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. Reimond,

    "Superdeterminism", i.e. hyper-fine tuning, i.e. denial of statistical independence for degrees of freedom universally regarded as free variables, is only possibly scientifically tenable if one posits retrocausation, which is an abstract idea that we don't even know how to implement in a tractable way in any theory. Without retrocausation, denial of statistical independence is denial of the presupposition of all scientific conclusions. It is like dealing with a scientific issue by saying that the entire apparent physical world is an illusion, or that all experimenters are liars who just make up their data. That is always a choice, and logic alone won't stop you. But if you make that choice you are just no longer doing science at all.

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  2. Tim,


    I was intentionally being vague about what is precisely meant by "measurement" here. As you know, all of these subtleties don't really matter for practical purposes, in the sense that the typical experimental physicist knows how to make extraordinarily accurate predictions and "measurements" without immersing himself in the literature on the foundations of quantum mechanics. He will effectively follow the original Copenhagen interpretation, even though if you were to press him on it he would be unable to explain precisely what it even means.


    I won't deny that there are all sorts of interesting and confusing issues regarding measurement in QM; what's not clear to me is whether they are directly relevant to black hole evaporation. That is, I think the question of the (non)existence of a unitary S-matrix is a question about quantum gravity that makes sense even without talking about how you would actually go about measuring the S-matrix.


    A bit more specifically: coming back to my analogy between black hole evaporation and a sealed lab that splits in two, my question is whether there is any distinction between these two setups insofar as the structure of the late time quantum state is concerned? I again want to press the question of whether the late time physical Hilbert space is or is not, according to you, a tensor product. It seems to me that one has to answer this question before one gets to the higher order question of how one would actually measure a quantum state, and to see whether the latter question is of any particular relevance.


    I believe my general argument should be clear. If you say that the Hilbert space is a tensor product, then I will appeal to my argument showing that this is incompatible with AdS/CFT. If is not a tensor product then I will call this radical because there is now a constraint limiting what the late time external observer can physically do, dependent on what is inside the horizon. I should note that versions of the latter option do appear in the black hole literature. For example, the "complementarity" proposal of Susskind et. al. involves an extreme version of this, and there are more recent less extreme (and much more precise) versions of this that have been argued to come out of AdS/CFT (though many are skeptical).


    Your example of recombining electron beams is cute, and I am interested to know where this is discussed as it would be a good challenge to give students. I can see the loopholes here depending on precisely how the experiment is setup (e.g. a statement like "now recombine the beams" is not as innocuous as it sounds).

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  3. BHG

    Okay, let's leave the measurement problem aside. I agree that this entire conversation about information loss has to be carried out only at the level of the universal wavefunction, which, in a globally hyperbolic spacetime, just means the wavefunction on a Cauchy hypersurface. (Since AdS is not globally hyperbolic we should have been more careful about what we even mean in this setting: we make up for the lack of global hyperbolicity by some choice of timelike boundary conditions, and we have not even discussed what those are.)

    If you want to ascribe a wavefunction to anything less than a Cauchy surface, then you confront a problem. The obvious thing is to trace over some part of the universal wavefunction, generically leaving an improper mixed state due to entanglement. (AMPS, as I understand it, gets off the ground by insisting that the external-to-the-horizon state derived in this way always be pure, so that the global wavefunction is a always a product state, which is a hell of a strong requirement, with zero physical motivation, that implies other weird stuff like firewalls.)

    Now, about your "dilemma". I am going with the claim that the solution space is not a tensor product space, basically because we are carrying this whole conversation out under the assumption that AdS/CFT is true, and you have told me that that is what AdS/CFT requires. Okay, so that's on you. Does the fact that the solution space is a not a tensor product put any "weird constraints" on what an experimenter in one region can do with equipment in that region? Not as far as I can see, for exactly the reasons articulated by Carl3. But we can also sidestep that question: even if I grant you (which I do not believe) that there is some weird constraint that follows, that very same wield constraint follows in AdS/CFT quite independently of any issue about evaporating black holes! I mean carry out your "separate the boxes" thought experiment in AdS. Your own argument implies that the solution space for the example is not a tensor product, and hence that whatever constraints follow from that are with us all the time in AdS! So if there are such constraints, and they are unacceptable, AdS/CFT is dead. And if they are acceptable, then what's the beef against my solution?

    You can't just take a generic problem about AdS/CFT and pin it on my solution in the setting of AdS/CFT in order to reject my solution!

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  4. Carl3,

    Let me explain the thought process behind the singlet constraint example. Suppose you are in possession of a proton in a box. And suppose I tell you that there is a neutron on the far side of the moon, and that the combined spin state of the proton and neutron is a spin-0 singlet state. That by itself is totally unremarkable. It implies that your proton has equal probability amplitude to be spin up or spin down, but if you want to you can measure its spin, thereby collapsing its wavefunction to be either spin up or spin down, and henceforth forget about the distant neutron. No problem here.

    But instead suppose I add the statement that not only are the proton and neutron in a singlet state, but further this singletness is a constraint on the physical Hilbert space. That would be bizarre. It implies that it is impossible to collapse the spin state of the proton to be either spin up or spin down, because such a state violates the singlet constraint. The underlying reason for the weirdness is that the constrained Hilbert space is not a tensor product. If such a constraint existed, you would definitely notice it -- it would not be subtle.

    Let's put this in the context of the lab which splits in two. Before the lab splits, imagine that some particle decays into a proton plus neutron in a singlet state. Then the lab splits, carrying the neutron to the far side of the moon. If you try to impose a constraint on the Hilbert space of the split lab system such that when evolved backwards in time it must give back the original proton+neutron singlet state then you are precisely in the bizarre situation I just described. The key distinction to make here is that between saying that the state of the world has some property X, versus saying that all allowed states in the Hilbert space must have this property.

    As far as I can tell, Tim's proposal for a non-tensor product late time Hilbert space will have these bizarre features, but my main goal at present is to understand what exactly his proposal is. One way to phrase the question is to ask whether a late time observer in AdS can tell in some obvious way whether the diffuse gas of quanta he is surrounded by came from black hole evaporation and the presence of a disconnected Cauchy surface, versus from some more mundane non-black hole process that produced the same density matrix. I say it would be extremely weird if a single observer could tell the difference, but also claim that this is precisely what will be the case if there is some constraint that renders the Hilbert space to not have a tensor product structure.

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  5. Tim,

    Let's put AdS/CFT aside for right now. Please understand that at the moment I am just trying to understand what your precise proposal is. After we get that sorted out we can ask whether it is or is not compatible with AdS/CFT.

    I believe we agreed that the usual structure of QM implies that at some late boundary time t the system is described by a state in a physical Hilbert space. The precise question I am asking is: is this physical Hilbert space a tensor product, with one factor associated to the region behind the horizon and the other factor associated to the component connected to the AdS boundary? Note that I am only asking about the structure of the physical Hilbert space at some particular late boundary time t. I am not asking about the structure of the "solution space" (in your sense of the term) associated to global solutions with some boundary conditions in the past. For a reason that eludes me, when I ask about the physical Hilbert space you respond with a statement about this solution space, which is a different object. It will take us a long way if we can come to a concrete proposal for the late time physical Hilbert space.

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  6. BHG-

    You now want to define a third Hilbert space, which is neither what I call the kinematical space nor the solution space for AdS? If I am following, what you are asking is this: suppose that the complete space-time at issue is isomorphic to the set of events spacelike separated from a boundary Cauchy surface at some very late boundary time in AdS, and suppose that the gravitational theory is imposed just on that. Well, then the relevant space-time is disconnected, and offhand I would assume the solution space just for it would be a tensor product space. So what follows? (Warning: if something completely unacceptable follows then that would be an argument against my offhand judgment, and I would retract it. Since I have not even considered this question before I am not wedded to the answer.)

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  7. Tim,

    Your message makes it sound like I am introducing some new concept (a "third Hilbert space"), but I believe I have been consistent throughout this exchange in saying that there is only one Hilbert space of direct physical relevance: the physical Hilbert space. Indeed, I would just call this "the Hilbert space". I am now puzzled, because we had a long exchange about the definition and interpretation of the elements of this HIlbert space, namely physical state wavefunctions. This began with my Jan. 14 post, which I quote from below:

    " 1) Tim asserts that the bulk obeys the ordinary rules of QM, and that we should take the standard Penrose diagram at face value. Good, so let us ask where this leads when we consider black hole formation and evaporation in AdS.


    2) There is a well defined notion of time t at the boundary assuming standard asymptotically AdS boundary conditions. We therefore have a Hilbert space of physical states at each time t, and the wavefunction describing the process is \psi(t, ...) where ... denotes the relevant configuration space. In the bulk description, the ... correspond to the values of the metric and matter fields on a spatial manifold (more details of this were going to explained in the line of argument I was developing, but we may never get to this).

    3) This wavefunction evolves according to Hamiltonian evolution, d\psi/dt = -i H\psi, where H is the boundary Hamiltonian of Regge and Teitelboim. The full H_bulk also includes volume terms, but these are constraints and so annihilate the physical state \psi, leaving just the boundary part of H_bulk.

    (2) and (3) just correspond to following the standard rules of QM "


    I want to emphasize that this whole discussion is framed purely in terms of gravitational physics in the bulk; let's just ignore the CFT for now. We discussed in detail how in the semi-classical regime a physical state wavefunction at time t describes physics in the WDW patch associated to time t. In AdS, a WDW patch covers only part of the spacetime, and as t evolves different regions enter or leave the corresponding WDW patch.


    I'd be happy to give more background or go over this again. I am confused because your message is phrased as if this lengthy exchange of ours never took place.

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  8. BHG

    I am completely at sea here. You want to discuss the structure of physical states "at time t" in AdS. That can mean two quite different things.

    1)Ttake the collection of all the solutions for the gravitational theory *for all times* and the collect together the set of states particular to time t. I'm not even quite sure what this means, but let's leave it, because it does not seem to be what you had in mind.

    2) Forget about the t-evolution altogether. Forget about the idea that the state at this "late time" t arose from the formation and subsequent evaporation of a black hole. At this late time, there is no event horizon at all. Just ask, at this late time t, which is no longer considered as a time after the evaporation of a black hole, what does the space pf physical states look like? Subject to what constraints? I mean, the space of physical states is *obviously* and *trivially* a tensor product space if we are allowing as solutions sets of disjoint space-times each of which is itself a solution of the gravitational theory!. That's just a trivial fact. So I assume you are disallowing that? By what principle? If the principle is that you just won't consider solutions that are collections of disjoint space-times, then I'm not sure why we are even raising this question. It is not relevant to our problem.

    To repeat: the full (i.e. all-t) solution for an evaporating black hole in AdS will, according to my suggestion, always be a single, connected space-time which has the peculiarity that some of the Cauchy surfaces are connected and others are disconnected. In the setting of AdS, what we have discovered is that this same peculiarity implies that the WDW patch at some times will be a connected space-time and at other times will be a pair of disconnected space-times. To my mind, this is just a distraction: if you want to understand black hole evaporation and only discuss one WDW patch, take a patch that has both pre-evaporation connected Cauchy surfaces and post-evaporation disconnected Cauchy surfaces, i.e. tale a WDW patch that has the same structure as the standard Penrose Diagram I have been discussing all along! Leave AdS out of it, or at least leave out this worry about late, disconnected WDW patches! If the physics in the patch I just mentioned is clear, so you understand how you get from the state on a pre-evaporation connected Cauchy surface to the state on a post-evaporation disconnected surface, the main puzzle has been solved. Further evolution to later t values is unproblematic.

    Do you dispute any of that? If so, what? This whole discussion of late-time AdS WDW patches seems completely beside the point.

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  9. Tim,

    Once again, at the moment I am not trying to advocate for any one particular position, I am just trying to understand what your proposal is. The starting assumption is that there is a theory in the AdS bulk that obeys the standard rules of QM. As i think everyone would agree, this means that for each boundary time t the system is in a quantum state which is itself a vector in a physical Hilbert space, and the t evolution of this state is governed by the Schrodinger equation, H\psi = id\psi/dt. To distinguish different scenarios we now need to get more specific. The AdS analog of asking whether or not there is a unitary S-matrix is to ask about the relation between states defined at early and late t, the former being before the black hole has formed, and the latter being long after it has evaporated. One basic question that needs to be answered as part of defining a specific scenario is what is the structure of the late t Hilbert space: is it a tensor product or not? The 1970s Hawking argued that (if we adapt his argument to AdS) the late time Hilbert space is a tensor product; the full final state is pure but information is lost to an external observer because he only has access to observables that are outside the horizon. This position is also taken by Unruh and Wald among others. On the other hand, advocates of firewalls/fuzzballs claim that the late time Hilbert space is not a tensor product, so there is no information loss to the external observer. I am asking what happens in the scenario that you have in mind? This is not an obscure technical question, but rather one that is at the heart of any argument for or against information loss. You ask me: " Just ask, at this late time t, which is no longer considered as a time after the evaporation of a black hole, what does the space pf physical states look like? Subject to what constraints?". My answer is that you should impose whatever constraints define the physical Hilbert space in your scenario. It is not for me to answer this, because I am not the one proposing the scenario.

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  10. BHG

    I don't see why were are having trouble here. Let me try yet again, and forgive this for being repetitive.

    In AdS, there is a "time co-ordinate " associated with the boundary. This is a global time co-ordinate on the CFT, and the equal-time surfaces on the boundary are Cauchy surfaces of the CFT. as time on the boundary advances, so does this Cauchy surface.

    Fix a particular time at the boundary, that is, fix a Cauchy surface on the CFT. For each such Cauchy surface, there exists the set of all points in the interior that are not timelike separated from the events in the Cauchy surface. We call that set of events the WDW patch of the boundary time. the values of boundary time t can be partitioned into three classes: 1) early times, in which every Cauchy surface of the WDW patch for that time is connected and the patch itself is connected, 2) middle times, in which the WDW patch is a connected space-time but some of the Cauchy surfaces of the WDW patch are connected and others are disconnected, and 3) late times, in which the WDW patch is disconnected and all of the Cauchy surfaces are disconnected.

    The standard Penrose diagram—the one I discuss in my paper—corresponds to one of the middle times in AdS. So I would have thought that analyzing that digram would also suffice to understand how things go in AdS: in order to deal with the black hole evaporation you have to work through the middle time. So I can't really understand why you are fixated on the late times.

    At a middle time, we can foliate the space-time into Cauchy surfaces many ways, Choose one such way (whichever you like) and parameterize the set of Cauchy surfaces with the variable alpha. So Cauchy surfaces with "early" alpha numbers are connected and Cauchy surfaces with late alpha numbers are disconnected. And we can ask: what does the "physical space" look like? In particular, is it a tensor product space. And you have convinced me (just by saying so, not because I understand) that the solution space is not a tensor-product space. Fine! If you insist! What do I know?

    I infer that even when focusing on the physical states on the disconnected Cauchy surfaces, it is not a tensor product space. Again, that sounds a little odd to me, but I am not expert in AdS. I concede to your understanding.

    The only question left is what about the late period when the entire WDW patch is disconnected? Offhand, you would think that the physical space in such a case must be a tensor product space. But it is here that the successive overlaps of the WDW patches comes into play. Take a middle time, where the the WDW patch is one connected space-time that imposes constraints on the possible physical states on the disconnected parts. Now at a later time, the WDW patch may be completely disconnected, but it also cannot ignore the constraints that were imposed on the disconnected Cauchy surfaces in the earlier patch. So I assume, even at these late times, that the two patches will be subject to the same constraints, which will be passed in this way down the line as far as you like. So my final answer is that the physical space, even at late times, is not a tensor-product. But I am strongly relying on your results in making that claim.

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  11. AdS/CFT (or maybe better the holographic principle) relies crucially on the assumption that all is entangled, i.e. it is NOT a product state. The essential paper for me is https://arxiv.org/abs/hep-th/0603001 which is heavily based on https://arxiv.org/abs/hep-th/9303048. The entanglement entropy given by tracing out the density matrix is all important.
    Thus, for BHG the question “to be or NOT to be a product state” is essential and BHG just gave a good and comprehensive summary of the opposing camps:
    “One basic question that needs to be answered as part of defining a specific scenario is what is the structure of the late t Hilbert space: is it a tensor product or not? The 1970s Hawking argued that (if we adapt his argument to AdS) the late time Hilbert space is a tensor product; the full final state is pure but information is lost to an external observer because he only has access to observables that are outside the horizon. This position is also taken by Unruh and Wald among others. On the other hand, advocates of firewalls/fuzzballs claim that the late time Hilbert space is not a tensor product, so there is no information loss to the external observer”.

    On the other hand, if all is entangled and gets even further entangled, how “on earth” are we able to prepare a state to perform a measurement. This prepared state is supposed to be unentangled with the rest of the universe, i.e. the prepared state and the state of the rest of the universe form a tensor product.
    In Tim´s example, the prepared state itself can of course be a superposition of spin up and spin down in z-direction and evolves unitarily. But when measured in z or x-direction the (unitary) information goes bye-bye and we gain (Shannon) information.

    (“ceterum censeo” the whole problem is to believe in an exclusively unitary evolution. Thus, solving the measurement problem resolves the paradoxes)

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  12. And my “Superdeterminism is definitely a tenable CHOICE ;-)” was intended to question Carl3´s “Do I notice my lack of freedom? Can I do any experiment to prove to myself that I am fully constrained by past states of the world? No.”
    Since Tim replied to my single sentence, here is what I originally wanted to write:
    “Yes, we can!” - with EPR and Bell´s theorem. Well, ok, if you believe in “Superdeterminism” (https://en.wikipedia.org/wiki/Superdeterminism) then you cannot, because the “freedom-of-choice loophole” stays open. But then you should also not be curious about anything, because it is already set, since the beginning of time. (And Darwinian evolution would be completely senseless - you will evolve and have your hot dog … ;-)

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  13. Tim,

    Good, this is helpful and I believe I can now shed some light on the discussion. Let me emphasize that right now nothing in the discussion directly involves AdS/CFT: everything being said about the structure of bulk states and their t evolution follows from applying ancient rules of canonical gravity and could have been said long before AdS/CFT was conceived.

    Let's consider your statement: "At a middle time, we can foliate the space-time into Cauchy surfaces many ways, .... I infer that even when focusing on the physical states on the disconnected Cauchy surfaces, it is not a tensor product space.". It seems like you are thinking of there being a quantum state defined on each Cauchy surface labelled by some value of \alpha. But this is not the case: at a given boundary time t, the system is described by a single wavefunction \psi{g_{ij},\phi;t]; at fixed t there is no 1-parameter family of states labelled by alpha, only a single state (which of course lives inside a larger Hilbert space). This single wavefunction assigns a complex number to any choice of 3-geometry and matter field configuration, and at an intermediate time t, the space of 3-geometries on which the wavefunction is of appreciable size will include both connected and disconnected ones. Note that I reviewed these issues in my post of Jan. 15, which you may find helpful to review. This intermediate value of t is a complicated and difficult one to understand for the reason that it involves both connected and disconnected 3-geometries , and it's why I want to simplify life by first considering very late times where all 3-geometries are disconnected. This is in the same spirit as in particle physics where one focuses on the S-matrix: the collision process itself is complicated, but things simplify at late time where particles separate from each other and one can draw strong conclusions just in this regime. That is not to say that ultimately one doesn't want to understand the full evolution in detail, just that it's wise to first consider regimes where things are under better control.


    Now, as to constraints "being passed in this way down the line", this is an argument that the quantum state does not take the form of a tensor product of two pure states, not that the Hilbert space itself is not a tensor product.

    If you take the Penrose diagram at face value, then I say that at very late times, when all surfaces are disconnected, the natural expectation is that the Hilbert space is a tensor product. That's certainly what the 1970s Stephen Hawking had in mind, as do current advocates for information loss. I really can't see any argument (without invoking AdS/CFT) that it should be otherwise. The situation at the intermediate t is much less clear, which is why I would rather leave it aside for the time being.

    Does this make it clear why I am asking for what your proposal is for the late time Hilbert space, and why I would expect that you would advocate for it being a tensor product?

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  14. BHG-

    Sure, I understand your reasoning. Let me reformulate it in an even clearer way.

    Forget about Cauchy surfaces for a moment. Your wavefunction at a given boundary time t is really the wavefunction for the WDW patch, which is a four dimensional thing. And the thing that differentiates both the early and middle times from the late times is this: at the early and middle times, the WDW patch is connected and at the late times it is disconnect. Not merely the Cauchy surfaces, but the 4-dimensional patch is disconnected. That is why every single Cauchy surface is disconnected: a Cauchy surface for the pair of disconnected space-times is just a pair or Cauchy surfaces, one for each disconnected part.

    Now your immediate intuition is that the *physical* space, which I would call the *solution* space, must therefore, at late times, be a *tensor product* space. The picture in your mind is this: let psi_in be a wave function defined just on the interior of the event horizon and psi_out be a wave function defined just on the outer disconnected piece, and let both of these satisfy the dynamics individually (i.e. be annihilated by the Hamiltonian. then if there are a set of such solutions psi_in and another set of solutions psi_out, then I can mix-and-match from these two sets as I like to get global solutions. But that means that the solution space is a tensor product space.

    Well, that has a plausible ring to it, but it could break down. All of the global solutions just mentioned are product states, with no entanglement. But very plausibly, any state arising from the evaporation of a black hole will be entangled between the two pieces. If so, the quick argument does not work.

    In fact, here is a direct argument that the "mix-and-match" argument *cannot* be right. Suppose I have solved the problem for a bunch of different evaporating black hoes in AdS. In each case, though, all of the initial matter is used to form a black hole: outside the black hole event horizon is a vacuum, or better, has only Hawking radiation. Each of the black holes radiates Hawking radiation until it evaporates, leaving (according to me) two disconnected pieces: the interior of the event horizon and the exterior region. So in late times, each of these is a pair pf disconnected space=times. Then why can't I just mix-and=match: pick any of the interior space-times and any of the exterior space-time to form a solution space that is a tensor product.

    The answer is that if the Black hole is initially (when formed) of ADM mass M, then it will radiate until it has radiated out the equivalent energy: Mc^2. As it radiates out the equivalent of mass M, that same amount of anti-Hawking radiation must be created as well. The event horizon disappears and the Hawking radiation stops exactly when the amount of Hawking radiation = M, at which point there is as much positive energy as negative energy inside the black hole. So even if the inner and outer bits of spacetime detach from one another. I can tell which inner part goes with which outer part. If the outer part has ADM mass M, then the inner part has a mixture of M regular energy and M negative energy. So even apart from entanglement, the mix-and-match Strategy doesn't work.

    Anyway, My off-hand opinion about this is pretty worthless. In a normal situation, I would also expect a tensor product, and the only reason I don't is exactly because you have told me that AdS has some very strange and highly constrained solution space, and I am trying to see what the consequences of that are. But I am just taking your word that AdS has these strange properties and trying to work out what follows. And given that, it looks like it is not a tensor product after all. So that's what I'm guessing here. As I said, I never gave such a question any thought. So I am just trying to figure out what the constraints allow.And it looks to me like it shouldn't be a product space. OK?

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  15. Tim,

    After a slight clarification, I don't see that anything in your message argues against having a tensor product Hilbert space at late times, in fact rather the opposite. You first say that we can mix and match solutions of the constraints in the two components to get global solutions. I agree. But then you go on to say that this is insufficient, since the state resulting from black hole evaporation is not a product state but rather displays entanglement. But there is no conflict here, as I now explain. The constraint operators are linear operators, so given any two solutions of the constraints you can add them together with arbitrary coefficients to get another solution. So the generic solution of the constraints is an arbitrary linear combination of all the mix-and-match solutions you mentioned. In fact, this is just the statement that a tensor product vector space has a basis consisting of product vectors, but that the the generic vector in the space is not a product. So I think your argument in fact supports the statement that the Hilbert is a tensor product, but that the state resulting from black hole evaporation is a non-tensor product entangled state, two statements which are perfectly compatible with each other.

    Two other points: instead of talking about the constraints you speak of states being "annihilated by the Hamiltonian." I advise against this language for two reasons. First, there are an infinite number of constrains (4 for each point x), but only one Hamiltonian. Second, the Hamiltonian does not annihilate the state because of the boundary term. Really, the constraints and the Hamiltonian are logically distinct objects, although it is of course the case that the volume integral part of the Hamiltonian is built out of the constraints. It's better to say that physical states are annihilated by the constraints.


    Second, you write "But I am just taking your word that AdS has these strange properties". I am not sure what strange properties you are referring to here. The only significant distinction between AdS and flat space for our purposes is that AdS has a timelike boundary, which I wouldn't call strange. Otherwise, as far as I can tell we're just discussing generic feature that would hold for black holes in any spacetime. My point is that if you take the Penrose diagram at face value and make other natural assumptions then you are lead to the statement that the late time Hilbert space is a tensor product, but the state is highly entangled, just as Hawking was led to this in his original work.

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  16. Black Hole Guy:

    You say:

    "Let's put this in the context of the lab which splits in two. Before the lab splits, imagine that some particle decays into a proton plus neutron in a singlet state. Then the lab splits, carrying the neutron to the far side of the moon. If you try to impose a constraint on the Hilbert space of the split lab system such that when evolved backwards in time it must give back the original proton+neutron singlet state then you are precisely in the bizarre situation I just described. The key distinction to make here is that between saying that the state of the world has some property X, versus saying that all allowed states in the Hilbert space must have this property."

    This is exactly the situation we are always in. If you evolve the state of the split lab system back in time, you had better get back the original proton+neutron state, otherwise determinism is dead. You are imagining that the observer in the lab can somehow cause the wave function to collapse, but then we are outside the framework in which we're talking about the information paradox. The observer is part of the wave function too. Nothing he does will change that. Nothing the observer does will make it so that if the full state of the lab is evolved backwards in time, you won't get back the proton+neutron state. To bring this over to the quantum gravity case: If you evolve a late time WDW patch backwards in boundary time, it had better give an earlier time WDW patch which represents the initial conditions under which the black hole formed and evaporated. Nothing an observer does inside the patch can possibly remove this constraint since he himself is part of the patch. So it simply doesn't matter whether it's a tensor product space or not. Nothing weird will be observed.

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  17. Travis,

    I agree that if you evolve the system back in time you recover the original state, but that wasn't my point. My point is that if the Hilbert space is subject to the constraint that the p-n system is in a singlet, then an experimentalist will be unable to measure the spin state of the proton (i.e. up or down). Proof: according to QM, the probability to find, say, spin-up is obtained by summing the squares of inner products of spin-up eigenstates and the state of the system. But spin-up eigenstates are not in the Hilbert space by assumption. Said differently, the operator (S_proton)_z is not an allowed operator because it doesn't commute with the constraint.

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  18. BHG

    The "strange properties" I was referring to are that the Eigenvalues of the Hamiltonian are supposed to be discrete and non-degenerate (up to symmetries). I feel a bit of vertigo at this point. My understanding of the dialectic so far is that

    1) we are restricting the discussion to AdS because you have an argument against this solution that only goes through in AdS
    2) the argument is more or less like this: in AdS, the Eigenvalues of the Hamiltonian are discrete and (effectively) non-degenerate but if we take the Penrose diagram seriously we are led to the conclusion that the physical space is a tensor product space and if the physical space is a tensor product space then the Eigenvalues of the Hamiltonian cannot be discrete and (effectively) non-degenerate.
    3) the argument that the Eigenvalues of the Hamiltonian are discrete and (effectively) non-degenerate holds in AdS because the gravitational structure acts rather like a square well potential, which also has a discrete spectrum.

    Now I found the propriety that the Hamiltonian has a discrete and (effectively) non-degenerate spectrum pretty incredible at the time, and tried to give arguments (with six electrons) against it, but that went nowhere. So not wanting to further slow things up I have just been granting that feature, and it is that feature that leads me to the conclusion that the physical space is not a tensor product space.

    Have I gotten part of this wrong?

    On a side note: I find your discussion with Travis also very perplexing. You can't be assuming that there is a collapse of the universal wave function on measurement: then of course information is lost. But if there is no collapse, then your argument does not go through. The experimenter can measure whatever spin he likes, and he will thereby become entangled with the proton. The proton never gets into any z-spin eigenstate, and the experiment does not have a definite single outcome. That is, if you rule out collapses and if the wave function is complete then you are stuck with a Many Worlds picture. I thought we all appreciated this.

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  19. Black Hole Guy,

    I don't understand your reasoning for why spin up eigenstates are not in the Hilbert space by assumption. There are eigenstates in which the proton is in a spin up state, it's just that those eigenstates also involve the neutron being in a spin down state. What's the problem?

