in Minkowski space doesn't count microstates.

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  180. Bee,

    If you think about what you are saying you will see how absurd it is. Apparently, if someone comes to you saying "I have developed theory X, and it successfully reproduces all known facts and further it makes the prediction y. I therefore have a reasonable expectation that y will indeed occur ", your response to this person would be "Your reasoning is circular because you are assuming X is correct". As I said before, if I shared your attitude I would quit physics. Yes, of course its ultimately not possible to fully logically justify expectations for an as yet unperformed experiment, simply because it's possible that there exist an infinite number of theories that agree on all experiments done so far, yet disagree on the new one. This leads to such deep thoughts as "how do we know the sun will rise in the east tomorrow just because it did everyday for the last billion years. "

    Good scientists learn to make judgements as to when a given theory can be extrapolated beyond its tested domain. In the present case, there is precisely one proposal on the table for a non-perturbative theory of QG, and as I have set it up there is no particular reason to expect that it would fail as energies go above the PLanck scale, so its certainly *reasonable* to take its predictions seriously here. This is not some circular assumption as you try to portray it, it is just science as usual.

    "Use your three facts and explain why the BH entropy in Minkowski space doesn't count microstates."


    No idea what you are talking about here. Like most people in the field, I do think that BH entropy counts BH microstates, something that can be proven explicitly in some cases.

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  181. BHG and physphil,

    Since we have reached a critical moment and neither of you seem to realize it, I ask this simple question:

    Start with enough infalling matter to form a black hole of 4 solar masses in an otherwise empty spacetime. Let the black hole form and then evaporate. What is the recurrence time for the initial state if Lambda = 0? What is the recurrence time if Lambda > 0? What is the recurrence time if Lambda < 0? In the last case, how does the recurrence time depend on Lambda?

    If you can answer these simple questions we can put a whole bunch of distractions to bed for good and maybe make some progress.

    ReplyDelete
  182. bhg,

    "Apparently, if someone comes to you saying "I have developed theory X, and it successfully reproduces all known facts and further it makes the prediction y. I therefore have a reasonable expectation that y will indeed occur ", your response to this person would be "Your reasoning is circular because you are assuming X is correct".

    No, that's not what I am saying. I am not sure whether your misunderstanding is sincere or an attempt at distraction, but I'll try to be favorable and assume you are still misunderstanding the point, so here we go again.

    You have developed theory X. Theory X reproduces fact y_1 but does not reproduce fact y_2, where y_2 is the statement "the BH entropy does not count black hole microstates". You chose to ignore fact y_2 and instead postulate that the prediction of your theory must be correct, hence y_2 is wrong. I am pointing out this isn't a proof, it's an assumption and hence circular: You could equally well say y_2 is correct hence your theory is wrong. Since you misunderstood this previously, let me emphasize that I am *not* saying y_2 is correct and your theory wrong. I am saying the statement is equally correct as the other one to show the argument is circular.

    Now look at what you do: You repeat the same mistake in your recent comment by explicitly stating that you "like most people in the field, ... think that BH entropy counts BH microstates, something that can be proven explicitly in some cases." We both know you can't prove it in Minkowski space. You are assuming your "thinking" to be correct and count this as a "fact".

    "As I said before, if I shared your attitude I would quit physics."

    I am not sure why you keep repeating this, but I hope you realize that this is why you rarely meet anyone who disagrees with you on the matter. It even has a name, it's called "survivor bias," and I go on about this in my book, right next to the paragraph about confirmation bias and groupthink. I am sincerely sorry to bother you, but I do indeed think that the logic which you exemplify (which I do know is very common among your colleagues) is fundamentally flawed and you'd be well advised to listen to Tim.

    (I also agree with Tim, btw, that most of the discussion on the BH infloss problem is nonsense, though I guess I outscore him in thinking that his solution attempt is also nonsense.)

    Best,

    B.

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  183. Sabine,

    I do not want to deflect attention from the present flow of discussion, since we are focussed down on absolutely critical matters. But I do note "his solution attempt is also nonsense" and intend to hold you to that claim after this has settled down. You might want to begin a defense of it by defining "nonsense". But no more of that question now.

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  184. Bee,

    Again, you are just missing the point. Your confusion is most easily exposed by your statement

    " Theory X reproduces fact y_1 but does not reproduce fact y_2, where y_2 is the statement "the BH entropy does not count black hole microstates". You chose to ignore fact y_2 and instead postulate that the prediction of your theory must be correct, hence y_2 is wrong. "


    Your error lies in calling y_2 a "fact", when y_2 is really a hypothesis based on an approximate computation with an unknown domain of validity. We don't know if y_2 is a true statement about any complete theory of quantum gravity, so by incorrectly calling it a "fact" you have twisted yourself into a logical pretzel. Furthermore, nowhere am I "assuming" whether y_2 is true or not. I am rather proposing a non-perturbative theory of QG that is known to reproduce standard perturbation theory but also makes predictions beyond the Planck. Once the theory is defined we can ask whether y_2 is true in this theory or not. As they say, you get what you get, and you don't get upset. It is not a question of "postulating" that my theory is correct. If you have some other theory of QG that also agrees with standard perturbation theory we could ask whether y_2 is true in that theory or not. We should try to develop all such theories as best we can, and then we can ask where they agree and disagree. So the situation is exactly the opposite of what you say.