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  20. Travis,

    "There are eigenstates in which the proton is in a spin up state, it's just that those eigenstates also involve the neutron being in a spin down state. What's the problem?". The problem is that the state |neutron down>|proton up> is not in the Hilbert space since it doesn't obey the singlet constraint, so it can't be the outcome of any measurement. Again, if the Hilbert space is subject to the constraint C|\psi>=0 then admissible operators must commute with the constraint. (S_proton)_z does not commute with the constraint, hence is not admissible, hence it makes so sense to talk about measuring it. Adding in the wavefunction of an observer and all that jazz is not going to change that basic fact.

    Perhaps a vaguely related example will be helpful. Suppose I localize a single particle in space by placing it in the good ol' infinite square well, and then I ask you to measure the momentum of the particle to arbitrary accuracy, including its sign. Clearly this cannot be done (e.g. it would violate the uncertainty principle). The reason is obvious: the operator P does not exist in this theory.

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  21. Tim,

    I think you have summarized the situation accurately, but let me recap

    1) If you forget about the existence of AdS/CFT, then by taking seriously the usual Penrose diagram for black hole evaporation in AdS you are naturally led to the conclusion that the late time Hilbert space is a tensor product, and that the spectrum of the Hamiltonian is hugely degenerate.

    2) It is a basic fact that the spectrum of the Hamiltonian in standard CFTs is discrete and nondegenerate (up to symmetry)

    Therefore: if you are committed to trusting the standard Penrose diagram then you conclude that AdS/CFT duality is impossible. Conversely, if you are committed to AdS/CFT then you are led to believe that the Penrose diagram must be fundamentally incorrect.


    My point has always been that there is an incompatibility between AdS/CFT and the standard Penrose diagram. Now, a separate (and of course very important) question, is which option is "correct" (I use quotes here, because it's not clear that the question makes sense, since there could be a number of different consistent theories of quantum gravity in AdS). Someone unfamiliar with the successes of AdS/CFT would likely place greater trust in the Penrose diagram and say that AdS/CFT, if it exists, must somehow define a pathological theory in the bulk. That sounds reasonable, until you start examining what has actually been found over the years. There is now very strong evidence that the bulk physics in AdS/CFT outside of black hole horizons look very conventional and respects the principles of ordinary Einstein gravity. The breakdown of standard physics seems to occur inside would-be black hole horizons, but we of course have no experimental data on what goes inside on an event horizon, or whether such things even exist, so this can hardly be used as a disproof.


    Finally, regarding the proton neutron example, please see my response to Travis. Bringing in the experimenter doesn't help, since the experimenter cannot become entangled with the proton. This is because the singlet constraint, which again is a constraint on all states of the Hilbert space, implies that the spin states of the proton and neutron are maximally entangled, and the monogamy of entanglement theorem then says that the proton cannot become entangled with anything else, including the experimenter

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  22. Black Hole Guy,

    Ok yeah the overall wave function doesn't have that eigenstate, but it can evolve into one in which it has that eigenstate from the perspective of the observer. Here's the situation we have. There is no constraint on what the wave function can evolve into; the only constraint is that, at time t = 0, the neutron-proton system must be in a singlet state. So we can write the wave function of the whole system at time t = 0 as

    |observer_0>(|nup>|pdown>-|ndown>|pup>).

    The constraint is that at t = 0, the wave function must be able to be written like this. Then, we let time evolve forward, and the observer interacts with the proton. Then we end up with

    |observer_sees_down>|nup>|pdown> - |observer_sees_up>|ndown>|pup>.

    From the perspective of the observer, the proton is in a spin up eigenstate, and yet the overall wave function remains entangled so that we can evolve it back in time and get back the original neutron-proton state. Does this not sufficiently satisfy the constraints in the quantum gravity case?

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  23. Travis,

    What's missing in your discussion is the essential distinction between imposing an initial condition and imposing a constraint on the Hilbert space. You write

    |observer_sees_down>|nup>|pdown> - |observer_sees_up>|ndown>|pup>.

    The notation makes it look like this state is the sum of two states, one in which the observers sees up, and so then we can apply standard MWI logic to analyze the situation. But that's not right: the state above is not the sum of two states since the object |observer_sees_up>|ndown>|pup> is not in the Hilbert space. So the usual interpretation does not apply, and unless you invent some new rules there is no sense in which the measurement can be carried out.

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  24. Black Hole Guy,

    Reading your last paragraph of your most recent comment, you seem to be saying that whatever the constraint is which prevents the Hilbert space from being a product space at a certain time, that constraint must remain the same for all times (and thus prevent the observer from ever becoming entangled with the proton in the analogous case). But the constraint that the WDW patch be one which evolved from something that formed a black hole will apply a different constraint on the Hilbert space at different times, no?

    Or are you saying that, within a particular WDW patch, the same constraint must hold, so the observer evolving with alpha in that patch can't become entangled with the analogous proton? But it's still unclear what would cause such a constraint to look the same to an observer at all values of alpha.

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  25. BHG

    Your last post to me contains a a misunderstanding. Here is you're recap:

    "1) If you forget about the existence of AdS/CFT, then by taking seriously the usual Penrose diagram for black hole evaporation in AdS you are naturally led to the conclusion that the late time Hilbert space is a tensor product, and that the spectrum of the Hamiltonian is hugely degenerate.

    2) It is a basic fact that the spectrum of the Hamiltonian in standard CFTs is discrete and nondegenerate (up to symmetry)

    Therefore: if you are committed to trusting the standard Penrose diagram then you conclude that AdS/CFT duality is impossible. Conversely, if you are committed to AdS/CFT then you are led to believe that the Penrose diagram must be fundamentally incorrect."

    This is not correct in a few different ways. One is this: I never said the Penrose diagram of the late time in AdS implies that the physical space is a tensor produce space, or that "trusting" the Penrose diagram leads to some conclusion. All I said is that my off-hand opinion would be that it is a tensor product state. Equally, my off-hand opinion, knowing just that we have a theory of gravity, is that the spectrum would be continuous and massively degenerate. None of these claims "follows" in any logical or mathematical sense from taking the diagram to be perfectly accurate. I mean, how could it: all the diagram accurately depicts is the conformal structure, and the issue about the eigenvalues of the Hamiltonian is about more than the conformal structure. Similarly, I don't see any way to get logically or mathematically from the conformal structure to the structure of the physical space. Indeed, that is why I kept saying, when you asked we whether the physical space is a tensor product space: "I have no idea. I never thought about it." But you kept pressing me to reply with something definite, for reasons I still don't understand. The reason I did not have any idea is not because the Penrose diagram is incorrect, but because this is just not something that can be read off of the Penrose diagram at all. There is just not enough information about the theory encapsulated in the diagram.

    So I do not endorse your conclusion here. I see no reason to conclude that "the Penrose diagram must be fundamentally incorrect". As far as what it actually depicts, it may well be perfectly correct. It is my own off-hand way of finding it suggestive about things it does not depict that may be faulty.

    There is another issue. The Penrose diagram, as it contains only conformal information, may be a perfectly accurate picture of the conformal structure of any evaporating black hole. But seeing as this is a quantum theory, we can imagine cases where the initial condition is a superposition of a state that will lead to black hole formation and subsequent evaporation and a state that will not lead to the formation of a black hole. What is the Penrose diagram for that? Who knows? To really understand that situation we need a quantum theory of gravity, which is what we do not have.

    Finally, I fail to see what work the CFT in AdS/CFT is doing. You say that the CFT has (effectively) non-degenerate eigenvalues. You can then appeal to AdS/CFT to conclude that the AdS theory of gravity does as well. But you also gave me another argument, completely independent of CFT, that the spectrum is discrete. That argument used an analogy to a square well potential. And that argument never mentioned CFT at all. So the appeal to AdS/ CFT looks to be an unnecessary step in your argument. If there is a conflict here at all, it is not between the Penrose diagram and AdS/CFT, but between the Penrose diagram and any theory of gravity in AdS. And, to repeat again, I see no conflict even in this case. So I see no objection to analyzing the information loss paradox by taking the Penrose diagram to be perfectly accurate. If that was the criticism behind your argument.

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  26. Tim,

    To me, the interesting question is whether there exists any concrete scenario for black hole evaporation which respects the "ordinary" rules of quantum mechanics plus gravity, which exhibits information loss to the external region, yet is compatible with AdS/CFT. Hawking presented a concrete proposal for black hole evaporation. He argued (persuasively) that by following trusted approximations the final state is a pure but entangled state in a tensor product Hilbert space, and there is information loss because external observers only have access to one component of the Hilbert space. What I have shown is that this scenario is incompatible with AdS/CFT. Perhaps we can at least agree on that.

    On the other hand, you don't want to commit to the late time Hilbert space being a tensor product or not, and indeed apparently do not want to commit to there even being a physical Hilbert space at all (I infer this from your steadfast non-use of the word "Hilbert"). If there is no physical Hilbert space then QM has broken down, so you are not even committed to whether or not QM holds. That's how I see it anyway.

    So, I do claim that no one has ever presented a concrete scenario for info loss that is compatible with AdS/CFT, but of course I can't say anything about ill-defined scenarios. At this point I am left scratching my head as to what new light you claim to have shed on this old debate. We are somehow coming at this from a very different perspective.

    As to your final paragraph, that I can answer. If you quantize a collection of particles on a connected Cauchy surface in AdS then the spectrum will be discrete and non-degenerate. Since such a situation arises in AdS/CFT, this has to be the case for AdS/CFT to survive this most elementary of checks. But if you quantize on a disconnected Cauchy surface the spectrum will be degenerate, at least if you apply any standard method of quantization. So the AdS/CFT "prediction" is that such disconnected Cauchy surfaces never arise: i.e. a connected Cauchy surface will never dynamically evolve to a disconnected one. This does not contradict any firmly established fact about bulk physics, because such a topology change in the Cauchy surface necessarily involves passing through a singularity.

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  27. Travis,

    I am saying that after the lab or universe splits into two parts the expectation is that the Hilbert is a tensor product, with the state being entangled. I claim it would be bizarre if the Hilbert space were not such a tensor product. I provided one example of how imposing a constraint -- the np system being constrained to be in a singlet state -- leads to bizarre consequences. There are many other possible ways one could think of imposing constraints, and I think they will all lead to similar oddities. But if someone advocates a non-tensor product Hilbert space at late times, the burden is on them, not me, to explain precisely how this works.

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  28. Black Hole Guy,

    Ok it's possible I'm just being obtuse here. As I understand it, all the constraints are doing is picking out which WDW patches are valid and which aren't. So one ray in the Hilbert space represents an entire WDW patch, with effective dynamics occurring inside of it parameterized by alpha that correspond to observers measuring things. In the example with the lab and the proton, we have a ray in Hilbert space evolving in coordinate time, so it makes sense that if you place some constraints for all time t on which rays are allowed then some weird things are gonna happen because you can't have just any ray evolving to any other ray. But this doesn't seem to be the case with the WDW patches. It seems like the very fact that we have chosen some valid ray in the Hilbert space means that nothing weird should be seen by the observer, or else it wouldn't be a valid ray.

    Moreover, we are always entangled with remote regions of space, and yet what is happening in Andromeda on a Cauchy slice that passes through me seems to have as much or as little to do with what I observe right now as it would if Andromeda were in a disconnected region. So it seems like we have two options:

    1) We are always in a non-tensor product Hilbert space, which raises the question of why it's big deal in the case of the disconnected Cauchy slice that the late time Hilbert space is a non-tensor product, or

    2) We are always in a tensor product Hilbert space, in which case Ads/CFT is ruled out by everyday experience.

    What am I missing?

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  29. Suppose the blackhole exhibits "hair" characterized by a set of parameters {Q}.

    There are an enormous number of blackhole states characterized by a particular set of values of {Q}.

    In an eternal, non-evaporating blackhole scenario, I expect the Hilbert space to be
    H_interior{Q} ⊗ H_exterior{Q}
    with H_exterior being states with suitable boundary conditions on the event horizon.

    In the evaporating blackhole scenario, now the hair Q become Q(t) or Q(α) where α was discussed somewhere above.

    Now it is not clear to me if the space of states simply morphs to

    H_interior(Q(α)) ⊗ H_exterior(Q(α))

    --
    Coming to CFT, it is not clear to me whether the CFT in the case of an eternal blackhole is reflecting
    H_interior{Q} ⊗ H_exterior{Q} or just
    H_exterior{Q}

    --
    If the CFT cannot go from a non-tensor product bulk space to a tensor product bulk space, i.e., blackhole formation then the CFT can only reflect H_exterior{Q}, and I don't see it being able to tell us anything about unitary evolution in the bulk.

    If the CFT can go from a non-tensor product bulk space to a tensor product, then I don't see why it can't go the other way as well.

    Thanks in advance!

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  30. BHG

    I'm not sure why I need to repeat this, but yet again take the issue of whether the physical space is a Hibert space off the table. I have said already that I am happy to accept that it is. I raised that issue in the context of distinguishing the kinematical space from the solution space. Nothing more than that. But this same distinction rears its head in this sentence of yours:

    " He argued (persuasively) that by following trusted approximations the final state is a pure but entangled state in a tensor product Hilbert space, and there is information loss because external observers only have access to one component of the Hilbert space"

    On one reading of "in a tensor product Hilbert space" I agree: the kinematical space is a tensor product space, and the pure, entangled state at the end is (of course) in that space. Every physical state is in the kinematical space. But not conversely: not every kinematical state lies in the solution space. Now you say that Hawking *argued* that the solution space is a tensor product space. I am completely unaware of any such argument. Can you cite it?

    At this point I am in complete agreement with Hawking as I understand him, which does not require a tensor product Hilbert space of solutions. So I think my scenario is as concrete as Hawking's. That's because it is Hawking's solution! As I said, Hawking's problem is not with his solution, but in thinking that being to the future of the evaporation event changes the situation. Hawking thought that at that point, you needed a Cauchy-to-Cauchy evolution that is pure-to-mixed. But that's not right. The evolution even to the post-evaporation Cauchy surface is still pure-to-pure. And you get the state on the bit connected to the boundary by tracing off over the bit interior to the event horizon. So no fundamental principle of QM is ever violated.

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  31. Tim,

    ". Now you say that Hawking *argued* that the solution space is a tensor product space. I am completely unaware of any such argument. Can you cite it?"


    Sure, I already pointed you to, for example, a quote from his "breakdown of predictability.." paper. Namely:
    " Given only the initial state one cannot determine the final state but only the element of the tensor product H_2 x H_3." Here H_2 is the Hilbert space inside the horizon, and H_3 ("the final state") is the Hilbert space of radiation that escapes to infinity. You can find plenty of other similar statements if you poke around in the literature for a bit.

    I am still baffled as to why you bring up the kinematical space: by definition, the universe can never be in anything but a physical state, so kinematical but non-physical states are of zero interest. As I have said before, they are just scaffolding used as part of the construction of the theory, but have no physical significance (e.g. one can have two completely equivalent theories that have wildly different kinematical spaces; this is in fact quite common). Any argument that rests on nonuniversal properties of the kinematical space is doomed from the start.

    Anyway, I think I have made my point: Hawking's scenario is incompatible with AdS/CFT

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  32. BHG

    What I meant was an argument that the physical space is a tensor product space. The quote from Hawking makes no such claim. We are having a complete breakdown of communication, and we should be able to clear that up.

    You say that the kinematical space, which is a tensor product space, is just a scaffolding. Fine: that is exactly how Hawking is using it here! Every physical state is in the kinematical space, but not every kinematical state is physical. It is true therefore that every physical state in an element of a tensor product space—namely the kinematic space. Nothing Hawking says requires that the relevant tensor product space be the physical space.

    So is there a passage where Hawking argues that the physical space is a tensor product? The example you give is not one.

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  33. Tim,

    When a physics paper refers to "the Hilbert space" they mean the physical Hilbert space, not some other fictitious convention dependent structure. Hawking means the physical Hilbert space -- nowhere that I can see does he ever mention a kinematic Hilbert space. In fact, he makes this explicit when he writes of the "Hilbert space of possible data". Non-physical states in the kinematic space correspond by definition to impossible data.

    Again, I am honestly perplexed about why you want to bring this unphysical space into the discussion or what you hope to gain from it. I again emphasize that even the term "kinematic Hilbert space" is essentially vacuous, as two physicists studying the same theory could ascribe wildly different kinematic Hilbert spaces depending on what formalism they happen to feel like using that day. But they will always agree on the physical Hilbert space. If Hawking's statement was just about a kinematic Hilbert space the referee would have rightly rejected the paper by writing, "come back when you have something physical to say".

    Can you point me to any place in the literature that makes the distinction you are making? Can you find any claim that applying the standard rules physics leads to a situation where the physical Hilbert space on a disconnected spatial surface is not a tensor product? I doubt it, and in fact I would go as far as to say that if you have some argument that leads to this conclusion then you should immediately publish as this would overturn a lot of conventional wisdom.

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  34. Travis,

    Any theory of gravity or E&M on a connected spatial surface will be such that the Hilbert space is not a product of Hilbert spaces corresponding to disjoint spatial regions. The reason is just that the gravitational or E&M field lines necessarily escape from one region to another -- the constraints require this. On the other hand, for a disconnected space this does not happen, so we do have a tensor product. Now, in either case we still have the cluster decomposition principle stating that experiments carried out at widely separated locations have negligible effect on each other. So the fact that Andromeda and the Milky Way cannot be related to separate Hilbert space factors is not of practical relevance, but is important conceptually. My neutron-proton example was intended to magnify the effect to make the distinction clear.

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  35. BHG

    We are going around in circles here. Yes, I can point to such a place, and already have: Thiemann's lectures "Introduction to Modern Canonical Quantum Gravity", which I take to be quite standard. It's on the ArXiv. Look at p. 43, section III, "Kinematical Measures": he says

    "Now one has to equip C with the structure of a Hilbert space H. [N.B.: Thiemann uses the letter H for this kinematical space exactly because it is a Hilbert space! It is what he means by "the" Hilbert space, as H indicates.] This will naturally be L_2 space for a natural measure mu_0 on C. Certainly, H is not yet the physical Hilbert space, as one still has to impose the constraints, however one has to start from such a "kinematical" Hilbert space H in order to quantize the constraints (the name "kinematical" stems from the fact that it does not know about the constraints yet, which capture the dynamics of the system)."

    Having the kinematical Hilbert space in hand, we can then formulate the constraints as operators on that space. (Well obviously it has to go like this: if you only had the physical Hilbert space then your work is already done!)

    Now if we look at section V (p. 45) there is a complication. The constraint equation is just the equation stating that the constraint operator annihilates the physical states (i.e. the physical states are Eigenstates of the constraint operator with Eigenvalue zero). In some cases, this is just a subspace of the kinematical space. Note: it is perfectly possible that the kinematical space is a tensor product space and the subspace is not. It is also possible that the kinematical space is a tensor product space and the subspace is a tensor product space. I suppose that the kinematical space is always a tensor product space. As to whether the solution subspace for a theory is a tensor product or not, I take that to be a generally quite a difficult question to answer if the constraints are complicated. In fact, in this case not only is it complicated, in fact the kinematical space does not even contain states that solve the constraint equation. This is similar to what would happen in QM if the constraint equation were P|psi> = constant. Properly speaking, P has no eigenstates, so we have to expand the original Hilbert space to include non-square-integrable functions, or distributions, or whatever. And having done that, there may be some work to make the set of solutions into a Hilbert space, which would not happen for a subspace of the kinematic space. That's what happens here.

    At this point Thiemann introduces the notion of "generalized eigenstates", and goes through a rather complicated construction to get a set of generalized eigenstates for the constraint operator, which he calls D*_phys. And it is exactly here that he says: "Notice that D*_phys does not carry a natural Hilbert space structure", which is why he uses a D rather than an H! That is why I have been saying all along that I did not take for granted that the solution space—the set of states that solve the constraint equation—even forms a Hilbert space.

    So I hope we can put this whole thing to bed now. I am no expert, and maybe I am being misled by having read Thiemann (because he is very clear). but the idea that you start with a kinematical Hilbert space and impose constraints on it to get to the physical space is hardly novel to me, or unusual. Or if it is, take that up with Thiemann.

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  36. Con't

    So how does all this play out in the Hawking passage? Hawking takes H_1 to be the physical space of possible initial conditions: every element of H_1 is physically possible. I see no reason in the world to believe that H_2 X H_3 should be the space of physically possible final states. That would require, in the first place, that H_1 be isomorphic to H_2 X H_3, and therefore that H_1 itself be a tensor product space. But why think that? Already, it seems, Hawking's whole approach would not be applicable to AdS.

    So I take it that H_2 X H_3 is the kinematical space for the final physical states. In order to figure out the physical space of possible final states you need to have the dynamics that connects the initial states to the final states, and impose the constraints. So H_1 need not even be isomorphic to H_2 X H_3, while the unitarity of the evolution implies that the structure of the initial condition physical space most be isomorphic to the structure of the physical space of final states. H_1 is not isomorphic to H_2 X H_3, then some of the states in H_2 X H_3 simply cannot be generated by evolution from any physical initial state.

    In Hawking's formalism, this would show up as some of the S_ABC's being zero. Are you aware of any proof that this can't happen?

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  37. I might be wrong, but it seems to me that a lot of the confusion that's happening here (aside from the kinematical vs. physical issue, which is really interesting) is due to mixing up the Hilbert space on a Cauchy surface with the Hilbert space of a WDW patch. Here's my understanding based on the discussion so far: In AdS/CFT, the relevant Hilbert space is the Hilbert space on which WDW patches live. So Black Hole Guy's argument that the Hilbert space can't be a tensor product applies to this Hilbert space. However, when Tim talks about the Hilbert space of the initial and final states, he's referring to the Hilbert space on a Cauchy surface. Such a thing must be a Hilbert space defined on a fixed background. The WDW patch, being a state in a full quantum gravity theory, can contain a complicated mess of many approximately fixed backgrounds, so it seems like the fact that the Hilbert space on which the WDW patch lives can't be a tensor product doesn't tell us much about the nature of the Hilbert space on one Cauchy surface that we might pick out of the patch.

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  38. Tim,

    What I was actually asking for was a reference in which someone brings up the kinematical space (as opposed to the physical Hilbert space) as an important player in a black hole evaporation context.

    Your summary of Thiemann is accurate, but you are drawing the wrong conclusion from it. His goal is to construct a quantum theory of gravity, which *requires* defining a physical Hilbert space. He tries to proceed by starting with a kinematical Hilbert space and then imposing the constraints. This fails because the inner product on the kinematical space is singular when restricted to the physical space. You appear to be attaching some physical significance to this, as if this means that there is no physical Hilbert space in quantum gravity. That is wrong -- this is just a breakdown of his formalism. Instead, one has go and search for a new inner product on the physical space. Actually, this is a excellent illustration of why the kinematical space is of no interest: at the end of the day, you can't even think of the physical Hilbert space as being a subspace of the kinematical space because the inner products are different. Once you've defined the physical Hilbert space, you should forget about the original kinematical space, as it has no physical interpretation.

    Perhaps the following passage from a Smolin paper hep-th/0209079 will be helpful :


    "7.1 A brief review of quantization

    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


    C|Ψ >= 0 (59)

    where C 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 constraints 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."



    Back to Hawking. Physical states can only evolve to physical states, so if the original state in H_1 is physical (as he indeed states) then the state in H_2 x H_3 to which it evolves is also physical. You ask if there is some reason that H_1 is isomorphic to H_2 x H_3. Of course it is, and indeed the isomorphism is provided by S_ABC. S_ABC defines a map between physical states, since it manifestly vanishes for any non-physical final state.


    I am not sure how to state this any more clearly, but I can guarantee you that all states that Hawking talks about are physical. He is not talking about irrelevant unphysical states. We would really move the ball forward if we could just banish talk of the kinematical space. As explained above, the physical Hilbert space is not even a subspace of the kinematical space due to the different inner products, so trying to frame an argument that relies on the kinematical space is a really bad idea as it confuses formalism with physics

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  39. Travis,

    There is only one relevant Hilbert space: the Hilbert space of physical states. At each boundary time t, we have a state in this Hilbert space. Now, the notion of a Cauchy surface only makes sense in the semi-classical regime where metric fluctuations are small. I explained in an earlier posting how the idea of a state defined on each Cauchy surface emerges in the semiclassical limit. So the idea of a wavefunction at a fixed time describing a WDW patch is in harmony with the idea of states defined on Cauchy surfaces in the semiclassical limit. It all hangs together beautifully, as my earlier post was meant to explain.

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  40. BHG-

    We seem to be inching forward bit by bit. Let's see what we agree about.

    1) The kinematical Hilbert space is used—and used essentially—in actually specifying a theory of quantum gravity. Both Thiemann and Smolin verify that. You want to call the kinematical Hilbert space mere scaffolding, that is of no interest in the end. That depends on what one is interested in, of course! Whether it is important to talk about the kinematical Hilbert space as opposed to the physical space depends on the point being made. So I have no grounds, or appetite, to banish talk of it. That's just silly. Neither Thiemann nor Smolin banish talk of it, and neither can even explain how to arrive at a quantum theory of gravity without it. Your proposal to avoid mention of it would simply make my ability to make my points impossible. As long as we are clear whether a statement about the kinematic space or the physical space (solution space) we are certainly going to be fine. I have tried to be careful about that, and will continue to be.

    2) The difference between Smolin and Thiemann here is worthy of note. They agree to this point: you start with the kinematical space (with no presupposition that all the states are physical!) and impose constraints on it to get the physical space. The constraints are generally referred to as the "dynamics" of the theory. They are, indeed, the physical guts of the theory. They tell us what could actually physically occur.

    3) This is an important observation: both the kinematic space and the solution space, as I am characterizing them, have elements that are complete descriptions of the system through time. So, for example, the kinematic space for classical mechanics is not phase space, it is the set of all time-parameterized continuous curves through configuration space. In classical Newtonian mechanics, every element of phase space is regarded as a physically admissible initial state of the system, and the dynamics associates one element of the kinematic space with each element of the phase space. But this does not happen even in classical EM. If we are generous with the configuration space—treating the charged particle locations and the values of the electric field as independent variables—then some of the elements of the configuration space are not physically possible. In such a case we impose a constraint, e.g. Gauss's law, on the initial data. Because it is always possible to impose such constraints, there is no problem that can arise from making the kinematic space too big: it can always be cut down by the dynamics and the constraints so the physical space is fine. In particular, if some mathematical property (e.g. being a tensor product state) is convenient for solving problems, you can just build that into the kinematic space at will. The dynamics and constraints will compensate later by kicking the unphysical states out again. This is just what Hawking does, which you still do not understand.

    Con't

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  41. Tim,

    " The kinematical Hilbert space is used—and used essentially—in actually specifying a theory of quantum gravity"

    Let me explain why this statement is incorrect. Weinberg (and others) famously showed how to "specify a quantum theory of gravity" without ever bringing in a kinematical space or any other unphysical space. In this approach, applied to QED, Yang-Mills, or gravity, one starts with a physical Hilbert space of free particles, i.e. in gravity this would be graviton states with only the two physical helicities. Then one trys to add in interactions while demanding that the basic axioms of QM be satisfied. One is thereby led uniquely to QED, Yang-Mills, or quantum gravity. Things like the equivalence principle are thereby derived from first principles. Nowhere does any "kinematical Hilbert space" ever arise. You can look in Weinberg's QFT treatise for a very detailed exposition of this procedure for QED. Your statement is incorrect because it only refers to one specific choice of formalism for specifying a quantum theory of gravity -- it is a statement about that particular formalism, not about physical principles. I should also note that even if you do start from a kinematical space there are a vast number of inequivalent choices for it; the quantum gravity literature is filled with such examples.

    As I already said, another compelling reason why the thinking about the kinematic space is a red herring is that even in the approach you mention the physical Hilbert space is not a subspace of the kinematic space because the inner products don't agree. That by itself should be enough reason not to build an argument that involves the kinematic space.

    Hawking's papers never refer to a kinematic space, only the physical Hilbert space. We can discuss the derivation of Hawking radiation if you wish. One starts from a physical state in the far past, usually the vacuum state, and then evolves it forward in time according to the dynamical laws, and thereby arrives at a final state which is itself physical. No unphysical states ever appear.

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  42. BHG

    Your aversion to even mention the kinematic space is becoming quite puzzling. It may be that some approaches to quantum gravity don't employ it, but as we have just seen both Thiemann's and Smolin's do. And where it is mentioned, one needs to be sure to sharply distinguish it from the physical space.

    Hawking's original derivation of the radiation, of course, was doing QFT on a fixed curved space-time. You can't use that approach to doing quantum gravity.