    ""like most people in the field, ... think that BH entropy counts BH microstates, something that can be proven explicitly in some cases." We both know you can't prove it in Minkowski space. You are assuming your "thinking" to be correct and count this as a "fact". "


    Once again, you are spreading misinformation. The original Strominger-Vafa computation was for a black hole in Minkowski space, so what you write is just false and you should retract it. Again, you seem to misunderstand the meaning of the word "fact".

    "It even has a name, it's called "survivor bias," and I go on about this in my book, right next to the paragraph about confirmation bias and groupthink"


    Every few weeks I get an e-mail from someone who has claimed to have disproven special relativity, and they complain that the physics community only believes in SR because of groupthink. Same goes with climate change deniers. You diminish yourself by cheaply hurling these accusations.

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  185. Tim,

    (note to self: avoid insults and be nice)

    "Start with enough infalling matter to form a black hole of 4 solar masses in an otherwise empty spacetime. Let the black hole form and then evaporate. What is the recurrence time for the initial state if Lambda = 0? What is the recurrence time if Lambda > 0? What is the recurrence time if Lambda < 0? In the last case, how does the recurrence time depend on Lambda?

    If you can answer these simple questions we can put a whole bunch of distractions to bed for good and maybe make some progress."



    First of all, these are by no means "simple questions", but let us all try to address them in a constructive way. When \Lambda = 0, the recurrence time is presumably infinite if the quanta are allowed to stream off to infinity. For \Lambda > 0 we have no idea what happens and can only resort to handwaving computations like that of Page. For \Lambda < 0 the problem is well defined if we have a specific CFT in mind. Let's take N=4 SYM. Now for your E= 4 solar mass example, for small |\Lambda|, the maximal entropy state is of course a gas of weakly interacting quanta. The entropy for this was worked out in various place (Horowitz and Ooguri come to mind), but I am too lazy to look this up right now. On dimensional grounds the entropy will be a function of y= E times L_ads, I am guess it's something like S = y^p, where p is some positive number; I can work out the details if pressed and if it's worth my time to do so. Now, the recurrence time for N=4 SYM on S^3 x R can in principle be computed or estimated numerically, but such information is not available. Based on what is known from other theories, a recurrence time of something like e^{e^S} is plausible. This of course goes to infinity as you L_AdS -> infinity.


    I am not sure what you want to take from this, but let me note the following. In the \lambda=0 case, if you put the system in a large sealed box I expect the recurrence time to be finite. I think it's reasonable to expect that the answer to whether black holes "lose information to disconnected surfaces" should be the same whether the sealed box is there or not.


    I think of AdS as a very convenient box. Now, whether the black hole mass is 4 solar masses or smaller, how are you visualizing the spacetime diagram all the way to the recurrence time? Do you agree that the standard Penrose diagram is not a good guide to the physics at such long time scale? Are you imagining that the disconnected surface somehow reconnect to AdS? I actually think you have a chance of overcoming the recurrence challenge if you are willing to modify your statements a bit. But I am not moved by your argument that since the recurrence time is very large we can just ignore it.

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  186. Bee,

    you write "like most people in the field, ... think that BH entropy counts BH microstates, something that can be proven explicitly in some cases." We both know you can't prove it in Minkowski space.

    But you CAN prove it "in Minkowski space" in some cases. I put "in Minkowski space" with quotes because black holes are not Minkowski space. But you can prove it with Minkowski asymptotics, or at least reproduce it precisely, in some cases in string theory.

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  187. Tim,

    I said my factor "R" is arbitrary and depends on definition of recurrence time. If you have an apple, it has some entropy S. If I put apple in a sealed box it will recur. Recurrence time will scale as exp(S). But actual time depends on size of box. If box is infinity, time goes to infinity. This is not deep or mysterious, it is very obvious. Same thing is true for black hole in AdS versus Minkowski.

    ReplyDelete
  188. Tim,

    About the "physical Hilbert space" I already quoted the following from a Smolin paper, which I haven't read, and I only bring up since it happens to have a concise definition of the sort you are looking for. Of course this fully agrees with what I have been telling you over and over and over... He follows Dirac. Note that Smolin emphasizes that physical states are annihilated by the first class constraints, and there is no mention of the Schrodinger equation here. The Schrodinger equation is not a first class constraint, and it is not relevant for *defining* the physical Hilbert space. If I could get you to understand this basic point I would be very pleased.

    Ref: http://www2.montgomerycollege.edu/departments/planet/planet/Numerical_Relativity/QLG/QGwPCosCon2002.pdf

    The approach taken here is Dirac quantization. This means that the whole unconstrained configuration space is quantized. This defines a kinematical state space Hkinematical. The constraints are imposed as operator relations on the states, as in

    ˆ
    C|Ψ >= 0

    where Cˆ stands for operators representing all the first class constraints of the theory. The solutions to the constraints define subspaces of the Hilbert space. A physical state must be a simultaneous solution to all the constraints.

    Often this is done in two steps. The kernel of the gauge and spatial diffeomorphism con- straints is called the diffeo-invariant Hilbert space, and is labeled Hdiffeo. The simultaneous kernel of all the constraints is called the physical Hilbert space, Hphysical.

    Generally, new inner products need to be introduced on these Hilbert spaces, because solutions to the constraints are not normalizable in the inner products on the kinematical Hilbert space.

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  189. bhg,

    "y_2 is really a hypothesis based on an approximate computation with an unknown domain of validity. We don't know if y_2 is a true statement about any complete theory of quantum gravity, so by incorrectly calling it a "fact" you have twisted yourself into a logical pretzel."

    You are almost there! Just one more step and you might finally understand what I've been telling you. y_2 can be established in the pert qg regime as long as you buy locality. Hence my point that your AdS-inherited cure necessarily relies on non-locality and a locality-based argument will not help you.

    "Once again, you are spreading misinformation. "

    To the extent that I am "spreading misinformation" it's because you leave me guessing as to what you are referring to. There aren't any such things as static, extremal black holes in our universe (and also, it's not supersymmetric).

    "You diminish yourself by cheaply hurling these accusations."

    I appreciate your concern, but please don't worry about me. Just pause for a moment and consider how much the large number of people who agree with you contributes to your convictions. Let me remind you that you are the one who brought this on with your declarations that you'd think people who lack faith in your research program would leave the discipline.

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  190. physphill,

    See my above response to bhg. I am sorry for the confusion, but if he doesn't say what he's referring to I can only guess. Sometimes that works badly.

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  191. BHG and physphill,

    Well you got the the right answer about the recurrence times! But then you didn't complete the rest of the exercise. recall:

    1) Calculate the recurrence time.

    2) Using the answer from 1), use your formula to calculate the entropy of the black hole.

    3) Using your answer from 2), use Bekenstein's formula to calculate the area of the event horizon of the black hole.

    4) Report any anomaly or puzzle that the result of 3) creates. Try to resolve that puzzle.

    Full marks on step 1! (Well, I'm being generous. If it is obvious that the recurrence time in Minkowski is infinite, it is all the more obvious that it is in dS). The exact number in AdS does not matter for my purposes, the dependence on the radius alpha does.

    For step 2 I remind you of physphill's principle:

    "Poincare recurrence time is exponential of entropy. The entropy of black holes is small when the mass is small. So, the recurrence time is not so long if the black hole is small.

    It is remarkable that you engage in this debate and do not know that."

    Using that principle, complete Step 2, and then 3 will be obvious. Your remarks for 4 will then be very, very interesting.

    Then we can have a discussion.

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  192. Bee,

    "You are almost there! Just one more step and you might finally understand what I've been telling you. y_2 can be established in the pert qg regime as long as you buy locality"


    I understand perfectly well what you are saying here -- and it is pure nonsense. What you should have written is "y_2 can be established by assuming that locality can be extrapolated to a regime where it has never been tested". The other point that seems to have sailed over your head is that I was only invoking locality at the AdS scale (and supporting it with explicit computations), which is a completely different matter than assuming locality at the scale of the black hole horizon. The crucial point, which again you are apparently missing, is that I make absolutely no assumptions about locality as the black hole scale; instead, once the nonperturbative theory has been defined I can go ahead and ask whether this is the case or not. Can you not see the point: if I can write down a non-perturbative theory of QG which respects locality and reproduces perturbation theory in the regimes where it has been tested, and then I find that it predicts nonlocality at the BH horizon scale, then I have an explicit counterexample to your assumption that locality can always be extrapolated to the inside of a black hole. This is an output, not an input, and it shows that your reasoning is flawed.


    As I said before, these accusations of groupthink just make you sound like a crackpot. I recommend you make scientific arguments instead -- it's more effective. And your last sentence is a fabrication, since I made absolutely no statement about what other people would do, only myself. Please don't put words in my mouth.


    Reading this over, I sound pretty grumpy here. Sorry about that, but man, I find this frustrating...