    You mention Weinburg trying to use an axiomatic technique. But that will fall afoul of Haag's theorem.

    Do you dispute the claim that very little is known about the solutions to the Wheeler DeWitt equation? Certainly not enough to say anything definitive at all about the structure of the solution space.

    And again, in the passage from Hawking you cite he does not claim that the tensor product space is the space of all solutions. It is certainly treated as a superspace of that space, which is all you need. But it is consistent with what he writes that the initial space H_1 not be a tensor product, and that every state in it be physical, and the same hold for the set of solutions on the final, disconnected surface. Why not?

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  43. I don't have time to catch up with this thread, but I see Thiemann being discussed as an exemplar of how to quantize gravity. Over ten years ago, when there was some blog discussion of how loop quantum gravity works, it came out that their quantization methods are highly nonstandard, cannot detect anomalies, and destroy the affine connection of Riemannian geometry. "black hole guy" might want to check, in particular, how, when, and why Thiemann imposes constraints on the "kinematic Hilbert space".

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  44. Mitchell

    How and why? This is just trying to follow standard quantization procedure: take the Hamiltonian and constraints from the classical theory, put the hats on to turn them into operators, and then let the operators act on something! The "something" is the kinematic space. Since the division of the space-time by Cauchy slices is now a "gauge" variable, instead of yielding a differential generator of temporal evolution, all of the various Cauchy slices through the space-time are regarded as gauge equivalent. The dynamics becomes "pure gauge", and the physical states are the eigenstates of the constraint/Hamiltonian operator with eigenvalue zero: the constraints annihilate the physical states. This is just standard stuff: the Wheeler-DeWitt equation.

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  45. Mitchell,

    Yes, thanks, I am aware of that. Thiemann is being referenced because Tim brought it up. We're not dealing with any of these technical issues here, just with Dirac's general approach to quantizing consrained systems. This is of course heuristic, as the Wheeler-DeWitt equation is hopelessly ill-defined when it comes to trying to actually compute anything.

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  46. Tim,

    My aversion is to confusing physics with formalism, which is a common error. The whole point of this discussion is to compare physics in AdS versus that in CFT. The CFT does not contain the bulk kinematical space (nor should it), and so focusing discussion on the latter is almost certain to muddy the waters. Of course, I have no objection to kinematical Hilbert spaces per se, as long one keeps in mind that they are just a technical construct that appear in certain approaches to quantization, and do not have any theory independent meaning. Also, I am still not sure how you intend to overcome the fact that the physical Hilbert space is not a subspace of the kinematical space because of the mismatch of the inner product.

    About solutions of the Wheeler-DeWitt equation: of course the equation isn't even well defined, so one can't conceive of exact solutions. This was my reason for wanting to focus on the very early and late time regimes. Here we have a diffuse gas of quanta, and we certainly expect that these provide an accurate and controlled set of solutions to WDW. And the physical Hilbert space of such solutions is a tensor product if the spatial slices are disconnected, since the gravitational constraints are local.

    I just don't follow at all what you are claiming about Hawking's paper, or what your thought process is here. It seems obvious to me that he is talking about the space of quanta streaming out to infinity, combined with the space of excitations behind the horizon, and everything we know about physics would lead one to believe that such physical states live in a tensor product Hilbert space of physical states. If you can point to any statement to the contrary in the literature, please pass it on.

    At this point it seems unlikely that we will agree on anything. To the extent I understand what you are saying I don't see how it is differs from what Hawking originally said. I have presented the argument for why Hawking's scenario is incompatible with AdS/CFT, but you apparently think I am confused about many basic issues (one example is your bringing up Haag's theorem in connection to my reference to Weinberg's work; surely you must see that the issues regarding Haag's theorem have no direct relation to the question of whether kinematical spaces are necessary or not, which was the point being discussed).

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  47. BHG

    You are missing my point again. I do not want to draw any inference at all from the structure of the kinematic space to the physical space! That's just the point I am making. To know anything about the physical space you have to know what the full set of solutions is, and that is, in general, very, very difficult. And even more so in the case of quantum gravity. That is precisely why when you asked me whether the physical space was a product space or not I just told you the truth: "I don't know". And that if you have an argument that in AdS/CFT if cannot be a tensor product, that's fine by me...then it's not a tensor product.

    You just kept refusing to take my "I don't know" at face value. You kept pressing me to answer one way or the other. And now I see why: you wanted me to say it was a product space so you could raise an objection. Well, things haven't changed: I don't know anything about the structure of the physical space. Whatever you can prove about it, I am happy to accept.

    So let's talk about what Hawking is doing in his paper. There are several points. First, he is not proposing a theory. It's not as if he has any equation for quantum gravity that he is analyzing. Rather, he is make a suggestion about how any theory must be.

    Second, he is not attempting to give any theory about what happens when a black hole evaporates. He is proposing no fine detail here. What he is doing instead is nothing more than scattering theory. Instead of any detailed time evolution from the formation of the black hole to and through the evaporation event. all you have is a set of in states and a set of out states and a scattering matrix that connects them.

    Now in the usual case, the scattering matrix uses one and the same Hilbert space for input and output. The S matrix is unitary and invertable. But Hawking is considering a non-stadandard situation, with the incoming beam represented on H_1 and the outgoing beam on H_2 X H_3. Since we can only observe what is on H_3, we want the rule for evolution from H_1 to H_3 alone,

    So what does Hawking do? He gives a completely general abstract account. Let H_1 be the Hilbert space of appropriate incoming free states. In the distant future, there will be outgoing free states whose physical space is some subspace of H_2 X H_3. Presumably, the physical outgoing state space is not H_2 X H_3, because scattering is an interaction and interaction causes entanglement, so the product states in H_2 X H_3 will not be physical. Hawking has made the wise choice if not begging any questions or presupposing any answers. All he needs is that the outgoing states form a subspace of H_2 X H_3. If that is true, then he is golden. Why presume that that subspace is the whole of H_2 X H_3?

    And then, what you really want is something that gives the outstate on just H_3 given the in state on H_1. SO you trace over H_2 and get a superscattering matrix.

    What does Hawking say that suggests otherwise?

    One of the key sentences in your post is this: "And the physical Hilbert space of such solutions is a tensor product if the spatial slices are disconnected, since the gravitational constraints are local." This appears to be the very heart of the problem, and we have been over it before. What is the force of "since the gravitational constraints are local"? I can see why that would imply for two disconnected space-times that the physical space should be a tensor product space, but why should it also be the case when the space-time is connected? In order to be a solution, and hence a physical state, a state must solve the constraints *everywhere*. And what mathematical reason is there to believe that the constraints can be solved *everywhere* just because they can be solved in a single WDW patch?

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  48. Con't.

    Here is a very toy example. Suppose that the black hole forms, and there are two incoming particle that reach the neighborhood of the black hole after it has formed. And lets supposed that the initial Hilbert space is restricted to be a space with two such incoming particles.

    OK. then intuitively there should be solutions where both particles pass the event horizon and fall into the black hole And there should be solutions where both particles miss the evert horizon and never pass through. So there are solutions with two particles inside and none outside and solution with 2 particles outside and none inside. But there are no solutions with two outside and two inside. So the physical space, the solution space, is not a tensor product of the outside and inside states.

    In short, constraints on the earlier states imply constraints on the later states. Even if the constraints are local and even if the later states are on disconnected Cauchy surfaces. This is especially clear, as I have said, in any of the critical WDW patches where the evaporation actually occurs, because there are events that lie both on connected and disconnected Cauchy surfaces. But obviously they lead to the same global solution whichever Cauchy slice you take. Similarly, the late-time physics in AdS can be constrained by the fact that it has arisen from a single connected space-time, even at a point where the WDW patch is disconnected.

    As far as I can tell, your certainty that the late time physical space is a tensor product just rests on an intuition that has nothing to back it up.

    Can Hawking's formalism implement a theory in which all of the outgoing states as entangled? Sure. Simple. For every ingoing state there is some outgoing state. Each out going state can be characterized by a mixed state, So at the end of the day, that is all there is to it. Hawking's Not-@ matrix needs to represent that transition. That's all.

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  49. Tim,

    Unfortunately, I disagree with pretty much everything here.

    " Presumably, the physical outgoing state space is not H_2 X H_3, because scattering is an interaction and interaction causes entanglement, so the product states in H_2 X H_3 will not be physical. "


    You lost me. Let's consider scattering of two particles in non-relativistic QM, mediated by some short range repulsive interactions. Just as you say above, the interaction causes entanglement so if we shoot the two particles at each other the final state will be entangled. But of course that in no way changes the fact that the Hilbert space is at all times a tensor product of the Hilbert spaces of the individual particles. You seem to be conflating a property of the Hilbert space with the properties of states that arise from scattering. It's the same story in Hawking's case.


    " but why should it also be the case when the space-time is connected? In order to be a solution, and hence a physical state, a state must solve the constraints *everywhere* "


    Here you seem to arguing that to determine the structure of the Hilbert space *now* I need to know about the past history of the system --- but that's not right. The time evolution to the past has to do with the Hamiltonian, but the structure of the Hilbert space is logically distinct from that of the Hamiltonian. One can determine the Hilbert space before specifying the Hamiltonian. Let me pose to you the following question which might help me to understand your thinking. Suppose at some time t I have a disconnected spatial surface on which live some charged particles coupled to E&M (no dynamical gravity). To determine the Hilbert space at time t, are you claiming that I need to know something about the history of the system, including whether or not these two disconnected components of the surface were connected at some point in the past? That is, just from the information given, do you say that the Hilbert is a tensor product, is not a tensor product, or that you need more information to answer?

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  50. I have a question for BHG related to Maudlin's comment about dimensional mismatch in AdS/CFT, if you don't mind the brief interjection. Is it at all appropriate to think of the scale factor in the CFT as corresponding to the "extra" spatial dimension in the dual theory?

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  51. BHG

    We are definitely crossing wires somewhere. I can't understand you at all. Let me articulate why.

    I am taking the terms "physical space" and "solution space" to be synonymous: they mean the space of all global solutions to the dynamical equations of the theory, or possibly all the global solutions that satisfy some stated property (e.g. with a given total mass, or charge, or containing an evaporating black hole, etc.) On the picture that I have been advocating (the same one as Thiemann and Smolin use) one arrives at this physical space by starting with a larger space—the kinematic space—and imposing some dynamical constraint on it. So there are states in the kinematic space that don't make it into the physical space. In classical theory the dynamics is encoded in the Hamiltonian function, and the physical solutions (which are time-parameterized curves through phase space) are solutions to the Hamiltonian. In quantum theory, one "puts the hats on" the Hamiltonian so it becomes an operator, and the solution space is the space whose basis is some eigenstates of the Hamiltonian operator. There is a peculiarity in quantum gravity, where the solution space is the space spanned by the eigenstates of eigenvalue zero, i.e. the states annihilated by the Hamiltonian operator.

    In addition to the Hamiltonian, there may be some non-dynamical constraints. These are also expressed as operators, and the solutions have to satisfy the constraints, i.e. be annihilated by the constraints.

    On this picture, the whole *point* of the Hamiltonian is to cut down some larger space (the kinematical space) to the solution space. But you write this:

    "The time evolution to the past has to do with the Hamiltonian, but the structure of the Hilbert space is logically distinct from that of the Hamiltonian. One can determine the Hilbert space before specifying the Hamiltonian".

    I can't make any sense of this remark. Since the physical space, A.K.A the solution space, just is the space of solutions to the Hamiltonian (and other constraints), your comment reads as self-contradictory to me. If what you are calling "the Hilbert space" in this sentence is not the solution space as I have defined it, then what exactly is it? You have spent a tremendous amount of time objecting to any talk of the kinematical space, so this Hilbert space isn't that. And nor is it the solution space. So what exactly is it? Until I understand that, I won't be able to make head or tails of your argument.

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  52. Tim,

    You write

    "There is a peculiarity in quantum gravity, where the solution space is the space spanned by the eigenstates of eigenvalue zero, i.e. the states annihilated by the Hamiltonian operator."

    Let me try to save us both some time here. In a space with an asymptotic region (like AdS or flat space, which what we are discussing) the above statement is just flat out false: as i have explained repeatedly, the Hamiltonian does not annihilate states, only its volume part does. The Hamiltonian is nonvanishing on the physical Hilbert space, and indeed its role is to generate time evolution with respect to asymptotic time according to H\psi = i d\psi/dt. Now, either your statement above was a typo and you meant to write "annihilated by the volume part of the Hamiltonian" or you disagree with what I am saying here. If the latter, please say so, and then I will respond with "thanks for the nice discussion, but I don't see any point in continuing".

    As for the rest of your message, what Thiemann and Smolin are doing is standard Dirac quantization, a subject I am very familiar with. What you describe in your message is something else, and I don't know where you are getting this from. In Dirac quantization one never appeals to the concept of "global solutions to the dynamical equations of the theory" when constructing the Hilbert space. In the absence of gravity, one quantizes on a specific Cauchy surface and never needs to make reference to "global solutions" away from this Cauchy surface, and in gravity one quantizes at fixed asymptotic time. At the risk of sounding presumptuous, let me suggest that you have never worked through examples of Dirac quantization at the level of equations (as diagnostic questions, are you familiar with such basic notions as first class vs second class constraints, and Dirac brackets?). If you like, we can go line by line through a standard example like Dirac quantization of E&M coupled to charged particles. This would reveal the following. Suppose we were to allow the particles to interact via some potential V(x_i) in addition to E&M interactions. You will see that to determine the Hilbert space we never need to make any reference to this potential, nor to the "global space of solutions" that would result from including it in the equations of motion. I know this because I have worked out such examples many times in complete detail from start to finish.


    As above, if your response is that I am just confused about Dirac quantization then let us just amicably end this discussion to save us both some grief!

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  53. Mark Gomer,

    Yes, that is part of story. Try googling "scale-radius duality" or "UV/IR in AdS/CFT" for more about this.

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  54. BHG

    Yes, I was just describing the standard methods of Dirac quantization, with which I am familiar, as applied to GR. I was leaving aside the extra complication of the t (I am reluctant to call it "time") dependence that appears in AdS because it seemed inessential to the question I was asking. I do wish you had made some effort to answer that question. Since you did not, let me ask again, leaving out the details that strike me as unimportant.

    I remind you of this sentence of yours:

    "The time evolution to the past has to do with the Hamiltonian, but the structure of the Hilbert space is logically distinct from that of the Hamiltonian."

    My question, to repeat again, is this. You state directly that there is some Hilbert space, which you here call just "the Hilbert space", whose structure is "logically distinct from that of the Hamiltonian". I understand that last comment to mean that the structure of this Hilbert space can be specified independently of the Hamiltonian, and is the same no matter what the Hamiltonian is. On the other hand, I take it that the solution space—the set of solutions to the fundamental equations—are not at all "independent of the Hamiltonian". I take that the structure of the solution space depends critically and essentially on what the Hamiltonian is. From these two propositions, I infer that whatever "the Hilbert space" is, it is not the solution space, which is what I have been understanding to be "the physical space".

    We have been round and round this merry-go-round, with me saying that although the kinematic space may be a tensor product space, I have no idea what the structure of the solution space—the physical space—is. And your response, over and over, has been that the kinematic space is unimportant and inessential and nobody cares about its structure. In fact, you were skeptical that I could even cite a work on quantum gravity that mentioned a kinematic space. To which I replied by directly quoting Thiemann, who speaks directly and extensively about the kinematic space. And then you yourself cited Smolin who also directly mentions the kinematic space. And, in case third time's the charm, the Wikipedia article on canonical quantum gravity also makes extensive mention of the kinematical space. So your insistence on the unimportance of the kinematical space, and your insistence that the physical space—by which I mean the solution space—is the only thing that anyone would ever refer to as "the Hilbert space", just seems odder and odder to me. And now we have this new shock: there you are, talking about something you call "the Hilbert space" which, for the reason I just gave, cannot possibly be the solution space.

    So can you please actually address the question I am asking, and not deflect to other issues. What is the "Hilbert space" of which you speak? Is it or is it not exactly the kinematical space that I and Thiemann and Smolin and Wikipedia refer to? Is it or is it not the solution space, i.e. the set of solutions to the fundamental equations? Is it or is it not the thing you refer to as the "physical Hilbert space"?

    Just answer this question as clearly and straightforwardly as you can. If you reply with more distractions then I will take a page out of your own book and say "let us just amicably end this discussion to save us both some grief!"

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  55. Tim,

    In an attempt to be constructive let me suggest the following. How about you outline how one carries out Dirac quantization for charged particles coupled to E&M, but with no gravity. In particular, please define the physical Hilbert space of this theory. What I hope this will reveal is that in defining the physical Hilbert space one never makes reference to any "global space of solutions of the dynamical equations".

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  56. BHG

    How about you stop deflecting from my very precise question and answer it?

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  57. Black Hole Guy,

    Is not gravity different from EM precisely in the sense that, disregarding the boundary time t which only shows up in AdS, the space of solutions are solutions which involve entire 4D spacetime dynamics? Hence "global space of solutions to dynamical equations"? It's possible that these are not quite the right words to use, but it seems nitpicky to focus too much on the exact words and not the content of what Tim is saying. I also am interested in seeing this conversation move forward. You seem to be equivoquating between the Hamiltonian which operates in the bulk, and hence annihilates solutions, and the Hamiltonian which operates on the boundary. Sure, there is a well-defined sense in which the Hamiltonian which operates on the boundary only operates on the physical Hilbert space, since its role is to evolve the state forward in boundary time. But it makes no sense to say that the Hamiltonian which operates in the bulk operates on the physical Hilbert space because as Tim has pointed out, this would just be equivalent to saying that the Hamiltonian is the zero operator, and thus contains no physics. It is the bulk Hamiltonian that should be focused on, since if the argument against Tim's solution to the information paradox depends crucially on the existence of a boundary time, then that is just an argument against AdS/CFT being applicable to our universe which, as far as we know, does not have a boundary time. There must be some kinematical Hilbert space on which the bulk Hamiltonian operates in order to produce solutions which are the zero eigenstates of the Hamiltonian, else the statement that the solutions are the zero eigenstates is meaningless. I agree it would be helpful if you clarify whether it is this space of solutions that you are saying is a tensor product Hilbert space, or the kinematical space on which the bulk Hamiltonian operates.

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  58. Tim,

    Why the hostile response? I gave you a completely precise definition of the physical Hilbert space, which is the standard definition in the textbooks. Isn't that what you asked? And it's different than the kinematical space, clearly. One could say that it's equivalent to the "space of solutions of all dynamical equations", in the sense that the only equation left to solve is H\psi = id\psi/dt, and since this is a first order equation every solution is specified by a wavefunction at some fixed time, and the space of the latter is the physical Hilbert space. But the key point is that the physical Hilbert space can be obtained without knowing all details of the Hamiltonian.

    That's as straightforward an answer as I can give. Now how about answering my questions, which you seem to be studiously avoiding? How about you tell me about Dirac quantization for E&M. And about the case I brought up where the Cauchy surface at some time is disconnected and you don't know whether in the past it was connected or not. It would be very helpful if we could reach a clear (dis)agreement about the nongravitational case, since this is much simpler.

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  59. Travis,

    1) The existence of a physical time t in terms of which physical states evolve is not specific to AdS. It occurs in any case in which the spacetime has a fixed asymptotic structure, such as asymptotically flat space. In such cases, a physical state wavefunction depends on the asymptotic time and evolves according to H\psi = id\psi/dt. There are important differences between AdS and flat space, but this is not one of them.

    2) The problem with your discussion is that you are completely obscuring the distinction between the constraints and the Hamiltonian. It is not the role of the Hamiltonian to reduce the kinematical space to the physical space; rather it is the role of the constraints! The theory has a bunch of constraints, C_i(x), and physical states obey (by definition) C_i(x)\psi_\phys = 0. Now, we also have an operator H in this theory which happens to have the structure of a bulk piece which is built out of the C_i(x) and a boundary piece. Prior to discussion of the Hamiltonian we can determine the physical Hilbert space as above. Then we can ask how the Hamiltonian acts on this physical Hilbert space, and the answer is that it acts as a boundary operator. Again, I stress that the constraint operators and the Hamiltonian are separate entities, and should not be conflated as you are doing.

    3) I thought I had made this clear, but my question (not claim) is whether Tim believes that the physical Hilbert space at late times is a tensor product. To move forward on this point, I am suggesting that we first examine the non-gravitational case.

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  60. BHG-

    No, I asked a series of precise questions, which you have still failed to answer. And going through the Dirac quantization of EM is a waste of time exactly because the time-dependence issue in quantum gravitation is quite different from the issue in other quantum theories: that is exactly why quantum gravity, but not quantum EM is said to have a "problem of time" and why in quantum gravity, and not in quantum EM, it is said that the dynamics becomes "pure gauge" and the Hamiltonian becomes a "constraint". These are issues that are not illuminated by EM; they are obscured by EM.

    I have been very, very clear: in cases where time is treated as a parameter (including as a parameterization of some specified foliation into Cauchy surfaces) a "solution" is a time-indexed sequence of states. There, the notion of the physical space of solutions and the *instantaneous* (relative to t) space of physically possible states gives rise to isomorphic structures: every "instantaneous" kinematical state corresponds to a unique solution (in my sense) and vice versa. But in these cases, the Hamiltonian *does not annihilate the state*, it rather is the infinitesimal generator of the time evolution, and so had better not annihilate the state: that would lead to no change of state at all. Thing just do not and cannot work like this in quantum gravity where you are trying to implement diffsomorphism invariance as a symmetry. So in the usual cases, "physical space" is ambiguous between "solution space" and "instantaneous kinematic space". You don't need the Hamiltonian to specify the latter, but you do need it to specify the former. The case of quantum gravity is different because the dynamics and the Hamiltonian are expressed as constraints.

    So I have every reason to resist your suggestion to send us off in an unhelpful direction. I do not understand your reason to continue to refuse to answer my questions. Here they are again:

    What is the "Hilbert space" of which you speak? Is it or is it not exactly the kinematical space that I and Thiemann and Smolin and Wikipedia refer to? Is it or is it not the solution space, i.e. the set of solutions to the fundamental equations? Is it or is it not the thing you refer to as the "physical Hilbert space"?


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  61. Just in case this thread one day ends without everyone having had all their questions answered, I have created a blog, Quantizing Gravity, as a kind of eternally open adjunct to the discussion.

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  62. Tim,

    I am perplexed by your message, as I did answer your questions in a clear and precise fashion. But I will do so again if that will move things forward

    1) "What is the "Hilbert space" of which you speak?"


    I use "the Hilbert space" synonymously with "the physical Hilbert space", as it is the only object that all of the many possible formulations will agree on. In the case of gravity coupled to a scalar field, it is the space of normalizable wavefunctionals \psi[g_{ij},\phi] that are annihilated by all the first class constraints of the theory, C_i(x)\psi=0. Here g_{ij] is a spatial metric and \phi is a spatial field configuration. Also, there must be an inner product defined on this space so that it is a Hilbert space. This is the standard definition of "physical Hilbert space".


    2) Is it or is it not exactly the kinematical space that I and Thiemann and Smolin and Wikipedia refer to?"


    Clearly not since the kinematical space contains unphysical states (those not annihilated by the constraints)


    3) " Is it or is it not the solution space, i.e. the set of solutions to the fundamental equations?"


    Since I gave you the precise definition of the physical Hilbert space, it is perhaps best if you decide for yourself whether this agrees with your definition of "solution" space. I will say this. In terms of your "fundamental equations", the only equation that has not been invoked in my definition of the physical Hilbert space is the Schrodinger equation H\psi = id\psi/dt. The definition of the physical Hilbert space manifestly does not require saying anything about the solutions of this equation. However, since it is a first order (in t) equation, its solutions are in one-to-one correspondence with the space of wavefunctions at t=0 (say), so in this sense your "solution space" is isomorphic to the physical Hilbert space.


    4) "Is it or is it not the thing you refer to as the "physical Hilbert space"?"


    Yes, see above. If I mean something other than the physical Hilbert space when I refer to a Hilbert space, I will add an explicit modifier.


    I really do hope you will accept these as answers. Now, I really do think it would be helpful if you reciprocate by answering my questions


    1) In E&M coupled to charged particles, if at time t the spatial surface is disconnected is the physical Hilbert space a tensor product or not, or do you need more information about the past history including whether the surface was connected in the past?


    2) In gravity coupled to matter, if at boundary time t all spatial surfaces are disconnected, is the physical Hilbert space a tensor product or not, or do you need more information about the past history including whether the surfaces were connected in the past?


    You say that the E&M case is a waste of time, but I am actually quite confident that there are some highly relevant elements of Dirac quantization that you do not understand properly, and this would bring them to light. I say this mostly based on two past episodes: 1) your expressed confusion about my statement that the Gauss law constraint generates gauge transformations 2) your expressed disagreement with my claim that in the canonical formalism for a particle in non-relativistic QM you treat the position and momentum as completely independent quantities

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  63. BHG-

    I am still not clear what is going on in your responses. Let's at least make this agreement: if you make a claim that is true only for AdS and not for, say, and asymptotically flat space-time then indicate that. Even more precisely, if you make a claim or are describing a technique that requires that the space-time have a timelike boundary, please mark that.

    For example, you mention the Schrödinger equation above. And that equation has a t variable. Now if I have been following this, that t variable is exactly a variable defined on the timelike boundary of AdS. As such, it has no analog at all in an asymptotically flat spacetime. The situation, as I understand it is this: for a fixed value of t, there is a solution of the constraint equations which gives a solution on the WDW patch associated with that boundary time. In that patch, of course, there is no canonical time at all, and no canonical time evolution. That is the problem of time that appears in canonical quantum gravity. So all of the complications that arise from the very existence of this Schrödinger equation have no analog in the asymptotically flat case. And the idea of using an S-matrix for AdS where the incoming states are defined on a WDW patch that lies to the past of the formation of the black hole and the out states lie on a WDW patch that lies to the future of the evaporation of the black hole, has no analog in the asymptotically flat case. In the asymptotically flat case, all of spacetime lies in one WDW patch.

    Why is this important, assuming it is correct? Because when Hawking first introduces the idea of a super scattering matrix, he does so in the context of the asymptotically flat case, not the AdS case. And hence the AdS problem of have a scattering matrix or super scattering matrix for the transformation from early AdS time to late AdS time is simply not the problem that Hawking was trying to solve. Nor would he have had any Schrödinger equation in mind at all, The "dynamics" of the asymptotically free case is all in the Hamiltonian constraint.

    Let me stop here and you can either agree with or dispute what I just wrote. Because it makes a lot of difference.

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  64. Tim,

    Thanks for raising this, since I was beginning to suspect that this was a source of confusion. I will be happy to indicate when something is special to AdS, but in this case it happens that the equation H\psi = id\psi/dt holds equally well in either AdS or flat spacetime. What is common to both cases is that there is an asymptotic structure. I can go through the logic at a level ranging from the very technical to the physically intuitive, and I'm not sure what's most appropriate. The canonical formulation of gravity in asymptotically AdS and asymptotically flat spacetimes have been extensively studied over many decades and are very well understood, so any details that I leave out here I will be happy to fill in upon request.

    The shortest answer I can give is to once again point you to the classic paper of Regge and Teitelboim, since the whole point of their paper was to work out the classical version of this statement about time evolution in asymptotically flat space. That is, they showed that the Hamiltonian, crucially including the boundary term, generates translations of asymptotic time. In a bit more detail: to say that a space is asymptotically flat is to say that as you go out to large spatial distance the metric goes to the Minkowski metric plus corrections that fall off at a prescribed rate. Because the asymptotic metric is fixed in this way, this imparts a physical and unambiguous notion of time out at infinity. Intuitively, this is the time that a standard clock at infinity would read, and its clear that this makes physical sense. What R&T showed is that given any function on phase space, its derivative with respect to this asymptotic time is given by its Poisson bracket with the Hamiltonian. The quantum version is that H becomes the operator that generates asymptotic time translation, meaning that wavefunctions obey the Schrodinger equation as above. One way of saying this is that there is no "problem of time" when it comes to asymptotic time, precisely because one is fixing the asymptotic metric to have a prescribed form.

    I think what is confusing you is the Penrose diagram, which shows spatial infinity as a single point. This is one case where the Penrose diagram is not helpful in terms of understanding the canonical structure, as the Penrose diagram is designed to encode the causal structure, which is different. Intuitively, it's clear that time passes in a well defined and unambiguous sense at spatial infinity, and it is the picture of spatial infinity as a single point that is misleading in this regard.

    Yet one more way of stating things is that anytime there is a well defined notion of the total energy of the system then there is a well defined notion of physical evolution, because the energy operator's role in life is to generate such evolution.

    Finally, I would find it really helpful if you would try to answer the questions I have posed. They are not trick questions, I assure you.

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  65. BHG

    I appreciate the spirit of this reply. I am, however, concerned about the content. I have been trying, in my own way, to come to terms with some of the technical apparatus used in these discussions, and I think I know enough to know that the characterizations that you are offering are not technically correct, and this in ways I find worrisome.

    For example, you write:
    "to say that a space is asymptotically flat is to say that as you go out to large spatial distance the metric goes to the Minkowski metric plus corrections that fall off at a prescribed rate. Because the asymptotic metric is fixed in this way, this imparts a physical and unambiguous notion of time out at infinity. Intuitively, this is the time that a standard clock at infinity would read, and its clear that this makes physical sense."

    If only it were that simple.

    Wald lists 5 conditions for a spacetime to be asymptotically flat, and they are much more restrictive—as far as I can tell—than what you have written here. Furthermore, your conclusion has a puzzling ring to it. Since there is no "physical and unambiguous notion of time" in Minkowski space-time (at least in an obvious sense: there are an infinity of global Killing fields), why should it even follow that if the metric approaches the Minkowski metric that there is any such unambiguous notion of time?

    Furthermore, even if there such a notion of time at infinity, that does not in any obvious way determine a global time function. Since it is critical for our questions to understand what the situation is in the bulk—not just at infinity—we need to discuss how the bulk description is generated.

    Even if there were some way to pick out a global time function "at infinity" (which I would need to understand better) is there some assumption that there is a unique way to complete that time function to cover the bulk, i.e. to foliate the space-time into spacelike hypersurfaces? If the t function at infinity supposed to be used to index the leaves of this foliation?

    In our earlier discussion there was a distinction made between alpha—a parameterization of the leaves in a foliation of a WDW patch—and t which was a parameterization of the CFT and hence the boundary part of the bulk states. This, of course, left us with a timelike boundary.So is it your intention that this t variable will parameterize some unique global foliation in the end? If not, how is one to deal with the bulk?

    I have some more questions, but maybe the answers to these are needed to know how to go. I don't want to start in on your questions until this part is put into better shape. For my understanding of what is going on depends on the claim—which I have read many times—that in GR the temporal evolution becomes "pure gauge" in a way unlike other theories, including EM. Since my concerns about GR arise from this difference, I would prefer to stay focussed on GR for the moment.

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  66. Tim,

    "If only it were that simple. Wald lists 5 conditions for a spacetime to be asymptotically flat, and they are much more restrictive—as far as I can tell—than what you have written here"


    As far as I know, this "simple" version is sufficient for all of the issues under discussion here. In particular, since you bring up Wald, note how he goes about constructing the canonical Hamiltonian for asymptotically flat spacetime (see latter part of appendix E). He does exactly what I am saying, namely taking asymptotic flatness to mean that the metric goes to the Minkowski metric with a prescribed falloff. This is no coincidence, as his discussion is based (as he states) on Regge-Teitelboim.


    "Since there is no "physical and unambiguous notion of time" in Minkowski space-time... "


    There is a simple answer to this. Given an asymptotically flat spacetime of nonzero energy, there is a preferred choice of coordinates at infinity corresponding to the rest frame; ie. such that the total spatial momentum of the system vanishes. This is implemented by requiring that the components g_{t x_i} of the metric fall off sufficiently rapidly. So by "time translation" we mean a translation of time as measured by a clock that is at rest in this preferred coordinate system. This falloff condition is used very explicitly in the Regge-Teitelboim construction.


    "Furthermore, even if there such a notion of time at infinity, that does not in any obvious way determine a global time function"


    Agreed. An asymptotic structure just buys you a well defined notion of time at infinity, and doesn't provide a preferred time in the interior. The point I have been trying to get across is that there is *a* well defined notion of the far past and the far future, as measure by this asymptotic time. The reason this is important is that we are discussing what Hawking's original claim is. He phrases things in S-matrix language. In the standard S-matrix context in particle physics one has a quantum theory with a physical Hilbert space. There are two natural sets of basis states: one set corresponds to well separated particles with well localized momentum in the far past; and the other set trades far past for far future. The S-matrix is the operator that relates the two sets of basis states to each other. In Hawking's case, the analogous story for the far past is clear. For the basis states in the far future he takes outoing particles escaping to infinity tensored with states behind the horizon. Again, there is an S-matrix, but it is only unitary if the behind the horizon states are included. That's his argument as I have always understood it.


    cont

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  67. "So is it your intention that this t variable will parameterize some unique global foliation in the end? If not, how is one to deal with the bulk?"


    No, I don't believe there will be any unique global foliation. Instead I believe the correct way to make sense of the bulk in general is to ask "relational" questions that are independent of choices of foliations and coordinates. This is usually stated somewhat metaphorically as saying that one needs to identify a physical clock among the degrees of freedom in the system, and then ask questions relative to a given clock reading. In general this is very difficult, and this is related to why we don't in general know how to recover approximately local bulk physics in AdS from the CFT. This is why I have tried to make my AdS/CFT arguments based on general principles, like the structure of the Hilbert space, the existence of a Hamiltonian and so on, and this is why I would claim that in AdS/CFT we "know" that the information comes out in the radiation, even if we have almost no idea how this comes about when phrased in bulk language -- that is, we don't know where Hawking's argument fails.


    I suppose the point we still don't agree on is my contention that in Hawking's paper he assumes a late time physical Hilbert space that is a tensor product of particles escaping to infinity and particles behind the horizon.

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  68. BHG-

    Yes, we certainly do not agree that anything in Hawking's general approach requires the late physical space—by which I mean the space of all solutions of *all* the dynamical equations (both the so-called admissibility and evolutionary constraints, and your extra Schrödinger equation) to be a tensor product space. But let me ask you this, because it puts everything in a slightly different light. We have been talking about AdS/CFT, but the issue is the transition from the pre-evaporation situation to the post-evaporation situation. As I have pointed out before, that transition takes place within single WDW patches for many different boundary times. Each of those individual patches for a fixed boundary time has a Penrose diagram that is effectively similar to the standard Penrose diagram that I discuss, because it has no timelike boundary. And since that is associated with a single moment in boundary time, we should be able to discuss that case and leave the rest of AdS out of it.

    Now: do you agree with this proposition: If it can be shown that the physical space for that single WDW patch—which has a connected space-like boundary at the bottom and a disconnected space-like boundary at the top—is not a tensor product space then it follows that the physical space for the whole AdS is not a tensor produce space? That is, this case has to be the same as the global case? If so, then we can just forget about the whole AdS space and go back to Hawking's own diagram. If not, please explain why not.

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  69. Tim,

    I am afraid I don't follow your train of thought, and what seems to me to be a strange focus on the space of solutions to all equations, including H\psi = id\psi/dt. You also appear to resist using the word "Hilbert", and I don't know why. Why don't we just focus on the physical Hilbert space, which every QM theory possesses, by definition. In every QM theory there is, when we construct the Hilbert space we don't ever need to discuss properties of solutions to H\psi = id\psi/dt, and thank god for that, since we likely have little chance of finding exact solutions to this equation. Dirac quantization proceeds in a uniform fashion for all known QM theories: We start from states defined at a given time, as the space of wavefunctions of the configuration space variables; in a nongravitational theory this would mean working on a fixed Cauchy surface, while in gravity we fix the boundary time. Starting from the action, we identify a set of first and second class constraints, and a Hamiltonian. The physical Hilbert space is the kernel of the first class constraints. *After* the physical Hilbert space has been thus identified, we can discuss time evolution of states according to H\psi = id\psi/dt. This is all straight out Dirac, and what pretty much everyone follows, including Thiemann, Smolin, etc.

    I am going to ask this question again, now in the simplest version. In a nongravitational theory, if the Cauchy surface at some time t is disconnected, is the physical Hilbert space of this theory at time t a tensor product? Yes, no, or do you say we need more information about whether Cauchy surfaces in the past or future are disconnected? Please use the word "Hilbert" in your response.

    Finally, I already explained why in the AdS black hole evaporation case working at an intermediate time is the most difficult case. It's much better to first focus on what happens at very early and very late times, as in the S-matrix approach. Once we agree on that we can circle back to what you ask.

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  70. BHG-

    If we are ever going to get through this, you have to just drop certain things. I have never resisted using the term "Hilbert space" for technical reasons: I grant that every important space we discuss—the kinematical space, the physical space and the solution space if we are going to have all three as different—is a Hilbert space. OK? The problem was, as you recall, you at some point wanted to deny both the significance and even the mention of the kinematical space when doing quantum gravity. I take it we have settled that: the kinematical space is both mentioned and used by both Thiemann and Smolin. The issue was that you kept asking whether "the Hilbert space" in AdS/CFT is a tensor product space, and I kept answering that it depends on *which* Hilbert space one was asking about! So yes: let's drop the term "Hilbert" from here on. Every space we mention will be granted to be a Hilbert space. From now on, identify the relevant space *by name* so we don't get confused about what is at issue.

    Let's enumerate. There is a space—which I have consistently called the kinematical space—which is the first one constructed when building the theory. The kinematical space is unrestricted by the dynamics, which is why it is called "kinematical". Then we identify some constraints. It is here that you discussion above gets sloppy, and sloppy in an absolutely critical way. But note: as soon as one is talking about "constraints", there is some space being constrained! I call that space the kinematical space.

    OK, now we have an absolutely critical bifurcation. The procedure you describe—which would be the proper procedure for standard QFT— has the two steps you mention, each of which produces a new object. Step one is to impose the first-class constraints on the kinematical space. Those constraints are expressed in this setting as operators, and satisfying the constraint means being a zero eigenstate of the constraint operator. The set of zero eigenstates itself forms a Hilbert space which we can call the constrained kinematical space or, if you like, the physical space (this is just a matter of clearly settling terminology: I don't really care what it is.) As you yourself say, in a non-gravitational theory, this physical space is defined on—and confined to—some given Cauchy surface. Furthermore, in a non-gravitational theory you start with a background space-time that has been already foliated by Cauchy surfaces, and that foliation has been parameterized by the variable t. A full solution—the thing we want—specifies the wavefunction on every leaf of that foliation. We arrive at a full solution by solving the Schrödinger equation you mention, where the Hamiltonian appears for the first time. So that gives you kinematical space, physical space and solution space, where the elements of the solution space are in 1-to-1 correlation with the elements of the physical space. A solution is derived by the Hamiltonian as generator of time evolution from an initial state in the physical space. Agreed?

    Further, as you say, solving the Schrödinger equation is hard. Keep that in mind.

    Con't.

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  71. Now: how does all this show up technically? In the context of canonical quantization of GR, the dynamics (which in a theory like quantum mechanics is encoded in the Hamiltonian operator) appears instead as a *constraint*, the so-called Hamiltonian constraint. The constraint equations themselves contain the Hamiltonian, and instead of having the Hamiltonian *generate the time evolution* (because there is no such univocal thing as "the time evolution": every different foliation yields a different account of the "time evolution" in that solution and all are equally valid) the Hamiltonian operator *annihilates the solutions*, that is, the solutions are all zero eigenstates of the Hamiltonian constraint. Now this works perfectly fine since we have already decided to call all of the Cauchy surfaces through the same solution "gauge equivalent". The Hamiltonian constraint should annihilate every singe state in the same "gauge orbit". And to reconstruct the full 4-dimensional space-time that is represented in a single gauge orbit we have to figure out how to reconstruct a single 4-dimensional space-time out of all those Cauchy surfaces! The "solution space" in this theory will the be set of gauge orbits that are annihilated by the constraints. That, as I understand it, is just how Wheeler-DeWitt is supposed to work.

    Let's note a few things. First, in this picture, the Hamiltonian, which in QFT appears as a Schrödinger equation, appears here as a constraint. That leads to the so-called "problem of time": if you insist on thinking of the time evolution as generated by the Hamiltonian, if the solution is a zero-eigenstate then you will conclude that there is zero time evolution: nothing moves! But that is a complete mistake. What we normally think of as time evolution has already been taken care of in the definition of "gauge equivalent". If you can reconstruct a full 4-dimensioal space-time, then you have a perfectly normal sense of time. It is also often said of this theory that the dynamics is "pure gauge". We now understand why.

    Another thing: if the space-time to be constructed is a manifold, then by Geroch's theorem all of the Cauchy surfaces are contained in the kinematical space. For if Geroch's theorem holds, the Cauchy surfaces cannot change topology: if one surface is a connected 3-dimensional manifold, then they all are. So applying the constraint to that kinematical space will generate all of the Cauchy surfaces. But if something like what I have been proposing for an evaporating black hole is true, and the Cauchy surfaces can change topology, then all bets are off. We have to think through the consequences of this for the kinematical space, and hence for the physical space.

    And another thing: the two-step, three-space process we talked about above is either changed or absent altogether. You have the kinematical space. You impose constraints on it. What results are already the solutions of the dynamical equation, at least as far as the Cauchy surfaces go. So either there is no distinction between the "physical space" and the "solution space" or else you get from the kinematical space to the physical space by imposing all of the constraints except the Hamiltonian constraint, and then you get to the full solution space by further imposing the Hamiltonian constraint. But in no case do you solve the Schrödinger equation as such. Nor is there any preferred parameter t in terms of which a single foliation emerges. For an asymptotically flat spacetime, that is pretty much the story.

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  72. Con't

    And, as you say, it is even pretty much the story even in ADS/CFT *for a fixed boundary time t*. Given such a t, you specify a WDW patch and then follow these instructions for the patch. But in ADS/CFT, the existence of the time-like boundary requires yet *another* step: solving a Schrödinger equation for the successive boundary states. This Schrödinger equation, as far as I can tell, uses a *different* Hamiltonian than the one that appears in the Hamiltonian constraint. I may be wrong about that: please confirm or deny. But however that comes out, I again insist that solving some WDW patch where the black hole evaporates—some WDW patch with a connected initial boundary and disconnected final boundary, according to me— is where all the action is. Running the boundary time first way back to before the formation of the black hole and then way forward to after the evaporation is exactly the wrong thing to do. Because it is only by solving for what happens at the evaporation that we even know what structure the very late WDW patch ought to have.

    So that's how I see things. If you agree, then we can settle on some terminology. If not, say why not.

    The answer to your question is "yes". And the answer to your question is irrelevant. Because in a non-gravitational theory we would never consider giving up the manifold structure, so disconnected Cauchy surface ever means disconnected Cauchy surface always. But the suggestion I am making for the gravitational theory exactly denies this. And, to meet your explicit demand,: Hilbert.

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  73. Tim,

    I will get to the main part of your message later, but first I want to address your response to my question:

    "The answer to your question is "yes". And the answer to your question is irrelevant. Because in a non-gravitational theory we would never consider giving up the manifold structure, so disconnected Cauchy surface ever means disconnected Cauchy surface always. But the suggestion I am making for the gravitational theory exactly denies this. "


    I think you are missing the point here, since it's easy to construct a counterexample to your claim. Instead of saying disconnected Cauchy surface, let me say that at time t our theory lives in two sealed boxes. At some earlier time, it may or may not have been the case that the two boxes were connected to each other. That is, there may have been one box at an earlier time t_0, and then someone may have placed a partition in the middle of the box, cut the box in two, and flown one of the boxes to Alpha Centauri. My question is, given our two separated sealed boxes at time t, without any knowledge of what happened at earlier times, is the physical Hilbert space a tensor product? Yes, no, or you need to know what happened in the past? In this setup, we are for the moment ignoring gravity. This is a highly relevant question, and once you answer it we can modify the setup to include gravity.

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  74. Tim,

    "The problem was, as you recall, you at some point wanted to deny both the significance and even the mention of the kinematical space when doing quantum gravity. I take it we have settled that: the kinematical space is both mentioned and used by both Thiemann and Smolin.:


    I am really disheartened to read this. I will interpret this in the most charitable way I can: that your memory is playing tricks on you. I never made the ridiculous claim that the kinematic space is not mentioned in papers on quantum gravity! Obviously it is, and I have used it myself many many times. My point, which I suppose you have still not absorbed, is that this kinematic space is a formal object that arises at an intermediate stage in one particular approach to quantization. It has no *physical* significance, and to drive that point home I noted that one approach to quantizing gravity works only with physical states from the get go and never invokes a larger kinematic space. The reason why this is crucial is that we want to make a comparison between the same physics as described by the bulk theory in AdS or by the CFT. To do so, we need to focus on physical quantities. Bringing up properties of the bulk kinematical space will only lead us astray, as this space does not exist in the CFT formulation.


    "The issue was that you kept asking whether "the Hilbert space" in AdS/CFT is a tensor product space, and I kept answering that it depends on *which* Hilbert space one was asking about!"


    Come on, I have made it abundantly clear that I am asking about the physical Hilbert space when I ask this question. Your complaint here doesn't even jibe with the one above, where you complain that I don't want to talk about about Hilbert spaces other than the physical one. Anyway, it should be obvious by context that if I ask a question about the Hilbert space in AdS/CFT I mean the physical Hilbert space, since as I noted above, the CFT has no analog of the bulk kinematical space.

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  75. BGH,

    maybe to get out of this kind of deadlock between Tim and you, do you also allow to frame your question in the more familiar setting of an EPR pair (Alice in one box and Bob in the other)? Then you can talk about the state of this EPR pair and whether you can sustain the entanglement or not. If you can not it becomes a tensor product. Because what matters for the question to be a $-matrix or not is whether two particles stay entangled across a black hole horizon or not.

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  76. Reimond,

    I wonder if you are on the right track here. Of course there is no unitary process that will eliminate the entanglement, but if you are restricted for physical reasons to measurement operations that lie entirely in one or the other subsystem, then FAPP you can use a density matrix for each. This does not mean that the entanglement disappears, but it becomes irrelevant fo the S-matrix. BHG: is that what you have in mind?

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  77. BHG

    Here is a simple question. Do you or do you not acknowledge this three-step process—Kinematical space, physical space, solution space—in the construction of the solutions to many quantum problems, especially in QFT? It is only due to this three step process that there even exists your "physical space": the space that is midway between the unconstrained kinematic space and the full class of real physically possible worlds. Surely you agree that any process that is not in the solution space or any state that does not appear anywhere in the solution space is not really physically possible.

    If we agree about that, then we have to consider carefully the status of the "physical space" in quantum gravity, because in that setting the three-step procedure is compressed into a two-step process: straight from the kinematic space to the solution space without any intermediate stops. Here is the characterization from Wikipedia, which corresponds to my own understanding:

    "In Dirac's approach it turns out that the first class quantum constraints imposed on a wavefunction also generate gauge transformations. Thus the two step process in the classical theory of solving the constraints C^_I=0 (equivalent to solving the admissibility conditions for the initial data) and looking for the gauge orbits (solving the `evolution' equations) is replaced by a one step process in the quantum theory, namely looking for solutions Ψ of the quantum equations C^_I Ψ=0 This is because it obviously solves the constraint at the quantum level and it simultaneously looks for states that are gauge invariant because C^_I is the quantum generator of gauge transformations. At the classical level, solving the admissibility conditions and evolution equations are equivalent to solving all of Einstein's field equations, this underlines the central role of the quantum constraint equations in Dirac's approach to canonical quantum gravity."

    Since the Hamiltonian constraint is the analog if the Schrödinger equation. and since you have to solve all the constrains at once, you have to do the hard thing you mention—analogous to solving the Schrödinger equation—just to get beyond the kinematical space.

    I have pointed this out many times before. Whenever I mentioned that in quantum gravity "time evolution becomes pure gauge", that is what I had in mind. That is the origin of the "problem of time" in QG. It is also why I have no interest in reviewing QED or anything like that. That theory does (I believe) have a middle space, a "physical space" that can be specified independently of the Hamiltonian and solutions to the Schrödinger equation. The full solutions—which are function of time—are then in 1-to-1 correspondence with the elements of the "physical space". Thus the physical space and the solution space have the same structure. That is not only not a good model for QG, it is positively a misleading model. Misleading as well are the "gauge degrees of freedom" that arise in electromagnetic theory. As I mentioned, in that case the very same physical state can be expressed differently in different gauges, so one has to take gauge orbits. But in QG, there is an entirely new freedom: foliation freedom. The freedom to foliate the spacetime in different ways means that again there are different mathematical representations of the same physical situation, but the situation here is the full 4-dimensional space-time, not just a single 3-dimensional Cauchy slice. States on Cauchy slices are used, but two such states in the same "gauge orbit" do not at all have to represent the same (3-dimensional) physical state: it is enough that they both represent Cauchy slices taken from the same 4-dimensional solution. The application of the mathematical techniques that are used in QED for dealing with gauge transformations to handle foliation-freedom in GR leads to all sorts of erroneous analogs and bad understanding. I want to avoid that.

    Con't

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  78. As far as I can tell, AdS/CFT adds yet another level of complication to all this. Due to the timelike boundary, there is yet another dynamical equation—a Schrödinger equation—for evolution of the boundary state in boundary time. The existence of this extra step may produce the illusion that QG is a three-step process just like QED. But there is no such extra step in solving for a single WDW patch, and, as I have said over and over, solving for the right WDW patch gives the key to the general problem. The further evolution in boundary time plays no conceptual role.

    Finally, I really can't figure out what you are after with your boxes example. All of that can take place in a regular connected space-time. If that is supposed to show that the physical space is a tensor product space a) I can't see how it does so and b) if it does then the result applies to AdS/CFT quite apart from any issue about black holes. But that (as I understand it) you deny. So where are you trying to go with this?

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  79. Sure, the density matrix is the tool FAPP. And yes, a unitary process will not eliminate the entanglement. And yes, the S-matrix is unitary. But then to get a not S-matrix, some magic must happen. In Hawking's 2015 paper, now compliant with AdS/CFT the information is not lost anymore, but stored “in a chaotic but deterministic manner”. “The information about the ingoing particles is returned, but in a highly scrambled, chaotic and useless form. This resolves the information paradox. For all practical purposes, however, the information is lost.”

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  80. Tim,

    The discussion is in danger of fragmenting. To maintain focus, I will answer one of your questions and then ask that you do the same of mine.

    "Here is a simple question. Do you or do you not acknowledge this three-step process—Kinematical space, physical space, solution space—in the construction of the solutions to many quantum problems, especially in QFT? "


    Yes, I agree that a common procedure is to 1) start with some unconstrained (or partially constrained) space 2) impose the first class constraints, thereby defining the physical Hilbert space 3) study time evolution of physical states via H\psi = i d\psi/dt. Note the very important fact that the physical Hilbert space is defined at stage 2, prior to any discussion of solutions of the time dependent Schrodinger equation.


    Now, I would find it very helpful if you would answer my box question, which I believe is clearly stated and well posed. I would hope that the answer to this question is uncontroversial, and then we can discuss what, if anything, changes when we include gravity.

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  81. BHG

    To answer your question, Supposing that the theory can be presented in the three-step fashion discussed above—kinematical space, constrained to physical space, time evolved to solution space—then yes, the physical space is a tensor product. Since you omit gravity, as far as I know there is not impediment. But if dynamics is encoded in a constraint of the form H |psi> = 0 rather than ihd/dt|psi> = H |psi>, then all bets are off.

    Question: do you disagree that the use of the Hamiltonian constraint rather than the Schrödinger equation to implement the dynamics is highly significant? If the Hamiltonian annihilates the solution, then time evolution will be much subtler business.

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  82. Tim,

    Great. Yes, I certainly agree that if we were in the situation of a closed universe where the Schrodinger equation reads H\psi =0 then we have to deal with all the "problem of time" issues, and things get very confusing fast. Fortunately, we are not in that situation: we are discussing cases with an asymptotic structure, so there is a well defined asymptotic time t, and a well defined Schrodinger equation H\psi = id\psi/dt governing evolution of the wavefunction. With that in mind let us proceed.

    Let's first focus on what the equations say, following the Dirac procedure. Suppose we are doing quantum gravity in a space with an asymptotic structure, so we have an asymptotic time t. At some specific time t we have a wavefunction, \psi[g_ij;t] (also with matter fields, which I suppress). Suppose that all of the 3-geometries appearing in \psi at time t are disconnected. Let x and y denote spatial coordinates on each component. What is the physical Hilbert space in this case? We carry out the 3-step process

    1) We start with unconstrained wavefunctions \psi[g_{ij}(x), g_{ab}(y) ;t ]. These live in a tensor product space, since I can choose a basis of product wavefunctions, and I have an independent choice for each factor in the product.

    2) Now we follow Dirac and apply the constraints. These are of the form C_i(x) \psi =0 and C_a(y)\psi =0. Each C(x) is an operator built out of g_{ij}(x) and derivatives d/dg_{ij}(x). Important: each constraint involves an x or a y, but not both, and the constraints do not have any dependence on t. So the equations C\psi = 0 do not mix up the two components, and the t variable is just a bystander. The point here is that if that if the space in (1) is a tensor product, then the physical Hilbert space, defined as the space of solutions to the constraints (endowed with an inner product) is also, simply because the constraints have no mechanism to alter this fact.

    3) Now that the physical Hilbert space at time t has been defined, we can choose a particular state in this space and then evolve it forward via H\psi = id\psi/dt.

    Any objections?

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  83. BHG

    No objections to what you have done, but I do have an objection to some of the remarks, which I think are misleading. Of course, if you (by stipulation) start with a case where all of the Cauchy surfaces are disconnected (i.e in AdS a boundary time so late that the evaporation event does not lie in the WDW patch), then as far as the math goes you could be dealing with two disconnected space-times, and they will stay disconnected at all future times. I should just remark—in case this is where things are going—that putting in the non-manifold point gives a fundamental time-asymmetry to the space-time, and you will not be able to run the dynamics backward to recover the initial connected state. Further, for states that do arise from a common past, the constraints will restrict the physical possibilities.

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  84. Tim,

    We can discuss time evolution in due course, but right now the key fact that I want to make sure is agreed upon is that according to Dirac quantization to determine the physical Hilbert space at asymptotic time t you only need to consider equations at time t. The actual state you are in will depend on what happened at earlier times, but the structure of the Hilbert space does not. Therefore, regardless of what happened earlier, if at time t all spatial geometries are disconnected then the physical Hilbert space is a tensor product. It is just a statement of fact about the equations of Dirac quantization, but I want verify that you agree with this.

    Assuming you do, we will then apply this to black hole evaporation in AdS. At very late boundary time all spatial surfaces are disconnected, hence the physical Hilbert space at that time is a tensor product.

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  85. BHG—
    No, if you have been following what I have been arguing, I do not accept this. AdS Has a particularly baroque structure due to the timelike boundary, and you have tried to rush over the critical points. To arrive at the physical Hilbert space you have to impose the constraints, and impose them globally (at all times). Unlike the non-gravitational theory that you started with (I told you this would be misleading) that includes the Hamiltonian constraint, which has to be imposed on every Wheeler-DeWitt patch. And since each such patch overlaps other patches, the effect of the constraints at earlier times get propagated up to later times. Therefore, a late-time pair of disconnected Cauchy surfaces can be subject to more constraints if they both arose from a single Cauchy surface earlier, and those extra inherited constraints can prevent the physical Hilbert space from being a tensor product space.

    The problem it that you are analogizing the t variable that represents boundary time to the t variable as it appears in the non-gravitational theory. That's not right. The function of the t variable and the Schrödinger equation in the non-gravitational theory is played by the Hamiltonian constraint in the gravitational theory, not by the t-variable and Schrödinger equation peculiar to AdS. The critical temporal evolution is from the connected Cauchy surface to the disconnected pair, and that is governed by the Hamiltonian constraints, not by the Schrödinger equation.

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  86. Tim,

    What I am saying follows directly from the Dirac quantization procedure. In this procedure, the physical Hilbert space at time t is determined entirely by equations at time t, and no other times. I.e. the full physical Hilbert space at time t is the space of solutions of C_i \psi(g_{ij},t)=0. Period. Now, you seem to be claiming that there are some additional constraints not accounted for here, corresponding to also solving the constraints at earlier times, or something like that. I ask you to try to turn these words into equations. What additional equations should a wavefunction \psi(g_{ij},t) obey to be physical, beyond the ones I have specified? You will find that there are no such equations. One of the main points of Dirac quantization is that physical states stay physical under time evolution (the proof is simple, and we can discuss it if you want), so if you impose the physical state constraints at one time they are automatically obeyed at another time. So these extra conditions you are claiming don't exist, or more precisely are redundant. Again, if you believe otherwise please state a precise equation to the contrary and we can discuss it.