    ReplyDelete
  193. Tim,

    I am still waiting for you to turn in your homework assignment in which you find a reference where a) physical states are required to solve the time dependent Schrodinger equation as part of their definition, and b) probabilities are computed using inner products between states that are not in the Hilbert space.

    I have no idea what you are asking for, since you can't answer (2) from (1). You are obviously confused about recurrence times, since this makes absolutely no sense.


    Anyway, I look forward to seeing your homework assignment!

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  194. bhg,

    "y_2 can be established by assuming that locality can be extrapolated to a regime where it has never been tested"

    Please specify what you mean by "regime where it has never been tested".

    "The other point that seems to have sailed over your head is that I was only invoking locality at the AdS scale"

    You were arguing that in a sufficiently large patch of AdS with sufficiently small |\Lambda| you can recover the physics of Minkowski space to a given precision. The latter is local. You seem to be using the word 'locality' in a different way here.

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  195. Bee,

    "Please specify what you mean by "regime where it has never been tested".


    The regime I refer to is inside a black hole horizon.


    "You were arguing that in a sufficiently large patch of AdS with sufficiently small |\Lambda| you can recover the physics of Minkowski space to a given precision. The latter is local."


    Ok, to clarify, what I said was that agreement with local Minkowski space physics can be established (not assumed) for perturbative processes. For super-Planckian processes involving black holes, I am indeed invoking locality at the AdS scale to argue that it is *plausible* that black holes in an arbitrarily large AdS space will behave the same as black holes in Minkowski. This seems to me like a very reasonable hypothesis, and it would be quite shocking if its false.

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  196. bhg,

    "inside the black hole horizon" is quite vague. Do we agree that Hawking's calculation shows the microcanonical interpretation of the BH entropy is violated without the need to know what's going on in the Planckian regime? If you want to hold onto it you need some way to deviate from that calculation (that being the usual conundrum). The only ways I know how to do that are nonlocal (or acausal, but I don't think the distinction matters). So where do you get these deviations from?

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  197. BHG

    Yes, *you cannot calculate recurrence times from entropy*! That's exactly my point! I was trying to make that point to physphil after his snarky comment about how I should not even be commenting if I did not know how recurrence times are related to entropy, and then figured you needed to think about the issue because you just as confidently asserted that you could make the recurrence time as small as you like by making the black hole smaller. I objected at the time, but you ignored it. In fact, I could not work out what you could possibly be thinking until physphil announced his formula, and then I thought that must be what you were thinking. If not, maybe you can explicate your thought process. And, I assume, take the whole thing back because is makes no sense at all.

    ReplyDelete
  198. BHG

    As for my homework: I already did b) in citing Shankar who nicely defines what he means by the "physical Hilbert space". And, by the way, I really find it curious that you lecture me about how your definition is so common and well known that you can find it in any text, mentioning Dirac, and then when I point out that you can find it nowhere in Dirac the best you can come up with is a paper by Smolin *that you have never even read*. I guess I will assert that Shankar's book is much, much, much, much better known that Smolin's paper, so if the usages differ it is Smolin that is non-standard, not Shankar.

    What about a)? Well,, as I said, the problem here is that you seem to think that there is some accepted usage for the term "physical Hilbert space" that applies *only* to gravity in AdS! Take gravity in asymptotic Minkowski, for example. As I have said (and you have not denied), that gives rise to the Wheeler-DeWitt equation if you follow canonical quantization and treat diffeomorphism invariance as gauge. And in Wheeler-Dewitt, the dynamics is not implemented by a Schrödinger equation but by a Hamiltonian constraint. In other words, in asymptotically flat space, the "Wheeler-DeWitt patch" is the whole space time. And saying that the "physical states have to satisfy the constraints" is the same as saying that the physical states solve the dynamical equation, and the the states "at different times" in a solution (i.e. the states relative to different Cauchy surfaces) are all "gauge equivalent". That is the source of the "problem of time" in Wheeler-DeWill, and it arises from diffeomorphism invariance.

    The weird thing about AdS is that because it is not globally hyperbolic in addition to the Hamiltonian constraint, which you again do impose on the physical states, there is the boundary part that is governed by a Schrödinger equation. And that can happen, by the way, only because you have strongly constrained the system.

    the fact is that all of your talk about the "physical Hilbert space" and "taking inner products" is empty anyway. What are you taking these inner product for? to predict collapses? but then the evolution of the wavefunction is not unitary or information-preserving. So information is lost all the time. and there never was a problem to start with.

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  199. Tim,

    "you just as confidently asserted that you could make the recurrence time as small as you like by making the black hole smaller"

    We are discussing black holes in AdS. For a black hole in AdS, the recurrence time goes to zero as the mass goes to zero (all else fixed). That invalidates your entire argument.

    Either you do not understand this basic fact or you are throwing dust as usual.

    ReplyDelete

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