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  87. BHG,

    I don't understand this at all: you are acting as if something I have written is novel and not just a boilerplate account of what happens when you try to quantize GR by casting it in a Hamiltonian formulation and following the usual procedures. So let's do this step by step. Do you deny that in this approach the set of constraints C_i \psi(g_{ij},t)=0 contains two different types of constraint: the spatial diffeomorphism constraint and the Hamiltonian constraint? Or do you deny that this very distinction, between a "spatial diffeomorphism" and a "time evolution" is not a fundamental physical distinction in GR because it is relative to a foliation and there is no preferred foliation? SO to get a theory that does not depend on a preferred foliation one needs to impose both constraints together, and the division of the total constraint in a "spatial" part and a "Hamiltonian" part is mere convention, reflecting an arbitrary choice of foliation?

    None of this is new to me: it is what you read everywhere. But you appear to be wanting to deny it. Can you explain what exactly you are denying and why? Do you literally want me to write down the Hamiltonian constraint equation?

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  88. Tim,

    Everything I am saying is straight out of Dirac, and the only thing I am denying is your statement:


    " that includes the Hamiltonian constraint, which has to be imposed on every Wheeler-DeWitt patch. And since each such patch overlaps other patches, the effect of the constraints at earlier times get propagated up to later times. Therefore, a late-time pair of disconnected Cauchy surfaces can be subject to more constraints if they both arose from a single Cauchy surface earlier, and those extra inherited constraints can prevent the physical Hilbert space from being a tensor product space."


    because this is false. As I have emphasized, it is a basic fact that the physical Hilbert space at asymptotic time t is determined by imposing the constraint equations at time t, and makes no reference to any earlier time. I asked you to write an equation corresponding to what these "extra" constraints are, but you did not do so.


    Sure, let's go through the Dirac procedure step by step so we can confirm what I am saying.


    1) Start from the Einstein-Hilbert action. Write the metric in ADM form. Note that this doesn't involve any choice of foliation, because we allow the lapse and shift to be unspecified. Demand that the metric and extrinsic curvature fall off appropriately at the boundary so that the variational principle is well defined.


    2) Compute the the canonical momenta, which leads to constraints. Compute the Hamiltonian, including the boundary term. Determine any additional constraints via the condition that all constraints commute (weakly) with each other and the Hamiltonian. The constraints so determined are C_i(x) and C_t(x), the latter being the Hamiltonian constraint. Both are local functions of the metric and its canonical momentum.


    3) At fixed asymptotic time t, start from the full unconstrained space of wavefunctions \psi[g_{ij};t). Now impose the constraints C_i (x) \psi = 0, C_t (x)\psi = 0. The space of solutions is the physical Hilbert space (assuming one can define an inner product on this space). Key point: the constraint operators C_i(x) and C_t(x) do not depend on t, so the space of solutions, and hence the Hilbert space is independent of t. No reference is made to any other time.

    4) If at time t the 3-geometries appearing in \psi_{g_{ij},t) are all disconnected, then the physical Hilbert space is a tensor product.


    So, there are no "extra" constraints of the type you bring up. Again, I ask that if you disagree you try to present your claim in term of precise equations. Of course, I am sure that you will be unable to do so, because the equations I have written are the complete set, according to Dirac. Once more for emphasis: it is a universal fact about the quantization of Hamiltonian systems that the structure of the Hilbert space at a given time t is determined without reference to any other time.

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  89. BHG

    You are, I believe mixing up two different variables there. You have the t in the Hamiltonian constraint C_t (x), and the quite different t that occurs in the CFT. When we first started talking about AdS, we called the first t "alpha" and spoke of alpha evolution in the WDW patch.Alpha parameterizes the leaves of a foliation of the WDW patch associated with a fixed boundary time t.

    Now: do you agree or disagree that the WDW patch associated with t will largely overlap the WDW patch associated with the nearby boundary time t + delta t? And if you agree that these two WDW patches overlap, do you agree that making the solutions in the neighborhood of overlap places another constraint on any global solution?

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  90. Tim,

    I assure you I am not doing anything as stupid as mixing up t variables. Let's call the constraints C_i(x) and C_0(x), the latter being the Hamiltonian constraints. The asymptotic time will be called t. C_0(x) is then an operator in which t makes no appearance.

    My question to you is: If a wavefunction \psi[g_{ij},t], at some specific t, obeys C_i(x) \psi =C_0(x) \psi =0, is it thereby a physical state, or are there some extra conditions that must be imposed? I (and Dirac) say no. You seem to say yes. If so, what are these extra conditions, as precise equations? I will keep asking this question until you see that there are no such extra conditions that characterize physical states.

    In the first part of your message you bring up \alpha evolution. But that is not a fundamental concept, and I only brought it up earlier to see how in the classical limit one can approximately recover the picture of fields evolving on a fixed background geometry. But there is no notion of \alpha evolution at the level of the basic equations of Dirac.

    "Now: do you agree or disagree that the WDW patch associated with t will largely overlap the WDW patch associated with the nearby boundary time t + delta t? And if you agree that these two WDW patches overlap, do you agree that making the solutions in the neighborhood of overlap places another constraint on any global solution?"


    As I keep saying, no, this places no constraint on the structure of the Hilbert space, since if a state at time t is physical then it is automatically physical at time t+delta t. As an equation, what is the extra condition that you say imposed? Your comment is equivalent to the statement that specific wavefunctions at different times are related by the fact that the time evolution of the wavefunction is governed by H\psi = id\psi/dt. Yes of course, but this has no relation to the issue of what the Hilbert space is at time t.


    I think what may be going on here is that I am making statements about the full physical Hilbert space at time t, whereas you are speaking about properties of specific t-dependent solutions of all the equations, including the Schrodinger equation. But let's leave that for later, and right now just ask about the structure of the Hilbert space at some specific late time t.

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  91. BHG-

    Your choice of nomenclature is just getting in the way here. This is the question that I am addressing: take a disconnected Cauchy surface *that has arisen from a connected Cauchy surface via the evaporation of a black hole*, as is postulated by my solution to the information loss "paradox", and ask what is the space of physically possible states that can occur on such a disconnected surface *consistent with it having arisen in such a way*. That is the only interesting question to ask if our topic is whether my solution is somehow inconsistent with AdS/CFT. Of course, if one does not demand that the disconnected Cauchy surface have arisen in that way, then trivially the space of "physical states" is a tensor product space: after all. the disconnected Cauchy surface could be the Cauchy surface of a pair of completely disconnected space-times! That is completely irrelevant to our question. If one is interested in the question before us, that observation is as irrelevant as the observation that the kinematical Hilbert space is a tensor produce space.

    So using your preferred usage of "physical"—which includes states that could not possibly have arisen from the evaporation of a black hole—yes, the "physical" Hilbert space is a tensor product. I acknowledge that. Now will you please acknowledge that given what you insist on meaning by "physical" that fact is completely irrelevant to our question? Can we agree on some terminology for the space of states on Cauchy slices that could arise—according to the theory—in a solution to all of the fundamental equations in the theory in a model that whose initial state has connected Cauchy surfaces and that contains an evaporating black hole? That is what I would prefer to call the "physical" Hilbert space, but since you have co-opted that term for something weaker we need a new term. You provide the term and I will use it. For the moment, let's just call it Hilbert space X. So I acknowledge that the "physical" Hilbert space is a tensor product space and I deny that Hilbert space X—which is a subspace of the "physical" Hilbert space—is a tensor product space. Do you disagree?

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  92. So here's the argument as I understand it: Black Hole Guy is saying that if, at any boundary time t, there is a WDW patch which contains disconnected 3-geometries, then the Hilbert space in which the WDW patch lives must be a tensor product space, and hence is incompatible with AdS/CFT. In other words, he's saying that there can be no WDW patches with disconnected 3-geometries in Ads/CFT. This argument may or may not be correct, but I think that's the argument.

    On the other hand, Tim is wanting to talk about the subspace of the physical Hilbert space which arise from certain initial conditions, but this doesn't seem relevant. If Black Hole Guy's assertion that any single WDW patch having a disconnected 3-geometry is incompatible with AdS/CFT is correct, then it is irrelevant whether we look at the whole Hilbert space or a subset of it; it suffices for Black Hole Guy's argument that there is some WDW patch in Tim's model which is disconnected. The physical Hilbert space is only brought into the picture for the purpose of demonstrating that it must be a tensor product space *if any WDW patch is disconnected*, and it is this fact that is supposedly in contradiction with AdS/CFT.

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  93. Tim,

    Let's at least be clear that I have been very careful to use standard terminology, and in particular my usage of "physical Hilbert space" is the standard one. E.g. recall the Smolin quote:


    " The kernel of the gauge and spatial diffeomorphism constraints 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."


    So we are now in agreement that the late time physical Hilbert space (defined in the standard way) is a tensor product, correct?

    You say that this point is not relevant, but as you will see it is hugely relevant since it kills your proposal, assuming AdS/CFT is valid. Of course, one can look at subspaces of the physical Hilbert (let's call this space X, as you do), and these subspaces may or may not be tensor products. I completely agree with that.


    Now, please let me complete my argument. I hope we can agree on the following assertion. The late time physical Hilbert space is a tensor product, H_in x H_out, and the Hamiltonian acting on this space is a boundary term that acts only H_out. That is, the Hamiltonian is of the form Ham = I_in x Ham_out. Of course, the Hamiltonian acting on the full kinematical space includes additional constraint terms, but these vanish by definition on the physical Hilbert space, so let's please not go down that alley again. I am restricting attention to the physical Hilbert space, and I am also being very careful in what I write.


    So do you agree with the above, which is essentially the content of Regge/Teitelboim?

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  94. Travis,

    Basically, yes. If a WDW patch consists *entirely* of disconnected surfaces then, according to the rules of Dirac quantization, the physical Hilbert space is a tensor product. Now, this by itself is not in conflict with anything, since the CFT Hilbert space can itself be factored in many ways. The other key point is that the bulk Hamiltonian, acting on the physical Hilbert space, is a boundary term that acts only on one component of the Hilbert space. Clearly, this implies a huge degeneracy in the spectrum of the Hamiltonian, essentially equal to the dimension of the interior Hilbert space. This contradicts the CFT.

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  95. Black Hole Guy,

    So just to clarify: this argument seems to also imply that AdS/CFT cannot describe two completely disconnected spacetimes, is that correct? (Unless both of the disconnected spacetimes are AdS, and then you can trivially have two CFT's which describe each of them separately?) If this is not an implication, what is the difference here? If the difference is that there will be a boundary Hamiltonian on each spacetime, so that there is no degeneracy in the spectrum, why can we not also assume that there is a boundary Hamiltonian for the interior of the black hole in addition to a boundary Hamiltonian for the exterior? This seems like a place where Tim's non-manifold point might be playing a crucial role, by essentially providing another boundary for the Hamiltonian.

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  96. Travis,

    Correct. Two disconnected asymptotically AdS spacetimess would be described by two decoupled CFTs. But AdS and a separate closed universe, say, -- that can't be described in AdS/CFT.

    I am not sure what exactly you have in mind with an additional boundary Hamiltonian or how it would help, but if you elaborate I will respond.

    It's really not easy to find loopholes in this argument. Which of course makes it fun!

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  97. Black Hole Guy,

    If AdS/CFT can't describe two disconnected spacetimes, then it seems useless for commenting on whether Tim's scenario is viable. Clearly there is nothing wrong with having two completely disconnected spacetimes, neither of which are AdS. So if the argument against Tim's scenario also rules that out, then the argument "proves too much", as they say.

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  98. Travis,

    No: AdS/CFT doesn't rule out having completely disconnected spacetimes, it just doesn't describe quantum gravity in that situation. Just as Maxwell's equations don't describe gravity, they don't rule it out either. AdS/CFT does of course describe gravitational collapse and Hawking evaportation in AdS, which is what is under discussion here.

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  99. BHG,

    But if AdS/CFT simply has nothing to say about anything involving disconnected 3-geometries, then how can it be used to rule out Tim's scenario? It seems like you're assuming no disconnected 3-geometries, and proving from that that there are no disconnected 3-geometries. How is this not what you're doing? If your argument involves the presence of any WDW patch which is disconnected being impossible, then you are ruling out all scenarios with disconnected 3-geometries, including ones where the geometries are disconnected for all time. If this isn't what you're doing, then you do in fact have to take into account somewhere in your argument how the disconnected Cauchy surface arose from initial conditions, like Tim has been insisting.

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  100. Ok. So the argument is that AdS/CFT doesn't allow disconnected 3-geometries; Tim's model contains a disconnected 3-geometry; therefore, if someone accepts AdS/CFT as a good model of black hole collapse and evaporation, then black hole collapse and evaporation doesn't involve disconnected Cauchy surfaces, and so Tim's model is ruled out.

    The point I was trying to make with the boundary Hamiltonian is this: we presumably don't know how the Hamiltonian operates on Tim's non-manifold point (do we?). Since that point is connected to the interior of the black hole, is it possible that that point can act like a boundary for the interior so that there is in fact a Hamiltonian associated with the interior, so that the spectrum is not degenerate? It actually seems like something like this would have to be the case if we are to take Tim's model seriously, since as Tim pointed out, we can create a connected Cauchy surface going from the interior of the black hole out to the AdS boundary (which will have an energy given by the boundary Hamiltonian), and we can also create a disconnected Cauchy surface going from the interior of the black hole out to the same boundary point which contains (almost) all of the same spacetime points except for the non-manifold point, and that Cauchy surface should have (nearly) the same energy associated with it. And also it seems unfair to insist that the Hamiltonian exists only on the boundary, and then to not associate a boundary with the interior of the black hole so that all states on the interior are technically vacuum states in a way that not all states in the exterior are.

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  101. BHG

    No, I'm afraid that a review of the way you have this set up reveals that you have been led astray. Let's go back a step or two.

    You claim that your usage of "physical space" is the commonly agreed one, and that by your definition the late-time physical Hilbert space is a tensor product space. Let's go back and examine these claims.

    Start with the Smolin passage that you cite to explicate the standard meaning of "physical Hilbert space". According to Smolin, you arrive at the physical Hilbert space by starting with the kinematical Hilbert space and imposing the constraints on it. Those include, of course, both the (spatial) diffeomorphism constraints, that generate diffeomorphisms confined to a spatial hypersurface, and the Hamiltonian constraint, that generates the evolution from one hypersurface to another.

    Now in the context of the paper, when Smolin asserts this, the constraints are the totality of the physical laws. That is, any state that is a solution to all of the constraints is a completely specified physical world, and the physical Hilbert space is just the set of all of the physically possible worlds. (At that point of the paper, Smolin is considering spatially closed universes.) In order to determine the structure of this space, one would have to determine the complete set of solutions of the fundamental equations, which in Smolin's case are the constraints. So Smolin's sense of "physical Hilbert space" is the set of all global solutions to the fundamental equations of the theory.

    In the context of AdS, or even of a spatially open asymptotically flat space-time, the totality of equations that have to be satisfied are richer than just the diffeomorphism and Hamiltonian constraints: in addition to satisfying all of these in the bulk, it also has to satisfy some boundary conditions and the wavefunction on the boundary has to satisfy another evolution equation. The physical Hilbert space would then be the set of solutions of all of those conditions.

    And now we have yet another question: the set of all solutions on what? Well, the standard approach is to settle that a priori: the set of all solutions of these equations on a given differentiable manifold, which is taken to be Σ×R. So applying this to our case, if we ask whether the “physical Hilbert space” contains solutions that correspond to my proposal, we can immediately say “no”. As I argued, in order to have an evaporating black hole solution, you have to give up on a manifold structure. And so if my argument is correct, the fact that the correct solution will not show up when doing AdS this way is already decided at step one: the very first decision that is made about the right topological space-time to work with is wrong. We can say without further complications or further ado that AdS/CFT makes an initial assumption about the structure of space-time that is inconsistent with the nature of my proposed solution. So if that solution is correct, we know from the get-go that AdS/CFT will not be capable of handling situations in which black holes evaporate, and the physical Hilbert space of AdS/CFT will not contain any states in which a black hole forms and evaporates. In short, the entire argument you have been trying to establish begs the question at the very first step. It is therefore incapable to creating any problems for my solution. On the contrary, the problems are for AdS/CFT, which would have to be shown to actually contain any evaporating black hole solutions. The only way to do that, of course, is to solve the constraint and other equations for an asymptotically AdS space-time and show that he solution contains an evaporating black hole, or else figure out the dictionary with the CFT to such an extent that one could prove that a solution to the CFT corresponds to an evaporating black hole. But neither of these has a prayer of being done at the moment.

    Con't.

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  102. In short, there is no reason at all to think that the “physical Hilbert space” of AdS, as it is defined, can contain any evaporating black hole solutions, so nothing about the structure of that physical Hilbert space can have any bearing on the question of evaporating black holes, without first ruling out the my solution to the information loss problem. The attempt to appeal to AdS/CFT to rule out the solution is logically backwards.

    I should say, in addition, that once one takes on board the proper definition of the physical Hilbert space, namely as the set of global solutions to all of the fundamental equations, including the Hamiltonian constraint, the claim that that space has the structure of a tensor product space of states on one of the disconnected parts of a space-like surface tensored with the space of states on the other cannot be defended without actually determining all of the detailed solutions, which is something that has never been done.

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  103. Tim,

    I will direct you to Dirac's wonderfully clear original writings on the subject. He specifies a procedure valid for an arbitrary starting Lagrangian, including the case of gravity, since that was indeed one his motivations. This procedure algorithmically produces a set of first and second class constraints. The physical Hilbert space is the joint kernel of the first class constraints. The time dependent Schrodinger equation is definitely not a constraint, and is not imposed to define physical states. Indeed this would make no sense: when we consider the Hilbert space for the harmonic oscillator, say, we don't ask about time dependent wavefunctions!


    In gravity the constraints are C_0(x) and C_i(x), one of each for each spatial point x. The physical Hilbert space is the space of solutions of these constraints, endowed with an inner product. No time variable enters into any of these equations. That's it. Please ask around if you don't believe me, or read Dirac for yourself.

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  104. Travis,

    I want to emphasize that AdS/CFT provides not just a particular "model" of black hole dynamics, rather it is a well defined non-perturbative theory of quantum gravity, and indeed is the only example of such that reduces to Einstein gravity at low energies. So in the only known framework where it is possible to sensibly ask the question, we know that the information comes out in the Hawking radiation, and Hawking's original scenario (equivalent to Tim's) does not occur. This is clearly significant, and any line of argument that simply ignores this will not generate much interest.

    As for your discussion of additional boundary Hamiltonians, let me say this. In standard treatments, boundary Hamiltonians arise when the metric is fixed in some way, so that the time variable t acquires a definite physical meaning. Near a black hole singularity one is in the opposite case where the metric is in a highly non-classical state. I don't know any sensible procedure for talking about boundary Hamiltonians in such a case. In AdS/CFT the only boundary Hamiltonian is at the AdS boundary.

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  105. BHG

    The C_0(x) constraint is, as you know, called the Hamiltonian constraint and is the place in this formalism where the Hamiltonian—the generator of time evolution—shows up. The other three constraints are the spatial diffeomorphism constraints, and are connected to diffeomorphisms within a particular spacelike hypersurface, i.e in spacelike directions. They are connected to the shift operators. The Hamiltonian constraint is connected to the lapse operator, as in lapse-of-time operator, as in diffeomorphism that takes points off of the given spacelike hypersurface, i.e. that generate a time evolution. Again, as I'm sure you know, the 0 in C_0(x) refers to the 0 component of a space-time vector, i.e. the timelike part. The reason that the time evolution shows up as a constraint, and not in a Schrödinger equation is because there is no canonical way to foliate a generic relativistic space-time into spacelike hypersurfaces, so the choice of a particular set of hypersurfaces is formally equivalent to a choice of gauge. And the reason that the time evolution at the boundary of an asymptotically flat space-time can be represented by a Schrödinger equation is because there one does enforce a way of foliating the essentially flat boundary part of the space-time by equal-time surfaces in a Lorentz frame. So t shows up as a parameter on the boundary Hamiltonian, but the time evolution in the bulk is imposed via the Hamiltonian constraint rather than by a Schrödinger equation.

    When you write "No time variable enters into any of these equations" I can't figure out if you are just making a rather irrelevant typographical observation or you are really meaning to deny that the time evolution in the bulk is implemented by the Hamiltonian constraint. If the latter, maybe you can answer a few questions like
    1) Why is the Hamiltonian constraint called by the name "Hamiltonian"? What does it have to do with the Hamiltonian of the system?
    2) Why is that constrain also referred to as C_0(x)? What does the "0" there connote if not a timelike direction?
    3) How is the time evolution in the bulk implemented if not via the Hamiltonian constraint? Obviously it is the time evolution from the pre-black-hole-formation state to the post-black-hole-evapoation state than we are trying to understand. What physics governs that evolution in your estimate?
    4) Why are the strict constraints on the behavior of the foliation at the boundary imposed if not to have a well-defined time coordinate at the boundary, in terms of which the Schrödinger equation can be framed?

    It honestly looks to me as if you are completely misreading the physical significance of the Hamiltonian constraint, and the reason no t variable appears when specifying the time evolution in the bulk and does show up when specifying the time evolution at the boundary. If that is not what you are thinking, please explain what you are thinking.

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  106. Black Hole Guy,

    Ok. But if we just don't know how the Hamiltonian is supposed to be calculated when we have a singularity, it seems like it makes more sense to assume that, however it is calculated, it will be sensitive to changes in the interior of the black hole, just as it is sensitive to changes in the exterior. For any given disconnected Cauchy surface at an early enough time in Tim's model, we can draw a connected Cauchy surface which is identical except for a small region around the non-manifold point, and the Hamiltonian of that Cauchy surface is sensitive to changes in the interior, so it seems reasonable that the Hamiltonian of the nearly identical disconnected Cauchy surface would also be. Then we can extend that reasoning to disconnected Cauchy surfaces at late times.

    Another way to think about it: if we have a Cauchy surface at a given boundary time t = 0, the Hamiltonian should evolve that surface in time so that it evolves through the entire future of the Cauchy surface as t goes to infinity. But if the Hamiltonian only acts as the identity on the interior part, then the Cauchy surface on the interior will never change, and so we will never reach the future of the initial Cauchy surface. So it seems like your assertion that it operates as the identity on the interior part must be wrong.

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  107. Black Hole Guy,

    I'd definitely be interested in understanding better in what sense AdS/CFT reduces to Einstein gravity at low energies. The measurement problem in quantum mechanics is an unsolved problem, so for ordinary quantum mechanics we don't even know exactly how quantum mechanics reduces to classical mechanics, but at least this is ordinarily not too much of a problem because we can separate the system from the environment and assume the environment is classical. This approach clearly doesn't work when talking about entire spacetimes. AdS/CFT uses unitary quantum mechanics with no hidden variables, which implicitly means some kind of many worlds solution to the measurement problem; but even so, what are the observables which reduce to the classical observables in the limit of low energy? Is it their expectation values that reduce to the classical values? If so, expectation values with respect to what state?

    I am admittedly pretty ignorant on this topic.

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  108. Tim,

    I am happy to explain these points, but first I want make sure that you concur with a basic fact about Dirac quantization, since much hinges on this. Namely: the physical Hilbert space is defined as the space of solutions of C_0(x)\psi = C_i(x)\psi =0, endowed with an inner product. Yes?

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  109. Travis,

    Occurrences of singularities and "non-manifold points" are just statements that the classical approximation is breaking down, nothing more than that. Obviously, the notion of a well defined metric and so forth breaks down near a black hole singularity. The quantum equations don't break down, or rather, if they do that just means we haven't been using the right equations! Fortunately, the statement that the on-shell Hamiltonian is a boundary term at infinity is valid even in the strongly quantum regime.

    As to how Einstein's equations emerge from AdS/CFT, that is a more involved topic than I have the energy to explain here. But you will find a copious literature by doing a little googling.

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  110. BHG—

    No. As far as I can tell, for the purpose of your argument the physical Hilbert space must be the space of solutions of all of the fundamental dynamical equations of the theory, that is, the space of all of the universes that are physically possible according to the theory. Why else would it be called the “physical” Hilbert space? Anything short of a solution to all of the fundamental equations may not be physically possible since it may not satisfy whatever equations have been left out.

    I think that this is, at bottom, the very thing that has been the sticking point the whole time. It is the reason we had all of that time arguing about whether the Hamiltonian in the bulk is the zero operator—it isn’t, of course, it is the Hamiltonian constraint—and it is the reason you have been stuck so long on their being a boundary term for the Hamiltonian (which you have been giving way too much significance too), and it is the reason you are talking about AdS in the first place (which is not a physically realistic model). It is the reason that we have had so many arguments about what to even call different Hilbert spaces. I finally realize recently that I had been reading the wrong paper by Reggae and Teitelbaum, and got the right one. I now understand what Reggae and Teitelbaum were up to, and why. So I think I am now in a position to explain what has been going on all these many months, and I am not going to be making pre-emptive concessions that will just land us back in the same arguments again. So let me put this whole thing together. We need to settle this—and the correct settlement is that in this context the space of solutions of the constraints *is not* the physical space, i.e. the space you need for your supposed “killer argument” to go through. It is exactly here that you have been misled by putting Dirac together with Reggae and Teitelbaum without paying attention to how Reggae and Teitelbaum are *rejecting* some of what Dirac did. For *Dirac*, the space of solutions to the constraints is (or is isomorphic to) the physical space. For Reggae and Teitelbaum, the space of solutions to the constraints *is not* and *is not isomorphic to* the physical space.

    The ultimate root of this entire endless argument is right here. Just a little while ago, I was in the strange position of having to suggest a name—a used “Hilbert space X”— for the set of solutions to all of the fundamental equations of a theory. That was a concession I should not have made. The “physical Hilbert space” of a theory, of any theory, is the set of all of the solutions to all of the fundamental equations of the theory. *All* of the fundamental equations. Anything that is a solution to less than that is not yet guaranteed to represent a real physical possibility, and therefore should not be denominated “physical”.

    Con't.

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  111. And there is a second bit of terminology we have to be careful about. Sometimes, as I have been noting, the term “physical Hilbert space” is used for the space of all global solutions to all of the fundamental equations of the theory. Other times, “physical Hilbert space” is used for a Hilbert space that is *isomorphic* to the physical Hilbert space as I have just defined it. If I cast the fundamental dynamics of a theory as a well-posed initial value problem, then the space of global solutions is isomorphic to the space of initial conditions. So for many purposes one can speak of the space of initial conditions as the “physical Hilbert space”, letting each possible initial conditions stand as a proxy for the global solution that comes from it. The 1-to-1 map from initial conditions to global solutions makes all of that talk clear. Of course since AdS is not even globally hyperbolic, we can’t possibly do that for AdS without some further discussion of the timelike boundary conditions. But the main point is that even where there is the 1-to-1 map, the primary meaning of “physical Hilbert space” is the space of all global solutions. It is only because we have a well-posed initial value problem that the space of allowable initial conditions itself my derivatively be called a “physical Hilbert space”.

    I am prepared to explain now how three things fit together—Dirac’s set-up, Wheeler and DeWitt’s set-up, and Reggae and Teitelbaum’s set-up— and why they are all different and why mixing them up (which is what has been going on all these many months) is so dangerous. But before going into the details, we have to settle the terminology. I intend to use “physical Hilbert space” to mean primarily the space of all global solutions to the fundamental physical equations of a theory. I may use it derivatively, if appropriate, to a space that is isomorphic to the physical Hilbert space if some exact 1-to-1 function from it to the physical Hilbert space as just defined is specified.

    Do you agree to this terminology, or do you want to argue that it is essential to regard some Hilbert space as “physical” even though it cannot be put into a 1-to-1 correspondence with the space of global solutions in a natural and completely specified way? If the latter, then let’s have that argument first. We can start by explaining why you want to reject my terminology.

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  112. Tim,

    I am glad you were able to find the correct Regge/Teitelboim paper, but note that I did supply you with the authors and title, which is typically sufficient. Note also that you are misspelling the names of *both* authors.

    Anyway, Dirac quantization is a widely known and well established procedure for quantizing an arbitrary Lagrangian system. In particular:

    1) One identifies a set of first and second class constraints, and computes a Hamiltonian

    2) The physical Hilbert space is the space of wavefunctions annihilated by the first class constraints, endowed with an inner product

    3) In GR, the first class constraints are C_0(x) and C_i(x), i.e. 4 constraints for each spatial point x.

    Now, do you a) disagree with Dirac, and if so where? b) think I am misrepresenting Dirac, and if so where?

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  113. BHG

    a) My own personal agreement or disagreement with Dirac is neither here nor there. What is critical for the discussion is the relation between Dirac, Wheeler and DeWitt, and Regge and Teitelboim. The problem is that you are jumping around between these various presentations without taking sufficient note of the differences between them. In particular, it is not my own opinion of Dirac that matters here, but the opening paragraphs of Regge and Teitelboim, which make clear that they consider what they are doing to be a correction or clarification of Dirac's procedure. In particular, if you just read that first paragraph of Regge and Teitelboim, you will not that precisely where they disagree with Dirac is one the question of whether is it essential or inessential to include surface term of the Hamiltonian that we have had so much discussion of. In precise detail, Dirac asserts (quoted directly by R & T) that the inclusion of the surface term is optional: throwing it away "does not disturb the validity" of the Hamiltonian, and De Will (quoted directly by R & T) says that including the surface term "leaves the dynamical equations unaffected". And the reason for this is clear: in an symptomatically flat space-time, you have put an a priori constraint on the stress-energy tensor as you go to spatial infinity, so nothing that happens "at spatial infinity" depends on the dynamics of the target system, which is in the bulk arbitrarily far away from the boundary. In our case the target system—the system whose behavior we want to understand—is the evaporating black hole.

    So both Dirac and DeWitt regard the surface term, which you have made quite a lot of noise about in our discussion, as physically uninteresting and irrelevant: the relevant dynamics is all in the constraint equations in the bulk. That is actually exactly the view that I came into this with, being unaware that R & T argued against Dirac and DeWitt. And truth be told, having read R & T I am still rather on the side of Dirac and DeWitt that the surface term is basically irrelevant in the case of an asymptotically flat space-time: as De Witt says, the surface term is not doing anything except fixing a convention about which energy to use when discussing the system of interest.

    But what I want to absolutely insist on is not who among Dirac, DeWitt and R & T is *correct* here, but the completely clear and incontrovertible fact that they *disagree* about the significance of the surface term. Dirac thinks you can just throw it away without any significant change to the analysis, DeWitt thinks that the only thing the surface term does is "change the definition of energy" (which is essentially a convention), but R & T take the completely opposed stance that without the surface term the theory "gives no well-defined set of equations of motion at all". This is about as strong a disagreement as one could find.

    To put it as clearly as possible, Dirac, DeWitt and R & T all agree that one can write down a Hamiltonian with a bulk term and a surfaces term, and they all agree that in the bulk the Hamiltonian for GR is expressed by a constraint equation, the Hamiltonian constraint. And they all agree that this constraint gives the dynamics in the bulk, so if you want to know what is happening in the bulk you have to find the solutions to the constraint equation (which is hard). Where they disagree is about the physical significance of the surface term. Dirac regards it as completely inessential to the important physics, DeWitt regards it as useful insofar as you want to choose a convention fo ascribing and energy to the entire universe, and R & T regard it as absolutely essential if one is going to have any chance at all to get any physics out of the theory.

    Do you agree or disagree with this assessment?

    Con't

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  114. The point is that as far as Dirac is concerned, all of the real physics is contained in the constraint equations, and in particular all of the real dynamics is contained in the Hamiltonian constraint. So it is perfectly natural for Dirac—and anyone following Dirac—to call the space of solutions to the constraints the "physical Hilbert space", as Smolin does. Even DeWitt could call the space of solutions to the constraints the "physical Hilbert space" if he regards the question of what the total energy of the universe is as merely a matter of convention rather than a matter of physical fact. But R & T would never call the space of solutions to the constraints the "physical Hilbert space". So if you pick up some locutions from Dirac and then import them into R & T you are going to get very confused. Which appears to me to be exactly what has happened.

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  115. Tim,


    I appreciate your effort to get into the details here, but a lot of what you write is not accurate. There are two pretty distinct issues. The first is the statement of Dirac quantization of arbitrary Lagrangian systems. The very well known and accepted procedure is that one 1) identifies constraints and computes a Hamiltonian 2) defines the physical Hilbert space as the kernel of the first class constraints 3) evolves physical states in time via H\psi = id\psi/dt. This basic framework holds for any theory, including gravity. I really doubt you want to argue about this.


    Next, we have the separate question of the canonical formulation of GR. Let's discuss this at the classical level, since that captures the issue. As we all know, the first class constraints are C_0(x) and C_i(x). Now, it is true that prior to the work of RT there was confusion about the correct Hamiltonian for GR. But everyone who works in GR now understands that the RT approach is the correct one, which for example is why that is what is discussed in Wald's book, which is the definitive reference for such things. The work of Dirac and DeWItt (on this specific topic) has been forgotten. As RT point out, the boundary term in H is not optional, since otherwise Einstein's equations do not arise from canonical equations of motion. That is, the entire point of the (classical) canonical formulation is the equations of motion should take the form dp/dt = - {p,H}, dq/dt = {q,H}, where (q,p) are the canonical variables and { , } is the Poisson (or more generally Dirac) bracket. What RT show in great detail is that the Poisson bracket of H with a phase space function is ill defined unless H contains a boundary term. That is, without a boundary term H is not a differentiable function on phase space; this is not a matter of "interpretation", but a cold mathematical fact. I am not trying to sound arrogant here, but this is really well known among people who work in gravity, and has been so for decades.

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  116. BHG—

    I'm afraid we do have to get into the details here, because I do not agree with your characterization of the situation, and one has to be especially careful. There are three different possibilities. One is that there are some first-class constraints, but they do not include a Hamiltonian constraint. The time evolution is then implemented via the Schrödinger equation in the usual way. A second possibility is that instead of a Schrödinger equation with a Hamiltonian operator all of the time dependence is folded into the Hamiltonian constraint. This is the only way to handle things in a completely general Relativistic setting. In this setting, there are no constraints at all on the foliation relative to which one defines the t-evolution. In other words, the slicing into equal-t slices is regarded and treated as a "choice of gauge", and all of the different g_mu,nu (x,t), for fixed t, are regarded as "gauge equivalent" because they arise from different coordinatizations of the same 4-manifold, and the choice of coordinates is merely conventional. In a typical standard presentation of the constrained Hamiltonian formalism (I can provide examples) the case of purely spacelike constraint equations plus the Schrödinger equation is strongly distinguished from the case of using a Hamiltonian constraint in favor of a Schrödinger equation. And we can see why. The Schrödinger equation requires a t-parameter, which in turn requires a preferred foliation of the space-time. In the absence of any canonical foliation, nothing like the regular Schrödinger equation can even be written down. As I understand it, Wheeler and DeWitt just went in whole hog here: they had no Schrödinger equation at all: they put all of the dynamics off into the Hamiltonian constraint. If you begin with data on a Cauchy surface, so the entire space-time lies in the domain of dependence of that surface and the Hamiltonian constraint yields the complete dynamics.

    Finally, there is the approach of R & T, which is a mixture: you split the Cauchy surface into two parts, one in the bulk and the other in the asymptotic regions that smoothly approach Minkowski space-time. You constrain the allowable coordinate systems to go over to Lorentz coordinates as you go to spatial infinity and then only have to worry about the Poincare transformations at infinity In this case you have a split dynamics: the Schrödinger equation for the wavefunction at infinity and the Hamiltonian constraint for the WDW patches in the bulk. As I mentioned, Dirac clearly thought that this former part, which is associated with the surface integral, was not important and could be just dropped, and R & T argue against that.

    Why is all this important? Because for someone interested only in the behavior in the bulk, all of this arguing about the boundary conditions has little point. By pushing the boundary far enough away I can render the bulk essentially insulated from to the boundary conditions. And having done that, the only physically interesting question is about the Hamiltonian constraint: how does it act in the bulk to generate the time evolution in a WDW patch? If the WDW patch is large enough, all of the interesting action occurs there, and the surface term is an inessential side show.

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  117. Now as I understand things, Dirac may be perfectly justified use to the term "physical space" for the solutions of the first-class constraints, because he was dealing with a theory in which the dynamics is implemented by the Schrödinger equation, and the set of solutions of all the equations of motion are in 1-to-1 correspondence with the data on a Cauchy surface that satisfies the spatial diffiomorphism constraints. And in Wheeler and DeWitt's original presentation, as I understand, it the "initial" data is given on an entire Cauchy slice and the Hamiltonian constraint plays the role of the Schrödinger equation for determining the time evolution (without preferring any particular foliation as the Schrödinger equation demands). So in that setting, the “physical space” is also the space of all solutions of all four constraints. That is consistent with what Smolin says.

    But In R & T’s approach to the asymptotically flat space and in the AdS spacetime (with its timelike spatial boundary) we have the dual dynamics: both a Hamiltonian constraint for the bulk and a Schrödinger equation for the wave-function at the boundary. So in this case, solving the constraints—even with the Hamiltonian constraint—alone does not yield the physical space. The physical space is the space of all complete 4-dimensional solutions of the dynamical equations. So although you may find contexts in which the physical space is the space of solutions to the spatial constraints (when the full dynamics is the Schrödinger equation) and although you may find contexts where the physical space is the space of all solutions to the constraints including the Hamiltonian constraint, it does not follow that the set of solutions to all the constraints (including the Hamiltonian constraint, but not the Schrödinger equation) is the physical space of AdS. Since your argument appears to require accepting that the solutions to the constraints without being solutions to the Schrödinger equation count as the physical space I reject your main premise. And for the reason given, citations from Dirac or Smolin or anyone working with Wheeler-DeWitt cannot be taken as evidence for your claim in the context of AdS or in the context of dealing with asymptotically flat space-times via the method of R & T.

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  118. Tim,

    Let's see whether your rather radical claims make sense at the level of equations, which is the standard of precision we demand in physics. There are two logically unrelated issues:

    1) Physical Hilbert space:

    Suppose I hand you a quantum system with canonically conjugate operators (q^i,p_j) (there could be a finite or infinite number of these), first class constraints C_a[q,p], and Hamiltonian H(q,p). For definiteness, let's take H=\sum_{ij f_{ij}(q)p^ip^j +V(q), to cover most cases of interest including gravity. Question: what is the physical Hilbert space? The textbooks tell us that this is given by the space of square integrable wavefunctions that obey C_a \psi(q) =0. Tim says differently. He says that to define the physical Hilbert space you need to first solve the equation H\psi = id\psi/dt to get the space of "global solutions" \psi_n(q,t). My questions to you: 1) what is the formula for the inner product on this space? 2) What is the rule for how the canonical momentum operator p_i acts on this space: p_i \psi(q,t) = ? Obviously, without answering these questions any claim to be defining the Hilbert space is vacuous. Please be mathematically precise, i.e. provide equations. I predict that in attempting to answer these questions you will either discover that your proposal is nonsense or land back at the standard prescription rephrased in an awkward way.

    2) Hamiltonian for GR

    Let's stick to the classical theory here so everything is perfectly well defined. The textbooks tell us that the Hamiltonian for GR in asymptotically flat space is H = \int d^3x N(x)C_0(x) + H_bndy. Tim says that we can drop the boundary term. My question: the most basic role of H is that should give us the equations of motion via dg_{ij/dt = -{g_{ij},H}, d\pi_{ij}/dt = [\pi_{ij}, H}, where { , } is the Poisson bracket. Please explain, at the level of equations, how to compute these Poisson brackets once you drop H_bndy. Again, I predict that you will find that your proposal is nonsense.

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  119. BHG—

    Radical claims? All I have been doing to quoting—often verbatim— the claims of Dirac and De Witt that are mentioned in the first section of R & T! It was Dirac and DeWitt who claim that in the case of asymptotically flat space-time the surface integral is essentially irrelevant for the equations of motion, not me. As for the rather odd claim that even in the case of gravity the dynamics is given by a Schrödinger equation rather than a Hamiltonian constraint (and that in such a case the Hamiltonian constraint somehow isn't a first class constraint) I can't at all imagine what you have in mind. As I asked you already, why do you think the Hamiltonian constraint is called the "Hamiltonian constraint" if not because it is a constraint that implements the Hamiltonian, i.e. the dynamics?

    As I have already said, if you have a Schrödinger equation *and no Hamiltonian constraint* and a well-posed initial value problem (so not with a timelike boundary such as AdS), then the space of initial states is in 1-to-1 correspondence with the set of global solutions, so the inner product on the space of initial states can be extended to the space of global solutions trivially. To calculate the inner product between two global solutions in this case just fix any time and take the inner product between the instantaneous states at that time. The unitarity of the Schrödinger evolution implies that this will be the same at all times.

    Regarding the boundary term you are making such a deal about. First, in a closed space-time there is no boundary term, of course. So you better not be too panicky about the boundary term. To repeat: it is Dirac and DeWitt who are dismissive about the physical significance of of boundary term.

    Tell you what: you seem not to recall what is in R & T. Why don't you sit down and reread it before we go on. You are very weirdly attributing to me as radical claims things that are coming straight out of the very paper you have recommended. And sit down and think through the difference between implementing the dynamics via a Schrödinger equation, and what that entails and requires, VS implementing it via a Hamiltonian constraint, and what that entails and requires. When you feel confident about all that we can make some progress.

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  120. Tim,

    You are avoiding the question. I am very familiar with all the technical details of RT, and in fact know this whole story inside out. Again: please explain how to compute {g_{ij},H} and {pi_{ij},H} for asymptotically flat space if H has no boundary term. Is this not a clear question? Show me some equations! If you really want to stand by your claim that the boundary term is anything but mandatory, then yes you are most definitely making a radical claim, (and a mathematically fallacious one). Dirac and DeWitt got it wrong originally, and RT corrected it, and everyone who knows anything about gravity has absorbed this -- just check the literature if you don't believe me. That's how science progresses.

    You also avoided the other question. Again: given a general quantum system as I stated it, do you really want to say that the physical Hilbert space is anything other than the space of square integrable wavefunctions \psi(q) that obey C_a \psi =0? If so, then you are definitely making a radical claim!

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  121. BHG- Look, it was you who strongly recommended that I go read Dirac! So if you want to say that Dirac got this wrong, that was a strange suggestion.

    And I completely answered your question about the inner product on the space of global solutions. What more do you want?

    How about answering my question about the necessity of using the Hamiltonian constraint when doing any quantum theory of gravity, or about how the freedom to foliate is being treated as a "gauge freedom", or about why the constraint is even called "Hamiltonian"?

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  122. Tim,

    My suggestion to read Dirac concerned his general treatment of the canonical quantization of constrained systems, not about his treatment of GR in asymptotically flat space. I apologize if this was unclear.

    Now I promise I will answer your questions after you answer mine. Once more:

    1) show me how to compute the Poisson brackets when H contains no boundary term? Or, just admit that what you are claiming is wrong, and then we can move on, as the entire GR community did some 50 years ago.

    2) Give me any example of a quantum theory where the physical Hilbert space is anything but the space of square integrable \psi(q) that are annihilated by the first class constraints. I haven't gotten a clear statement about this yet.

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  123. BHG—

    1) Obviously there is a way to compute the Poisson brackets for a closed universe with no boundary, yes? That is what R & T say. As for what to do in an open universe, I am not questioning R & T, so I have no idea what you are after here.

    2) Gravitational theory in AdS, where the entire space-time is not in the domain of dependence of any spacelike hypersurface. That kind of the point I am making.

    Now my answers please.

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  124. Tim,

    1) Great, so you now agree that in asymptotically flat spacetime the boundary term in H is absolutely crucial, since otherwise the Poisson brackets don't exist, and hence Hamilton's equations are ill defined (that is the content of RT)? I am glad that you now agree, even though shortly before you were arguing that the boundary term was something other than mandatory. If you agree that the boundary term is vital then I am happy to move on, so please indicate that you do agree with this.

    2) I need to press you a bit further here so I can understand your claim. If, in the general class of QM theories I was considering, (which includes gravity) I hand you a wavefunction \psi(q) which is square integrable and obeys C_a \psi =0, are you saying that there is some *additional* condition I need to impose in order that \psi(q) is in the physical Hilbert space, and if so which one? Note that the \psi(q) I handed you is a function only of q and nothing else. The reason I am baffled by what you are after here is that you have made reference to "global solutions", but this adds nothing new, since every \psi(q) as I have defined it above yields such a global solution (i.e. it is initial data for such a solution).

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  125. Tim,

    I believe the following addresses all your questions, but if not, ask again. It's useful to distinguish the classical and QM theories

    1) Classical GR: We have canonical variables (g_{ij}(x),\pi_{ij}(x)), constraint equations C_0[g,\pi] and C_i[g,\pi], and the Hamiltonian H = \int d^3x (N_0 C_0+ N_i C_i) +H_{bndy}. Einstein's equations are equivalent to the set of equations C_0=0, C_i=0, dg_{ij/dt = -{g_{ij},H}, d\pi_{ij/dt = {\pi_{ij},H}. The infinite number of constraints C_0(x) (one for each x) are called Hamiltonian constraints because the smeared constraint \int d^3x f(x) C_0(x), with f(x) of compact support, generates a local time deformation of a hypersurface; i.e. a displacement by t -> t+ f(x). Alternatively, the "Lie algebra" (quotes because it's not really a Lie algebra) of the constraints is related to the Lie algebra of diffeomorphisms (on-shell), with C_0 corresponding to diffs by timelike vector fields.

    2) Quantum GR: We have canonically conjugate operators (g_{ij}(x), \pi_{ij}(x), constraint operators C_0[g,\pi] and C_i[g,\pi], and the Hamiltonian H = \int d^3x (N_0 C_0+ N_i C_i) +H_{bndy}. Physical state wavefunctions \psi[g_{ij}] obey C_0 \psi = C_i \psi =0, and the physical Hilbert space is the space of such wavefunctions that are normalizable relative to an appropriate inner product. Physical state wavefunctions evolve in asymptotic time according to d\psi/dt = -iH \psi. Solutions to the constraint equations C_0 \psi = C_i \psi =0 encode all information about the physics occurring in a WDW patch at a given asymptotic time. The Schrodinger equation then allows you to go from one asymptotic time to another.

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  126. BHG—

    Excellent! So we agree that in a WDW patch—which is a 4-dimensional chunk of space-time, and can be taken, for the appropriate boundary time, to contain the entire history of an evaporating black hole from formation to evaporation—all of the dynamics is contained in the Hamiltonian constraint, and none of it in the Schrödinger evolution. And we agree that any global solution that contains such an evaporating black hole must contain such a WDW patch, with the solution before and after the patch being consistent with the solution in the patch. In particular, if the maximal space-like hypersurfaces go from being connected to being disconnected, then this will show up in the relevant WDW patch: earlier maximal space-like hypersurfaces being connected and later ones being disconnected. And if we want to know what are the physically possible states on the disconnected surfaces we have to determine which states can arise through the evaporation process.

    Now all along, I have been trying to understand why you are giving so much weight to the surface term in the Hamiltonian, and I still can't understand at all why you think it is very important. In essence, the requirements on an symptomatically flat space-time are so severe that the surface term really only carries a very, very limited amount of information about what goes on in the bulk, which is what we care about. In fact, all that can be recovered from the surface term is the net mass, linear momentum and angular momentum of the bulk, as referred to a given Lorentz frame "at infinity". So basically seven numbers, all of which are conserved in the asymptotically flat case. But what we want to understand about evaporating black holes has nothing to do with those seven numbers. What we want to know depends on how the constraints can be satisfied in the WDW patch, which is a problem you will get no help on at all from the surface term.

    I think that this is all rather obscured by this sentence: Physical state wavefunctions evolve in asymptotic time according to d\psi/dt = -iH \psi. The "t" in that equation is only even defined at infinity. If you pick a t you can calculate the surface term of the Hamiltonian, but if you ask "What is going on in the bulk at t" it is not merely that you can't calculate the answer but that you have not even asked a sensible question. What is "going on in the bulk at t" depends on how the hypersurface of t = constant, which is well-defined at infinity, is continued into the bulk, and that can be done infinitely many ways corresponding to different foliations of the WDW patch.

    And I can only repeat, yet again, that it is not true that the Hamiltonian, thought of as the generator to temporal evolution, is just the surface term. To get the full temporal structure of a complete global solution one needs the surface term to deal with the evolution of psi at infinity but also the form of the Hamiltonian that is encoded in the definition of the Hamiltonian constraint to get the temporal development in the WDW patch.

    I somehow anticipate that you are going to object to what I just wrote, but I cannot see a reason in the world to. As for the definition of the Poisson brackets, I repeat again that it can evidently be done in a closed universe with no surface term, so the surface term cannot be essential. Indeed, we could always replace an asymptotically flat space-time with a closed space-time by identification of points "at infinity", can't we? And in that case the Poisson brackets must still be fine, right?

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  127. Tim,

    What you write is basically accurate. But does it help advance the narrative? The following sounds like a great strategy: 1) choose an asymptotic time such that the WDW patch includes the black hole formation/evaporation event 2) solve the constraint equations to find the wavefunction describing this 3) extract the physics. Awesome. Just a few tiny problems. First, the constraints are hopelessly divergent as quantum operators, and giving them a precise definition is a very hard problem by itself, and probably impossible since we expect a metric based description to break down near the singularity. Even if you could define the constraint operators, you face the problem that the equation C_0 \psi = 0 is a nonlinear functional differential equation, so your chance of solving it exactly is essentially nil. Even if you could solve it, to extract physics you will have to find a way to identify an appropriate local time variable, which is itself very difficult. We can dream that someday these steps will be carried out, but for now it is beyond reach (many many papers on this notwithstanding) and so wild speculation runs rampant. If you want to make progress now you need a backdoor approach, which is what AdS/CFT provides. Given what we currently understand, to leverage this you have to focus on general constraints implied by the existence of the duality, and furthermore focus on the structure of quantum states at early and late times, where things simplify. As I have explained, once you go down this route you see that standard principles of quantization plus the existence of AdS/CFT rule out your (i.e. Hawking's) scenario, but if you don't want to engage with this then I can't force you, obviously.

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  128. BHG—
    You really need to think about the position you are advocating here. We are trying to figure out what may happen when a black hole evaporates. Your position is that it is essentially impossible to solve that problem by direct computation from the dynamics of what will happen, but nonetheless there is a "backdoor" approach that somehow can arrive at the answer without doing the hard work of solving the equation. Even more, it can somehow arrive at the answer without even taking any account of the dynamics in the bulk. The only dynamics you seem to want to consider is the Schrödinger equation as it applies at the boundary, but the wavefunction there simply does not carry information about the dynamics in the bulk. So this approach is not analysis of physics: it is just magic.

    As I have already said, one way to implement the solution is to postulate that at the evaporation event the manifold structure fails. It is that feature that defeats Geroch's theorem and allows the solution to make sense. But as far as I can tell, this whole approach simply requires that the space-time be a manifold. So you have ruled out my solution at step one, without any consideration of dynamics or constraints or anything else. On what basis are you allowed to do this? Can you prove, in a non-question-begging way, that the space-time is a manifold at the evaporation event? Whatever the evidence for AdS/CFT is, does it bear at all on this question?

    To be specific, is there any evidence at all for AdS/CFT that pertains to the situation where a black hole evaporates? That is the only situation in which there are grounds for thinking that the manifold structure must fail. (Of course, many approaches to quantum gravity would insist that the space-time is not a manifold anywhere: it is just well-approximated by a manifold at the appropriate scale. Do you have any reason to think otherwise?)

    R & T don't mention AdS as far as I can tell: just the asymptotically flat case. And AdS has some special characteristics, including not being globally hyperbolic, that make it more complicated to deal with in any case. But I can't see any plausibility at all that consideration of the evolution at infinity of the wavefunction can shed a bit of light on our problem.

    Maybe you want to explain what grounds we have to believe AdS/CFT in general, and then the grounds to believe it in this specific case. If you simply assert that there is nothing special about this case, despite the fact that it contains an evaporating black hole, then again the entire argument is question-begging.

    Further, we have established (I hope) that the physical space should at least be isomorphic to the space of complete solutions to all the dynamical equations. If you want to continue to dispute this, please offer a non-question-begging reason. For example, do not just give examples where the space of elements of the kinematic Hilbert space that satisfy some constraints is itself isomorphic to the space of global solutions. Those examples can in principle do nothing to establish your definition of the physical space.

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  129. Tim,


    I have explained in detail how your proposal is inconsistent with AdS/CFT, and I frankly don't see any counterarguments, just noise. For example, it is a universally accepted fact about Dirac quantization that the physical Hilbert space is the space of normalizable wavefunctions annihilated by the first class constraints. You are trying to obscure this simple and well known fact by making some either irrelevant or wrong (it is hard to tell which because you won't supply details) that you need to consider the "global space of solutions" to define the physical Hilbert space. No. How about this: show me any reputable reference which argues that this is needed as opposed to what I and every standard reference I know of claims? As soon as you sign on to the standard statement, it becomes obvious that the physical Hilbert space at late times in your approach is a tensor product since the slices are all disconnected. And since you have (finally) agreed that the Hamiltonian acting on physical states is a boundary term, it is inevitable that the spectrum of the Hamiltonian is highly degenerate, in contradiction to the CFT. You call this argument "magic", but in fact it is just logical reasoning.

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  130. BHG—

    If you don't see any counterarguments then you are not paying any attention. I keep giving them.

    The "physical space" should be the space of all things that could actually happen according to a given theory. If you you want to deny that, please supply the argument.

    To be something that could physically happen according to a theory, there must be a global solution to all of the dynamical equations of the theory in which it happens. If you want to deny that, please supply an argument.

    Therefore the "physical space" is in the first place, by definition, the space of global solutions. Then, derivatively, if the space of global solutions is isomorphic in some specified way to some other space (e.g. the space of initial conditions in a well-posed mathematical problem), one can study this second space to get information about the physical space.

    If you want to deny this, then you are in the bizarre position of trying to claim that for some X, X is physically possible even though there is no global solution in which X occurs. Do you really want to make this argument?

    What we are interested in here is the physical space of maximal post-evaporation states. That is, the set of physical possibilities for maximal space-like hypersurfaces post-evaporation that are the sorts of things that could conserve information about the pre-evaporation situation. Any post-evaporation state that is not maximal in this way is not of interest: of course it could lose information on account of the non-maximality. And any state that could not arise as a post-evaporation state is also of no interest.

    I don't want a "reputable reference" here: I want an argument. If you disagree with anything I have written above, locate it and make an argument. As for your "reputable references", I will wager that every single one that identifies the space of solutions to the first-class constraints as the physical space is one where the set of solutions to the first class constraints is isomorphic to the space of global solutions to all of the dynamical equations. And in such a case, the reputable reference is not rejecting my definition in favor of yours, it is just supplying examples that meet both definitions. So what you would need in any case is a "reputable reference" that explicitly rejects my criterion in case where the two definitions come apart. You can at least supply such an example.

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  131. Tim,

    The term "physical Hilbert space" has precise mathematical meaning, and its construction in Dirac quantization is defined precisely as I have stated it. You simply deny this fact. That appears to be the basis of your argument. You are wrong, and seemingly unperturbed that every reference on this topic agrees with what I am saying.

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  132. BHG—

    Do you want to a) deny either of the principles I annunciated? or b) Provide an explicit example where an example given in a reference work identifies a "physical Hilbert space" that is *not* isomorphic to the set of global solutions?

    If you don't want to do either a or b then you are conceding everything I have said. AdS is an unusual case, and you have to think about how to extend the notion to that case, not just blindly follow an algorithm without thinking.

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  133. Tim,

    All I am doing here is insisting on using standard terminology so that we don't get into Gauss' law vs. Gauss' theorem type confusions (remember that?). For reasons that remain baffling to me, you want to introduce a notion of "the space of global solutions" into the construction of the physical Hilbert space, even though this is totally unnecessary, nonstandard, and will only serve to confuse. Bottom line: the argument I am presenting does not refer to such a thing. I am going to stick with the standard definition and construction of the physical Hilbert space, since I see no reason to do otherwise. Now, getting back to the argument, if the spatial slices are all disconnected, it follows that the space of solutions of C_0(x)\psi= C_i(x)\psi=0 is a tensor product, hence the physical Hilbert space is a tensor product.

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  134. BHG—

    Obviously, the issue is not terminological, it is conceptual. You can use the phrase "physical Hilbert space" however you like, but if what you you have defined does not correspond to the space of global solutions—i.e. the space of things that could physically occur in space-time according to the theory—then who cares what it is? You ironically fail to see that the sort of dismissive attitude you took about the kinematical Hilbert space can be repressed here with just as much validity about what you choose to call the "physical" Hilbert space. If it contains state that could not arise from the evaporation of a black hole then there is not a reason in the world to care about its mathematical structure. And if it contains states that are not late-time states in any global solutions then it contains unphysical states (in the relevant sense) no matter what you choose to call them.

    You have managed to convince yourself that there is some almost trivial end-run around the hard physical problem of working out what they theory implies about evaporating black holes without actually bothering to specify even a single actual solution containing an evaporating black hole. That has advantages, as Russell would say, all the advantages of theft over honest toil. By simply appending the words "physical Hilbert space" to something that has not been shown to reflect the structure of the actual relevant global solutions to all of the physical laws you think that you have avoided dealing with those laws. But it is obvious that you have not.

    In short, there is a space of solutions to *merely* the constraints on a disconnected spacelike hypersurface. That space sure is a tensor product space. But since we have exactly zero reason to think that that is the space of late-time global solutions that arise from an evaporating black hole, the fact that it is a tensor product space has no significance at all. And no amount of arguing about nomenclature will change that a whit.

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  135. Does this help:

    BHG sees no reason to "do the hard work" Tim is talking about here, because what he cares about is showing that AdS/CFT is incompatible with the Hawking/Maudlin scenario, and he thinks that to show this, it is sufficient to point out that the late-time Cauchy slices are disconnected. Why is this sufficient? Because the boundary Hamiltonian doesn't care what happens on the disconnected piece, which means that there will be degeneracy in the spectrum of late-time solutions, which is not possible if we trust the AdS/CFT correspondence, because the spectrum of CFT solutions is for sure non-degenerate. What this shows for BHG, I take it, is that *given AdS/CFT, the late-time Cauchy slices are simply not disconnected*. So the Hawking/Maudlin proposal is off the table. From an earlier post of his: "[T]he AdS/CFT "prediction" is that such disconnected Cauchy surfaces never arise: i.e. a connected Cauchy surface will never dynamically evolve to a disconnected one."

    Tim by contrast thinks that one should not assume that there is a true degeneracy in the spectrum of the late-time solutions, because what matters is the space of late-time solutions *that evolved from earlier black hole formation and evaporation*, and we don't know that the structure of THIS space permits the energy degeneracy that rules out correspondence with the CFT.

    I can't weigh in on the question of whose reasoning here is correct from a physical perspective. But if both agree that these are the two positions, two sorts of progress are possible. (1) Shake hands and call it a day (year); or (2) look for new ways to cast doubt on the opponent's line of reasoning.

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  136. Tim,

    You are evidently missing the point of the argument, but perhaps a little rephrasing will help. Forgetting about black hole evaporation for a moment, let's ask about the physical Hilbert space of quantum gravity in AdS, and let's phrase this in the WDW language of wavefunctions of metrics on spatial surfaces. A basic question we need to address is: do we allow the wavefunction to have support only on connected surfaces or do we allow disconnected ones as well? Suppose that we do allow a subspace consisting of wavefunctions for surfaces with two disconnected components. This subspace of the physical Hilbert space clearly has a tensor product structure, and it then follows that the Hamiltonian has a hugely degenerate spectrum just from the states in this subspace. Next, we should ask whether we are in fact required to include this disconnected subspace. That is, might it be that there are superselection sectors, so that if we start with a state in the connected sector, evolution in asymptotic time according to the Hamiltonian will preserve this. If so, then we can construct a self consistent theory in which only connected surfaces appear in the wavefunction. However, suppose we start in such a connected spatial surface and then form a black hole that evaporates. According to the standard Penrose diagram the spatial surfaces will dynamically split into disconnected surfaces. So if we trust the standard Penrose diagram, we are forced to include the disconnected surfaces into our Hilbert space. But then, as above, the Hamiltonian must have a hugely degenerate spectrum.

    So, irrespective of AdS/CFT, if you believe the standard Penrose diagram then you will conclude that the Hamiltonian has a degenerate spectrum. Note here that I never needed to say anything about whether the "global space of solutions" corresponding to black hole formation evaporation is or is not a tensor product space; this is not a good way to think about these matters.

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  137. Carl3:

    That is spot on. And all I can add is that the space of possible final states after the evaporation is *obviously* the only one that could be relevant to BHG's argument. As we know, the argument turns on certain peculiarities that are claimed for the AdS theory of gravity, namely that there is either no or very little degeneracy in the spectrum. I have actually never been convinced of this claim, and would go back to explaining why it is at least prima facie physically unacceptable, but let's grant it for the moment. What follows from the low degeneracy of the spectrum is that that Hilbert space of physically possible initial state is not a tensor product space. So if the evolution is unitary, the space of physically possible final states—but note! final states *that evolved from the initial Hilbert space*!—must also not be a tensor product space. Well OK, even if I grant all that, no harm and no foul to my solution. Because to determine the structure of the space of possible final states of the evolution through an evaporating black hole, one actually has to figure out how the evolution goes! There is, in particular, no grounds to believe that every possible physical state on the final hypersurface (whether it be connected or disconnected), i.e. every state that satisfies the constraints that can be applied at that time, is a state that could have arisen by physical evolution from a connected initial state through the black hole formation and evaporation. This is *certainly* not a well-posed initial value problem with a reversible dynamics: AdS is not globally hyperbolic so that is not in the cards. If it were such a problem, one could try to argue that there must be an isomorphism of structure between the space of initial states that satisfy the constraints and the space of final states that satisfy the constraints. But that is just no good here, since AdS precisely *does not* have the right form for this.

    The conceptual trap arises because the time evolution in a normal case is very unlike the AdS situation in lots of ways. One is that AdS is not globally hyperbolic, as has been mentioned. Another is that the time evolution in AdS gravity theory is implemented in a dual way: part by imposing the Hamiltonian constraint and part by solving a Schrödinger equation at the boundary. Usually you would do either one or the other but not both. As BHG acknowledges, nobody can solve the Hamiltonian constraint! But solving the Schrödinger part is child's play: the boundary situation is stipulated to be especially simple. If you get distracted, you could think that the whole of the time evolution just was the Schrödinger evolution at the boundary, and the Hamiltonian constraint evolution in the bulk is just irrelevant. But that makes no sense: is the the evolution in the bulk that contains the formation and evaporation of the black hole. Until you confront that directly you cannot draw any conclusions about the space of physically possible end states of the black hole evaporation.

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  138. BHG,

    Please read my reply to Carl3, so I don't have to repeat myself. Of course it is trivial that the spectrum of all states that satisfy the spacelike constraints on a pair of disconnected hypersurfaces will be massively degenerate! No one would dream of denying that, and, as you say, that has nothing to do with AdS/CFT. But so what? First of all, it is patently clear that that space of states is not isomorphic to the space of states that could arise from a connected initial hypersurface via black hole formation and evaporation. For example, a vacuum state on both of the disconnected surfaces is going to solve all the space-like constraints, but that can't possibly be the outcome of black hole formation and evaporation: the Hawking radiation, at least, must still be around somewhere!

    That is proof positive that the set of states that obey the space-like constraint equations on disconnected hypersurfaces is not the space of states that can arise via black hole evaporation.

    And that blows up the argument. You have a perfectly good argument that an irrelevant space of states has a tensor product structure and no argument at all that the relevant space of states—the states that could be the end states of the evaporation—has that structure. But your argument relies on the relevant space having that structure. So this ambiguity about the precise meaning and significance of calling the space of solutions of the space-like constraints "physical Hilbert space" is the Achilles' heel of your argument. If you insist on stipulating that all "physical" means is the space of states that satisfy the spacelike constraints, then I certainly grant that the so-defined "physical Hilbert space" is massively degenerate with respect to the Hamiltonian. But since the so-defined "physical Hilbert space" is not the one you need for your argument, I don't care. And if you go the other way (the way I preferred to use the language, but who cares about that?) and call the space of physically possible end states of a black hole evaporation the "physical Hilbert space", then I grant that proving it to be massively degenerate would at least raise a problem. But you have come nowhere in the vicinity of proving that.

    And this logical situation changes not a whit even if I grant you AdS/CFT and the Holographic Hypothesis to boot, even though I am extremely skeptical of both of them. Even if you can prove that the Hamiltonian of the CFT is non-degenerate at all times, all that shows is that relative to the space of bulk states *that could evolve from the initial set of states on a connected hypersurface that solve the constraints* the Hamiltonian of the bulk is non-degenerate. But, as we have seen, that space of states certainly is *not* the space of all states on disconnected hypersurfaces that solve the space-like constraints. So no matter how you decide to use the term "physical Hilbert space", the space you can prove to be a tensor product is not the space you need for your argument. and the space you need for your argument is not the space you can prove to be a tensor product.

    The dream of coming to terms with this without actually solving the bulk equations is just that: a dream. And even granting the AdS/CFT full duality it is still a dream. Until a dictionary that allows one to reconstruct the situation in the bulk is developed and proven to be accurate, AdS/CFT just does not address the relevant question at all.

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  139. Tim,

    You are still missing my point. Whatever "quantum gravity in AdS" means, it is a quantum theory. Every quantum theory has a physical Hilbert space, by definition. The issue at hand is: what is the structure of this Hilbert space? For this or any theory, a completely well defined question is the following. Does the physical Hilbert space have as a subspace a tensor product space in which the Hamiltonian acts only one one factor? How would you answer this question for gravity in AdS, given that, according to you, disconnected surfaces arise? That is what I am asking. I am not asking a question about the structure of the space of states arising from black hole evaporation, which is what you are focusing on. I am asking about the full physical Hilbert space. In my argument, the role of the standard Penrose diagram for black hole evaporation is simply to show that disconnected surfaces arise dynamically. If I could get an answer from you on this that would definitely move things forward. As you know, my answer is that since the physical Hilbert is by definition the space of solutions of the constraint equations, it immediately follows that the physical Hilbert space does have a subspace of the type noted above. If you have a contrary argument, I am very receptive to hearing it.

    P.S. your comments about the non global hyperbolicity of AdS are way off-base, as you would quickly learn by glancing at any the AdS/CFT textbooks, of which there are now several. All this means is that you need to impose boundary conditions, and indeed there is a unique (slight but irrelevant lie here) choice of boundary conditions that preserves the SO(4,2) symmetry. Time evolution is then unambiguously defined. Fact is, AdS actually has a much simpler structure in this regard than asymptotically flat space, because it functions as a box. Anyway, if you think there is anything mysterious or poorly understood about time evolution in AdS you are deeply mistaken. At the very least, please try to look into the textbooks treatments of this point before raising it again.

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  140. Carl,

    I agree with your synopsis. Indeed, I am arguing that if quantum gravity in AdS includes disconnected spatial surfaces, then its Hilbert space contains a subspace which is a tensor product, with one factor for each connected component. That just follows from the standard quantization rules. I am missing something, or has Tim, or anyone else, failed to offer any counterargument to this?

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  141. BHG—

    We are back again to the issue about the Hamiltonian! You ask: "Does the physical Hilbert space have as a subspace a tensor product space in which the Hamiltonian acts only one one factor?" Well, the answer to that question is, of course "No". The Hamiltonian, as the generator of time evolution, acts on everything! It is just that in the bulk, that action is implemented by the Hamiltonian constraint, and at the boundary by a Schrödinger equation. You have, from the very beginning, been under the misapprehension that the Hamiltonian constraint is somehow not part of the theory, or not part of the way that time evolution is implemented in the theory. I asked you a while ago why you think the Hamiltonian constraint is called "Hamiltonian", and what relationship you think it has to the Hamiltonian,and why it is represent by H_0, and you never answered.

    The fact that the time evolution in AdS gravity is implemented both by the Hamiltonian constraint for the WDW patches and a Schrödinger equation at the boundary (with boundary conditions) has everything to do with AdS not being globally hyperbolic. Since it is not globally hyperbolic, there is no spacelike hypersurface whose domain of dependence is the entire spacetime. If there were, and we put data on that hypersurface, then the whole dynamics would be in the Hamiltonian constraint. Since there isn't, there has to be more than the Hamiltonian constraint, hence the Hamiltonian at the boundary. R & T argue—contra both Dirac and DeWitt—that the surface term is important even in an symptomatically flat space-time which is globally hyperbolic. That may be. But even if it is, there is obviously no prospect of just using the Hamiltonian constraint in a non-globally-hyperbolic setting, yes? Are you really disputing this?

    I did not say anything was mysterious. I don't find the need to use both the Hamiltonian constraint and a Schrödinger equation at all mysterious. What I find mysterious is why you think that anything you have pointed out constitutes an argument against my solution.

    I really don't understand what question it is you think I am avoiding. Every quantum theory has solutions to the dynamical equations and constraints that contain disconnected inextendible space-like hypersurfaces: just take the union of two disconnected solutions! According to you, does the Hamiltonian in this case "act on only one factor"? Or not? If you are not concerned with global solutions, which you insist you are not, then your questions are all trivial, and every theory will contain a physical space in your sense that is a tensor product space. So you apparently have in mind some unstated extra condition, but that condition is not (as I would have it) that we restrict attention to states that could arise from an evaporating black hole. What is this extra constraint? If there is none, then yes the "physical Hilbert space" as you define it for a pair of disconnected hypersurfaces is a tensor product space. So what? That is both obvious and trivial. How do you claim it bears on our problem?

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  142. Tim,

    First, I certainly did answer the question you claim I was avoiding -- go back and look for yourself. I have to say I find your comment that I am under the "misapprehension that the Hamiltonian constraint is somehow not part of the theory," and similar comments unnecessary and insulting (and of course wrong), but I will not respond in kind. Parenthetically, I will note that the term "Hamiltonian constraint" is not a very good one, since it may suggest to the unwitting that the Hamiltonian itself is a constraint, which of course is only true in a closed universe. But the term is there for historical reasons I guess.

    I will rephrase my question in a technically precise way, since apparently my wording was not sufficiently clear. Consider the physical Hilbert space of our theory, and the Hamiltonian operator, Ham. We ask: does the physical Hilbert space have a subspace, call it H_sub, with the following properties. 1) H_sub is a tensor product, H_sub = H_1 x H_2. This means that we can choose a basis of states of the form \psi_1 x \psi_2 and linear combinations of such states span H_sub. 2) The Hamiltonian restricted to this subspace has the form Ham = 1 x Ham_2, where 1 is the identity operator. More precisely, let \psi = psi_1 x \psi_2 and \chi = \chi_1 x \chi_2 be two basis states for H_sub. Consider the matrix element <\psi|Ham|\chi>, The question is: do we have <\psi|Ham|\chi> = <\psi_2 | Ham_2 | \chi_2> for all such states \psi and \chi? If yes, then I define this as saying that "the Hamiltonian in this subspace acts only on H_2".


    Given this precise statement, do you concur that that physical Hilbert space of quantum gravity contains a subspace which is a tensor product with the Hamiltonian acting on only one factor? Given what you have said about disconnected surfaces appearing, and given what we know about the structure of Ham, it seems clear to me that you will answer "yes", but I want to make sure.

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  143. BHG-

    I apologize for the tone. I will try to avoid that.

    The problem with your question is that your use of the term "the Hamiltonian" strikes me as misleading, in exactly the way that has led to so much lack of progress. The point of my asking about the Hamiltonian constraint is fairly simple. If one starts with, or restricts one's attention to, some coordinate system or coordinate systems, with all of the wavefunctions under consideration sharing the same t-coordinate (that is, we fix a spacelike foliation and parameterize it by t), then the dynamics of the theory will be represented by an operator, the Hamiltonian operator. That operator is the generator of time translations, and so given data on one hypersurface it will generate the t-development of that data within the domain of dependence of the initial hypersurface.

    On the other hand, if one does not start with, and is not restricted to such a foliation, then the time development of the theory must be handled with a different mathematical tool. This is what happens in GR, where all possible spacelke foliations are on the same footing, at least within the bulk. In that case, one does not use a Hamiltonian operator but a Hamiltonian constraint. This is mandatory because there is no univocal foliation into spacelike hyperplanes, and so no univocal parameterization of such a foliation. There just is no univocal t-coordinate to work with. This is just a different method of implementing the dynamics of the theory, appropriate to a setting with diffeomorphism invariance. (The diffeomorphism invariance in GR is referred to as a "gauge freedom". And your references to "small gauge transformations" and "large gauge transformations" is another reference to this: the group of all diffeomorphisms is divided into a set that leaves the boundary regions untouched, i.e. goes over to the identity as one goes to spatial infinity or to the timelike boundary, and the rest that do not leave the boundary invariant. The reason we restrict attention to the "small gauge transformations" is so there can be a univocal t-parameterization at the boundary, and we can then construct a Hamiltonian operator there, instead of needing the Hamiltonian constraint.)

    In the first case, the Hamiltonian operator operates on hypersurfaces to generate the state on later or earlier hypersurfaces, all indexed by the variable t. In the second case, the constraint is applied to complete wavefunctions Psi(x,t) defined over all x and t and the those complete wavefunctions that satisfy the constraint are annihilated by the Hamiltonian constraint, i.e. they are eigenstates of eigenvalue 0.

    In the first setting, it is natural to call the Hamiltonian operator "the Hamiltonian". In the second setting, it is natural to call the Hamiltonian constraint "the Hamiltonian". The two are intimately related mathematically.

    Con't

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  144. AdS or asymptotically flat space-time requires them both. In the bulk, the temporal evolution is generated from a constraint, with no restriction on the coordinates used. At the boundary, the allowable t-coordinates are restricted so there is a Hamiltonian that takes the form of an operator. As nearly as I can tell, your question is about this operator and this operator alone. It operates only on the boundary states, and generates the time evolution at the boundary. It cannot generate the temporal evolution in the bulk. So the operator Ham is only part of the full Hamiltonian. This is what I keep pointing out. The full Hamiltonian is Ham together with the Hamiltonian constraint.

    So based your narrow reading of "the Hamiltonian" as just the boundary operator, yes, I agree. Based on my wide reading of "the Hamiltonian" as including both the boundary operator and the bulk constraint, no, I disagree.

    As long as we are perfectly clear about what I agree to and what I do not agree to, we are fine. Just be sure that whatever argument you are trying to construct appeals only to the boundary operator and not what I would call the full Hamiltonian. Without the full Hamiltonian, we do not have a theory that can handle black hole evaporation.

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  145. Tim,

    1) In GR we have the option of imposing coordinate conditions (choosing a gauge) and solving the constraints to get a description purely in terms of physical degrees of freedom. In practice, this is what is done in most real world applications of GR; e.g. in astrophysics and cosmology. It is justified as long as one's choice of coordinates do not become singular. A possible objection, which I am sympathetic to, is that this approach might break down in the presence of strong quantum fluctuations, as near a black hole singularity, and I am sympathetic to this. However, away from such exotic regimes, it is perfectly acceptable to work with a formulation in which constraints are absent, having been eliminated at a prior stage.

    2) In AdS or flat space, I want to again stress that the boundary term is a mandatory part of the Hamiltonian, and so there is no sense in which the Hamiltonian is a constraint in these instances, if by Hamiltonian one means an operator which generates time translations including those which are nontrivial at the boundary. This is what RT showed. It is of course true that the Hamiltonian includes constraint terms, but it also has a non-constraint term. It would only be justified to drop the boundary term if you decide that you want to work at one fixed boundary time which does not change.

    3) Now to my main point. In a quantum theory of gravity, all objects that have physical meaning can ultimately be expressed as matrix elements of physical operators (i.e. those that commute with the constraints) between physical states. A matrix element between unphysical states can never be measured and has no definite meaning. So, given our Hamiltonian H = \int (N_0 C_0 + N_i C_i) + H_bndy, all properties of H are expressed via matrix elements <\chi_\phys|H|\psi_phys>. So I'm certainly happy to include the volume terms in H, but do insist that only physical matrix elements are ... physical.

    With all this in mind, the statement is now that if the theory includes disconnected surfaces, then the spectrum of the full Hamiltonian H = \int (N_0 C_0 + N_i C_i) + H_bndy is hugely degenerate. If we can agree on that, then definite progress will have been made.

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  146. BHG—

    Some time ago I mentioned that this is an area I have thought a lot about. So expect things to get highly non-trivial here. The main point is this:

    3), although it is commonly asserted, is false. Indeed, it is 3)s falsehood that led to the so-called "problem of time" in quantum gravity, which was asserted before R & T was written and when Wheeleer and DeWitt were using a formal apparatus that had only the Hamiltonian constraint with no boundary term. (BTW: how do you think you handle gravity in a closed universe, with no boundary?)

    I am prepared to defend this claim in great detail, but I don't have the time to start that project at the moment. So this is just to get us thinking about the right issues. And the main issue is this: when Dirac started this whole business of having the constrained Hamiltonian formalism, and was thinking about gauge transformations, he was thinking about EM gauge transformation of the scalar and vector potentials. Those transformations yield a new mathematical representation of *the very same physical state*. So you want to demand that all of the gauge-related mathematical representations be treated as physically equivalent: it is the equivalence class under gauge transformations that represents a physical state. (It is that, or else fix gauge completely by some condition, if there is one that will do the job.)

    Now in GR people quite blindly took all of these techniques and applied them to re-coordinatazations of the space-time, as if that were physically the same thing as a gauge transformation in EM. And just as one can talk, rather sloppily and misleadingly, of the EM gauge freedom as arising from a "symmetry", they talk about this freedom to choose coordinates on space-time as due to a "diffeomorphism symmetry". And all that has led to the very claim 3) which is conceptual disaster.

    That is all I have time for now, but this is a warning that we have finally gotten to the conceptual errors that lie at the center of a lot of mistakes in discussing quantum gravity. So be prepared for a lot of very detailed and precise pushback about things you are used to thinking of as gospel. I will be raising points that are never discussed in the literature you are familiar with.

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  147. BHG—

    I wrote a brief response that has not yet been posted. Perhaps Sabine will find it and post it yet. If not, I will try to reconstruct it. In short, it said that we have now hit the center of a bunch of conceptual confusions that have made discussion of quantum gravity well nigh incoherent for decades. So you should be prepared to have many claims that you take to be well-established questioned, and be wiling to actually defend them rather than just cite someone else.

    Your point 3) provides a perfect example. You write "In a quantum theory of gravity, all objects that have physical meaning can ultimately be expressed as matrix elements of physical operators (i.e. those that commute with the constraints) between physical states" and add " So I'm certainly happy to include the volume terms in H, but do insist that only physical matrix elements are ... physical." Well, what you insist on I flatly reject. So let's get into the brass tacks here.

    As I have said, the time evolution in the WDW patches is expressed in the Hamiltonian constraint N_0 C_0. It has to be expressed as a constraint rather than as a Schrödinger equation because the coordinatization is being treated as a gauge freedom, and hence the foliation into space-like hypersurfaces of constant t value is being treated as a gauge freedom, so there just is no univocal t variable to frame a Schrödinger equation in terms of. You can use a Schrödinger equation at the boundary because the coordinate system/foliation has been regimented there.

    So to demand that anything with "physical meaning" be represented by an operator that commutes with the constraints, including the Hamiltonian constraint is to demand that it commute with the Hamiltonian, which in turn means that it is a constant of the motion, at least in the WDW patch. And to demand that every meaningful physical quantity be a constant of the motion is to require that there is no physically meaningful notion of change. And that is absurd. That leads to the so-called "problem of time" in WDW. And there is no justification for such a constraint on physical meaningfulness. None.

    Physically meaningful observable quantities do change in time, such as the area of an evaporating black hole. Give that up, and you give up the entire subject matter here: what happens when a black hole evaporates! So I absolutely refuse to accept what you intend to insist on. If you continue to insist, then we need an argument, and a response to the point I just made.

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  148. Tim,

    I disagree with some of your comments, but since my argument does not rely on these issues I would rather not address them now. Instead, let me backtrack: ignore my last message and replace it by the more restricted and technically precise statement:

    The generator of time translations is the operator H = \int (N_0 C_0 +N_i C_i) + H_{bndy}. The statement is that this operator, restricted to the physical Hilbert space, has a hugely degenerate spectrum, in particular coming from the subspace of the Hilbert space describing disconnected spatial surfaces? Agreed? I assume you do agree since you have previously stated that this subspace of the Hilbert space is "obviously" a tensor product, and since H_bndy acts only on one factor the conclusion follows. But I want to confirm your agreement.

    A couple of comments. First, in the expression for H given above we have the ambiguity corresponding to the freedom in choosing the functions N_0 and N_i; however, since these functions multiply the constraints, the spectrum of H in the physical Hilbert space is independent of this choice. We just demand that N_0 go to a nonzero constant at infinity, so that H generates translations of asymptotic time. Second, you might wonder whether there is some other ambiguity in defining H. To see that there is not, we note that with standard boundary conditions quantum gravity in AdS is invariant under the conformal group, which means that besides H we can define other operators corresponding to rotations etc, and the commutator algebra of these is the Lie algebra of the conformal group. For the explicit construction see the 1985 paper "asymptotically AdS spaces" by Henneaux and Teitelboim. Since H appears as a generator in this algebra this removes any ambiguity in its definition.

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  149. BHG—
    Sure, the set of solutions to that Hamiltonian from acceptable data on a pair of disconnected surfaces is massively degenerate in the spectrum. Of corse, that alone does not mean that all of these solutions are asymptotically AdS. Indeed, it is not even clear what that extra condition comes to: is it that one of the disconnected surfaces is asymptotically AdS, or that both are? But in any case, what is sure is that not all of these solutions have an evaporating black hole in them. Not all of the solutions emerge from a single connected surface. So what possible difference can it make that the Hamiltonian is massively degenerate in this set of solutions to the Hamiltonian, since that set if not the set that emerges from an evaporating black hole?

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  150. Tim,

    Excellent. The point is that we are trying to establish some structural properties of the theory as a whole, i.e. properties which go beyond those of specific solutions. In particular, we want to know about the structure of the physical Hilbert space and the spectrum of the Hamiltonian acting on this physical Hilbert space. What we have concluded is: in a theory of quantum gravity in AdS, the Hamiltonian, uniquely defined as the operator that appears in the conformal algebra describing asymptotic symmetries, has a hugely degenerate spectrum in the physical Hilbert space.

    The assumptions that went into this were quite mild: we assumed that the physical Hilbert space is the space of solutions of local constraint equations, that the Hamiltonian is the sum of volume terms proportional to constraints and a boundary term, and that black hole formation/evaporation leads to the dynamical "splitting" of spatial surfaces as in the standard Penrose diagram. In particular, it is this last point that forces us to include disconnected surfaces in our physical Hilbert space, since they arise dynamically from connected ones. All of these assumptions are standard ones that appear in standard discussions of quantum gravity in a metric based, WDW type, framework.

    To reiterate, we have been considering black hole evaporation, but are then using this to make a general claim about the Hilbert space, including the existence of states that do not themselves arise from black hole evaporation. To make an imperfect analogy, suppose you had two particles in ordinary non-relativistic QM. You perform scattering experiments by sending them towards each other from infinity, and thereby deduce an attractive 1/r potential between them. The general structure of QM then implies that the Hilbert space for this system will include "hydrogen atom" bound states, even though such bound states of course never occur in the scattering experiments performed (since the total energy is positive).

    Good. Now suppose I have a CFT, of the type that appears in the AdS/CFT correspondence. Such a CFT has a uniquely defined Hamiltonian appearing in the conformal algebra, and it is a basic and intuitively obvious fact that the spectrum of this Hamiltonian is discrete and non-degenerate (up to obvious symmetries, a qualifier I will henceforth suppress). The intuitively obvious part comes from the fact that the CFT is defined on a spatial sphere (since the conformal boundary of AdS is a sphere), and quantum theories on compact spaces generically have discrete energy spectra, with degeneracies only occurring by virtue of symmetry. Therefore, if someone were to make the claim that the CFT is "equivalent" to quantum gravity in AdS, you should say: "nonsense: the spectra of the Hamiltonians are completely different in the two cases, degenerate vs. non-degenerate. The CFT has no chance of accurately describing gravitational physics in AdS, and certainly not black hole formation/evaporation".


    So an AdS/CFT duality must be impossible, right?

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  151. BHG—

    No, your logic fails, and at just the point I have been insisting on all along. The spectra that you are referring to are the spectra for the Hamiltonian operating on the space of *global solutions that arise from initial connected hypersurfaces*, including hypersurfaces with initial data that lead to black hole formation and evaporation. If, as I claim, some of those possible initial data lead to solutions whose late in-extendable hypersurfaces are disconnected, then those disconnected hyypersurfaces with their data, and only those disconnected hypersurfaces with their data, are elements of the physical Hilbert space. The fact that there are other solutions to the Hamiltonian which do not arise from a connected in-extendable hypersurface via an evaporating black hole is neither here nor there

    You keep being inconsistent about the rules of the game that you want to play. On the one hand, you insist that it is essential to restrict one's attention to the Hilbert space of "physical" states. But of course there are states that are physical in the sense that they solve both the constraints (including the Hamiltonian constraint) and the surface Schrödinger equation but are not AdS. These you are leaving completely out of account. And there are solutions of the Hamiltonian and surface Schrödinger equation that are pairs or triples of quadruples of disconnected space-times, one of which is open and the rest of which are closed. And again, you want to rule these out of account, otherwise the spectra will be massively degenerate again. So you have to make up your mind about—and implement consistently—the precise conditions that make a state "physical".

    Here is my definition: A state on a spacelike hyoersurface (including disconnected hypersurfaces) is "physical" for the purposes of this analysis iff it can be embedded as an in-extendable spacelike hypersurface in a solution that arises from a single connected spacelike hyoersurface that is asymptotically AdS. Now whether the spectrum of the Hamiltonian is massively degenerate on this set of physical states is not at all obvious. It is obvious, for reasons I already gave, that not every state on a disconnected set of hypersurfaces can arise in the manner specified. So without a hell of a lot more analysis of the solutions of the equations nothing at all can be said about the viability of AdS/CFT.

    Furthermore, as far as I know, AdS/CFT assumes that the space-time of the evaporating black hole is indeed a manifold. No provision has been made for cases where the solution can be continued in time, but only by abandoning the manifold structure. Since that is precisely what I am claiming, that would mean that there are or could be lots of physical solutions that are being left out of your discussion.

    In short, nothing in the claim that the Cauchy surfaces split either entails or refutes AdS/CFT. The inclusion of some disconnected states in the space of physical states relevant for our question does not require the inclusion of all such states.

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  152. BHG—

    Just to be more concise: you write "What we have concluded is: in a theory of quantum gravity in AdS, the Hamiltonian, uniquely defined as the operator that appears in the conformal algebra describing asymptotic symmetries, has a hugely degenerate spectrum in the physical Hilbert space." Now: what does the phrase "a theory of quantum gravity in AdS" even mean? To be even more precise, what does the qualifier "in AdS" mean?

    If it means "The theory specified by this Hamiltonian and boundary conditions, where the physical state "at a time" is always on a connected spacelike surface that is symptomatically AdS", then according to me such a "theory" cannot model evaporating black holes at all. If it means "The theory specified by this Hamiltonian and boundary conditions, where the physical state "at a time" is always arises from a connected spacelike surface that is symptomatically AdS by the action of the Hamiltonian" then according to me such a "theory" can model evaporating black holes, but we have not shown that the Hamiltonian has a hugely degenerate spectrum in the physical Hilbert space. Indeed, in neither case have we shown that the Hilbert space is massively degenerate. If you have some other reading of "in AdS" please specify what that is.

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  153. Tim,

    I will restate things to make it clear that I am just following the standard rules of QM. You are the one who is proposing a departure from the usual rules, and your specific departure is almost surely mathematically inconsistent.

    According to the standard rules, to define the physical Hilbert space we work at some fixed boundary time t and impose asymptotically AdS boundary conditions, meaning that our 3-geometry metrics and lapse and shift functions must approach a specified form at infinity. We then define physical states as solutions of C_0 \psi[g_{ij} ]= C_i \psi[g_{ij}] = 0. The only issue here is whether we allow disconnected 3-metrics. Since we have already fixed asymptotic boundary conditions, the question is whether we allow disconnected components that themselves lack any asymptotic region. Since black hole evaporation produces these, the answer is that we should indeed include them or our theory will be inconsistent. So our physical Hilbert space consists of all solutions to the constraints, connected or disconnected, such that there is a single asymptotic AdS boundary.

    You are instead proposing some sort of modification of QM where to define the physical Hilbert space you first need to solve the Schrodinger equation and then use to this to decide which states to keep; i.e. you throw out some of the states in my description. I have never before seen such a modification proposed, and of course it is usually very difficult to modify QM without running into inconsistencies. I would like to ask what you propose for measurement rules in this formulation? Note that since you have thrown out most of the states in the Hilbert space of the standard formulation, your state space is highly incomplete with respect to the standard inner product, and so the burden on you is to define a new inner product on your reduced space. I would like to see the details of this before commenting further. Of course, my skepticism will be evident: I think this is just a mathematically inconsistent modification of QM, but if you present a precise proposal I will examine it with an open mind. The point here is that you can't just decide by fiat that a certain subspace is the physical Hilbert space as you have done: you need to define an inner product such that physical observables are Hermitian etc. I am not asking for mathematical rigor here, but some definite proposal is needed.

    A couple of more comments: you ask whether AdS assumes a manifold structure for the bulk? No, it doesn't, and neither does any sensible approach to quantum gravity. No one expects the notion of a smooth manifold to survive in the quantum theory, and indeed the appearance of singularities is telling us that this breaks down. In AdS/CFT one can do detailed and precise computations that show how this happens; i.e. how certain singularities are resolved by departing from Riemannian geometry.

    You refer to a "surface Schrodinger equation", and the implication seems to be that this only encodes some very limited information about physics near the boundary, but this is very misleading. For example, in the classical theory if you first solve the constraints and then insert the solution into the boundary Hamiltonian you get a Hamiltonian that governs physics everywhere in the bulk via Hamilton's equations. I.e. it descries a planet orbiting a star deep in the bulk and so on. The point is that although the boundary Hamiltonian is defined at the boundary, the constraint equations connects data deep in the bulk to that at the boundary, so they are not independent. I again refer you to RT for details.

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  154. BHG—

    I don't see what issue you are raising about the inner product. Start with a connected spacelike hypersurface that is asymptotically AdS, specify the boundary conditions and the Hamiltonian. The Hamiltonian determines both the Hamiltonian constraint in the bulk and the Hamiltonian operator for the asymptotic part. The set of possible states on the hypersurface that satisfies the spacelike constraints forms a Hilbert space with an inner product. Now let the states evolve in accordance with the Hamiltonian. We want this evolution to be unitary, and we can get this by fiat: Given any pair of states that are generated by this evolution, define the inner product between them as the inner product between the states on the fiducial hypersurface they evolved from. What can go wrong with that?

    This proposal allows for the possibility of states on disconnected hypersurfaces if they arise. By the way, you somehow seem to have concluded that the piece that is disconnected from the AdS boundary "lacks any asymptotic boundary". How do you arrive at this conclusion? The disconnected part is clearly open. Offhand, I have no idea what the metric structure of the hypersurface is as you approach that open edge. How do you know what it is?

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  155. Tim,

    The problem is that you haven't addressed the obvious issues of how to make sense of measurement theory in your setup. Here I am just asking for a concrete set of rules that could be handed to a practically minded experimentalist -- I am not asking about "deep" philosophical issues.

    In particular, suppose I ask: "long after the evaporation, what is the probability for the radiated particles in the external region to be at such-and-such positions?" (Here I will use "position" in a general sense, since we might want to instead specify, e.g. angular momenta or use wavepackets). The experimentalist can make such a measurement by distributing particle detectors throughout AdS. Here is how the prediction would go in standard QM (i.e. what I am espousing). The state of the full system is |\Psi> which is some entangled state in the tensor product Hilbert space describing the interior and exterior components. Let |\psi_out;x> be an eigenstate of the outer particle positions; similarly let |\psi_in;y> be an eigenstate of inner particle positions. Then, P(x,y)= |<\Psi | |\psi_in;y>|\psi_out;x>|^2 is the probability for the inner particles to be at y and the outer particles to be at x. Since we can only observe the outer particles we sum over the inner particle locations to get P(x) = \sum_y P(x,y). This is our probability distribution that our experimentalist can try to verify.

    Now, every step of the above fails in your setup. The states |\psi_in;y>|\psi_out;x> are not in your physical Hilbert space since they are tensor product states which will not arise from time evolution starting from a single connected component. By the same token, the inner product I used is not defined in your setup. In the normal rules, after the measurement the experimentalist would say that he has collapsed the wavefunction to one in which the outer state is |\psi_out;x>. Again, this cannot be the case in your setup, since such a state (which is unentangled) is not in your Hilbert space.

    So, the question is, how do things work in your setup?

    As to what spatial geometries appear in my wavefunction, the answer is that I include everything that has a single asymptotic region, with the latter describing an asymptotic AdS region. The slice behind the horizon is compact -- it closes off at a singularity, but of course we expect this singularity to be smoothed out in some way in the full quantum theory.

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  156. BGH—

    Why is there any problem at all for the question you ask? We know how to predict the Hawking radiation with a thermal spectrum, so that gives the experimentalist all that could be wanted. The tracing off procedure is well defined even if the universal state for the case of disconnected surfaces is always entangled. Similarly, I see nothing preventing the calculation of a scattering matrix. What difficulty do you foresee?

    As for how much information is carried by the surface term, I quote R & T:

    If one wants to examine the behavior of the theory under asymptotic Poincaré transformations within the Hamiltonian formalism, one must enlarge the phase space of the system by introducing, besides the original variables g_ij , pi^ij (and the matter variables) a new set of ten independent canonical pairs which describe the asymptotic location of the spacelike surface on which the state is defined. One gains in this process ten new constraints which must be included in the Hamiltonian following the general method of Dirac [6]. The new Hamiltonian obtained in this way vanishes then weakly and it is this quantity which is the analog of the expression

    [Constraint equations],

    for compact spaces.Thus the configuration space of general relativity for open spaces is not Wheeler’s “Superspace” but rather the product of Superspace with the space spanned by the ten additional variables describing the location of the surface. That the asymptotic coordinates also play a role for open spaces is not new (it was already noticed by Higgs [15]). However what does not seem to have been previously done is the introduction of the “boundary conditions as canonical variables,” a procedure which has the advantage of permitting one to build a Hamiltonian formalism which is manifestly Poincaré-covariant at infinity."

    Note that the addition of the surface term to the constraints results in just ten new degrees of freedom—ten—which is hardly sufficient to characterize the state of the bulk! The main work is still being done by that constraint equations.

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  157. Tim,

    Your version of the theory does not specify rules for predictions regarding the state of the radiated particles, or at least I am waiting for your to explain (with formulas!) how this works. I request a concrete answer to the following question. If the full state of the system resulting from black hole evaporation is \Psi, what is the probability for an experimentalist to measure, say, precisely N photons at locations (use wavepackets here if you like) x_1, ... x_N in the region connected to the AdS boundary? Your formulas should contain only elements that exist and are defined according to your rules. The formulas I gave do not apply to your case, because the tensor product states I used are not in your physical Hilbert space as you have defined it, and (by the same token), the inner product I used does not exist in your formulation since the tensor product states do not evolve back in time to a state on a connected surface, which is your requirement. So I am very interested to know what formulas you are going to use to predict probabilities! Until you specify these, your version is not good for anything.

    As for the rest of your message, I am not sure what you are getting at here. Of course, if you work with the full unconstrained phase space then you need both the constraints and the boundary Hamiltonian to define dynamics; H_bndy by itself is not very useful. On the other hand, R&T also discuss, with several explicit examples, how you can first solve the constraints and then insert the solutions into the boundary Hamiltonian. This gives you a formulation of the theory in which there are no constraint equations left, just a Hamiltonian. The two versions of are physically equivalent (at least under certain conditions), and both are used in concrete applications of GR. So it is a choice whether you want to work with a constrained or unconstrained formalism.

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  158. BHG—

    Your request is trivial, and contains a rather basic confusion.

    Suppose I hold the following ontological claims. 1) The only quantum state that fundamentally exists is the universal quantum state. This is an extremely plausible—indeed almost inevitable—claim. 2) The only physically possible quantum states for the entire universe are represented by pure states. That is, the universe cannot physically exist in a mixed state that is represented by a non-pure density matrix. This is a very, very plausible claim. As a matter of what is physically possible, in fact, I think most physicists of any stripe would accept these postulates.

    But if you ask me how to make empirical predictions about subsystems—e.g. about one of the particles in an entangled pair—I am perfectly free to make use of density matrices! Even if those matrices cannot, according to my theory, represent a physically possible free-standing physical state of the electron, as mathematical objects they are perfectly acceptable for computational purposes. Predictions about observable phenomena in the region external to the event horizon are always treated like this, and there is absolutely nothing preventing me from doing what everybody else does for the purposes of assigning probabilities to possible outcomes in the case you outline. You seem to be suggesting to rather bizarre and completely unargued view that the mathematics used in calculating predictions must be restricted to those mathematical items that represent physical possibilities. The use of non-pure density matrices for making predictions about subsystems is a direct counterexample to this odd proposed restriction. I simply reject the rules of the game you are proposing because nobody else plays by them and they have no obvious justification.

    Until you explain and defend your requirement that "Your formulas should contain only elements that exist and are defined according to your rules" I see zero reason to try to abide by it. It is, in the first place, not even grammatically well-formed: my formulas, just like everybody else's formulas, contain mathematical expressions, not elements that physically exist. So at best you mean "Your formulas should contain only elements that represent entities that could exist according to your theory." But so stated the demand is just silly. It is like saying that someone who defines the Hilbert space in non-relativistic quantum mechanics as the space of normalized square-integrable functions is forbidden from using Fourier analysis because the sines and cosines are not part of the Hilbert space! In short, you have run together the possible physical states with their mathematical representations, and then tried to restrict the use of mathematics only to those mathematical items that represent possible physical states. Nobody else plays by these rules, and I can't think of a single reason anyone should.

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  159. Tim,

    As I said, I am asking what rule you would provide to a practically minded experimentalist who wants to make predictions about observations carried out in the late time external region -- so far you haven't provided that. At the moment, I will be satisfied with some mathematically well defined formulas. Again, given the state \Psi how do I compute the probability that I will find a bunch of radiated particles at specified positions? I don't see how you are going to make this work given your statement that inner products are defined by evolving everything back to the original connected surface, but I will suspend judgement. Just state the precise rules and we can go from there. You say this is trivial, so this should be easy to do.

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  160. BHG—

    This is getting ridiculous. You ask me how to put an inner product on the physical states and I tell you how to put an inner product on the physical states. Then you tell me to provide a computation using a restriction on the available mathematics that makes no sense and you do not even try to defend that restriction when I point out how unmotivated it is. Now you ask for advice for a "practically minded experimentalist". OK: I would tell the practically minded experimentalist just what everyone else would: look for thermal radiation whose temperature is given in the usual way by the surface gravity of the black hole as long as the event horizon exists. That thermal radiation will be represented by a density matrix because the radiation will in fact be entangled with radiation in the interior, but the practical minded experimentalist in the exterior could not care less about that. What else do you want? The radiation that comes out of the Evaporation Event? I don't know: that requires some real mathematical work in characterizing that non-mainofld point and the physics there. Do you have any examples to offer of a theory that does any better than that? I have been defending a general approach to solving the so-called information loss paradox, not constructing a complete theory of quantum gravity.

    You said you had some sort of principled objection to that solution. At this point you are not even pretending to offer any objection, you just keep asking questions that have no bearing on the nature of the solution.

    Look, I understand what you wanted to argue. You wanted to argue that the only constraints on the physical states on the late-time disconnected hypersurfaces are the spacelike constraints, and that the space of states that satisfy those constraints has a tensor product structure with a massively degenerate Hamiltonian spectrum, which is supposed to conflict with the claim that the space of initial physical states, on a connected hypersurface before the formation of the black hole, is not a tensor product space and does not have a massively degenerate spectrum of the Hamiltonian. And if the evolution is unitary, that would seem to be a problem. But once one sees that if you are going to restrict the initial physical states to being states on a connected hypersurface that is asymptotically AdS, then you have to restrict the late-time physical states on the disconnected hypersurfaces to those that could evolve from the space of initial states, and that is a much stronger restriction than just the spacelike constraints. And given this stronger constraint, there is nothing standing in the way of the set of possible late-time physical states also not having a tensor product structure. So your whole argumentative strategy has collapsed.

    I note also that the supposed AdS/CFT duality has dropped out of the discussion as irrelevant.

    Tell you what: if you have some better advice to give your practical minded experimentalist about what to expect, how about you give it as an example of what you are looking for from me. Maybe that would make the relevance of the request more evident.

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  161. As a long time lurker in this thread, maybe I can help out here.

    Black hole guy believes that Tim's prescription for how to define the physical states post-evaporation runs afoul of the usual rules of quantum mechanics. For instance, suppose an experimenter in the asymptotically AdS region at late times (long after the hole has evaporated) performs a measurement of the positions of all the Hawking particles. The textbook rules of QM say that after that measurement, the wavefunction is projected onto the corresponding eigenstate of the position operators for those exterior particles. By locality that projection has no effect on the interior state (after all, the projection is just the effect of detectors which are all located in the exterior region).

    The full state after that projection is no longer entangled, because it is that single external position eigenstate times some state in the interior. In other words after the projection the exterior state went from a density matrix to a pure state when you trace out the internal state.

    The question for Tim is whether you include such unentangled states in the Hilbert space. If you do then there is obviously the large degeneracy black hole guy has been talking about, which is not consistent with AdS/CFT. But if you don't you cannot apply the standard rules of QM as I just did.

    I'm not sure what Tim thinks is the loophole here. All I can think of is that he could deny that the projection does not act on the interior state. But if you carry that to its necessary conclusion to avoid degeneracies, you need the internal state to be completely determined by the exterior state. But then you might as well throw the interior away, it is redundant and plays no role in the information paradox.

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  162. physphil,

    I understand your suggestion, but I don't see how that could possibly be the issue. if you believe in a real, physical collapse of the wavefunction or quantum state, then information is irretrievably lost all the time, and there is no paradox at all. Take whatever quantum state you like for a system and "measure O". By the von Neumann axioms, after the "measurement" the system is in some eigenstate of O. From this information, what can I retrodict about the pre-measurement state of the system? Just that it was not orthogonal to the eigenstate, nothing else. So on this theory the wavefuntion just does not evolve unitarily or deterministically, or retrodictably. The paradox cannot get off the ground.

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  163. Tim,

    You are clearly exasperated by my question, but I am not sure why. I am honestly confused about what formula you want to use to predict the probability for a measurement outcome in the external region. I would greatly appreciate seeing an explicit formula. As I explained, the standard formula does not apply to your formulation since it uses an inner product between states, one of which is a tensor product state, but your inner product is ill defined in such a case because the product state does not evolve back to one of your connected states. That is, if the full state is \Psi, then to compute the probability for external particles to be at locations x one thing I need to do (according to any rules I am familiar with) is to first compute inner products <\Psi| |\psi_x>_out|\psi_y>_in, but these inner products are simply not defined according to your rules, because the state |\psi_x>_out|\psi_y>_in is a product state. An equivalent way to state this is that to compute the probability to find outer particles at x we should define a projection operator P_x that projects onto specified locations of outer particles. But such a projection operator does not exist within your formulation, because it takes an entangled state to a product state, and the latter are not in your Hilbert space.


    I think it would save us both a lot of time and grief if you would humor me and present a formula for computing probabilities for measurements in the outer region which only uses the inner product that you defined. Shouldn't take you more than a minute or two of your time. Of course, I am skeptical that such a thing exists.

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  164. physphil,


    Well said -- that is indeed the point, and I am equally baffled as to Tim's objection here. My only comment on your remarks is that I have been trying to avoid talking about

    a measurement "projecting" the wavefunction, as that sounds a lot like "collapsing" the wavefunction, and I don't want to go down a rabbit hole filled with irrelevant philosophical

    issues. But I completely agree with the content.

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  165. BHG—

    Ok, here's the answer. I should have just given it immediately, but I was not being careful, and accepted the unjustifiable restriction you put on the answer. We are, in fact, right back where we started months ago.

    When you construct the theory, as we have been over and over again, you start with a kinematic Hilbert space with an inner product. There is no presupposition that all of the elements of this kinematic space will end up as physical states. If there are constraint equations, as there are in this case, you cut down the kinematic Hilbert space by imposing the constraints.And if you have further constraints—such as requiring that the state be part of a solution that is a some time a state on a connected hypersurface that is asymptotically AdS—you impose those too. The set of states that satisfy these constraints form a subspace of the original space. The kinematic space is a tensor product space, but the the cut-down space may or may not be. But the cut-down space inherits its inner product from the kinematic space. The inner product between any physical states and any element of the kinematic space—physical or not—is well defined.

    Now the story about the measurement theory is just the usual one. In your example, construct the projection operator to the particular subspace of the kinematic Hilbert space of the hypersurface connected to the AdS boundary that corresponds to the distribution of radiation you are interested in, and take the tensor product with the identity operator on the disconnected piece. Operate with this projection operator on the physical state you are interested in—which will certainly yield an element of the kinematic space—and take the inner product of that with the physical state. Square the result. Done.

    If you now try to complain that I used a non-physical state in the calculation, namely the projected state, well sure I did. And the question now is simple: so what? In fact, this is the very same thing one would do to figure out the probability of an outcome for a measurement made on a restricted region of the connected hyperplane before the black hole evaporation. Obviously. You would take projectors for the region where the measurement occurs crossed with the identity on the rest. This is exactly the same procedure: the fact that there is a disconnected hypersurface is neither here nor there. And the fact that one takes the inner product with a non-physical state is also neither here nor there. The procedure gives the probability of the outcome,

    This could only cause problems if you thought that the post-measurement state had to be the projected state, via collapse. But, as I mentioned to physphil, you certainly do not, and cannot, believe that.

    So: done and done. Now, one more time, what exactly is the nature of your objection to my solution? As far as I can tell, there isn't one.

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  166. Tim,

    Here is the problem with your proposal: to compute probabilities you are using precisely the Hilbert space in my description, not yours, since you need to take inner products involving states that are not in your Hilbert space. So you are saying one thing but actually doing another. Think of it this way. The following are the general axioms of QM

    Physical states are in 1-1 correspondence with states (really rays) in a Hilbert space

    Observables correspond to Hermitian operators acting on this Hilbert space

    The outcome of a measurement is one of the eigenvalues of the Hermitian operator

    To compute the probability for an outcome X you evaluate ||^2, where |X> is the eigenstate corresponding to eigenvalue X

    The question of interest is, what is the quantum gravity Hilbert space that will appear in the above? Clearly, your proposal for this Hilbert space cannot fill that role, since, among other things, the states of interest |X> are not contained in it, and the associated inner products are not defined on it. So the actual Hilbert space you need to use is the standard one that I and everyone else uses. Hence your statement about "physical states" means nothing more than that the states that arise from black hole evaporation lie in a subspace of the fill Hilbert space, a statement I certainly agree with.


    This standard Hilbert space is the Hilbert space that is supposed to be isomorphic to the CFT Hilbert space, with the Hamiltonians identified as I have said. But then we have the paradox about the clashing (non))degeneracies of the spectrum. So that kills your scenario (or alternatively kills AdS/CFT if you believe your scenario).


    Also, physphil's objection also shows why what you are claiming cannot work. A Copenhagen collapse interpretation should certainly work as a rule of thumb, if nothing else, for our practical experimentalist. But such an interpretation would be untenable in your setup since the product states do not exist. Maybe you want to make the claim that black hole evaporation is incompatible with a Copenhangen collapse scenario, but you will need to make a much stronger case for that than you have so far.

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  167. I tried to post a few comments but I guess they got lost. Tim, thanks for the response. I agree that projections don't preserve info. But still, I think your answer conflates two separate questions: (1) are the early and late time states of a black hole formation/evaporation related by a unitary transformation, and (2) what is the correct way to describe measurements in QM? Unless you think black hole evolution is a measurement, your answer can be "yes" or "no" to (1) independently of your answer to (2). But if I choose your way of answering "yes" to (1), apparently I cannot use the textbook rules of QM on the final state. If I can't follow textbook rules, that sounds to me like a radical modification of QM.

    Apart from that, suppose you use a unitary model of measurement instead. Then the measuring device would end up entangled with the state, and the state would decohere to something very close to a sum of projected states as above, none of which are in your Hilbert space. That certainly sounds dangerous. Are you claiming it's OK?

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  168. physphill,

    This is the only comment I have from you. There's none in the spam folder either.

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  169. BHG—

    Now we are really in the are of foundational questions, so this may explain quite a lot. The "general axioms" you write down would appear to be the axioms of a collapse theory, and indeed of a rather naive collapse theory, but it is hard to tell exactly what you have in mind. For example, in many cases the mathematical gadget used to make predictions for "measurement" outcomes is a POVM, rather than a Hermitian operator. And, as I already mentioned to physphil, anyone using a collapse theory will reject the Information Loss Paradox at step 1, since collapses lose information all the time. The basic set-up of the puzzle requires a non-collapse theory, so either a pilot-wave sort of theory or a Many Worlds theory. If you really want to hold some set of statements up as the "general axioms" of QM, then you are going to have to declare which "interpretation" you are defending, and then we can get down into the nitty-gritty of trying to make sense of it. But no matter which way you go, you are not going to end up with this list of "general axioms". I can guarantee that.

    So we have a choice point here. When you said that you wanted to know what the "measurement theory" in my solution was, I thought all you wanted was a practical technique for making predictions, a technique that your "practically minded experimentalist" can use. Well, practically minded experimentalists have, throughout the history of QM, used a logically incoherent mixture of principles—sometimes using collapses and sometimes not—with good taste and discretion to make predictions. That is, if what a "measurement theory" is supposed to be is a *principled physical solution to the measurement problem*, then your practically minded physicist has never had a measurement theory and a fortiori has never used one to make predictions. If you would like to dispute that, I'm all for it: we are completely on my turf now. And if by a "measurement theory" you mean a principled physical solution to the measurement problem, then let's put our cards on the table. I will defend a pilot wave solution, in which there are no collapses (and hence no loss of information in the wavefunction) but which is not a Many Worlds theory. In such a setting, Hermitian operators play no fundamental role at all in solving the measurement problem, and your "general axioms" go out the window. And in calculating the probabilities, I never need to make use of any wavefunction outside my physical Hilbert space. Indeed, I never have to make use of anything but the single, precise wavefunction that obeys the fundamental dynamics and whose initial state is given at when specifying the initial conditions of the problem. I am 100% prepared to detail all of this completely. I assume that you will be willing to do the same for your preferred solution to the measurement problem.

    If all you meant by a "measurement theory" was the specification of some mathematical technique for deriving numbers that are treated as probabilities—without any principled physical story explaining why you are justified in so using them—thenI have given that. And in doing this, I can use whatever mathematics I like. So, as I said, done and done.

    So: shall we get into the solution to the measurement problem now? I am raring to go. Please declare how you think that problem is supposed to be solved so we know which issues to discuss. Or are the usual, physically unprincipled rules of thumb for deriving statistical predictions sufficient? Then you have no grounds for complaint. I'm willing to go either way on this. Your call.

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  170. Physphil,

    As I wrote to BHG, my preferred solution is a pilot wave solution in which the quantum state (the physical referent of the wavefunction, which is only part of the physical ontology) always evolves deterministically and unitarily. In such a theory a measurement outcome is not determined by the state of the final wavefunction—which will be the same in a Schrödinger cat situation whether the cat lives or dies—but by the state of the additional variables, e.g. the particle positions. And one derives the usual Born probabilities for the various possible outcomes by the usual techniques of statistical physics applied to the additional variables.

    I see exactly zero "danger" in your observation about decoherence. What is at issue, to start with, is only approximate decoherence, because that's all you will ever get by Schrödinger evolution. And the fact that you get it is just a mathematical fact about Schrödinger evolution given some Hamiltonian. There is no new physics in that at all, and every non-collapse solution has the approximate decoherence automatically. In a pilot wave setting, the approximate decoherence plays no fundamental role in the physics, If you defend Many Worlds, then you may well appeal to this approximate decoherence in a much more serious way, as part of the account of "branching".

    The "textbook rules" are imprecise and unprincipled rules of thumb that do not comprise any coherent physics at all. As far as I am concerned, you can follow them all you like, but the job of Foundations is to supply a properly formulated, precise physical theory that validates their use (within epsilon).

    As for your question: the entire "Information loss paradox" is predicated on the quantum mechanical answer to 1) being "yes". And in my solution it is indeed "yes", but taking care to note that the "late time states" are states on a pair of disconnected hypersurfaces rather than states just on the hypersurface connected to the AdS boundary. If you want to ascribe a state just to that piece then you need to use an improper mixed state that you get by tracing over the other, disconnected hypersurface. The evolution from the initial state to that improper mixture is, of course, not unitary. It is that evolution that Hawking wanted to describe by his superscattering matrix.

    I am not quite sure what you have in mind by the "textbook rules" of QM. Why do you say that they can't be used on my final state? I am completely puzzled by that comment. As I said, it is incumbent on any "interpretation" of the quantum formalism to validate the usual way of making statistical predictions, at least within experimental error. The pilot wave theory does precisely that. So please clarify what claim you are making.

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  171. Tim,

    Thinking about this a bit more, I believe the following is a good way to capture what I am saying in a way that avoids what I think are irrelevant conceptual questions. Suppose you want to program a computer (assumed to be arbitrarily powerful) to compute predictions for measurements that can be performed by the late time external observer. We either hand the computer the late time quantum state, or we specify the dynamical laws and let the computer compute the final state itself; that part is immaterial. We of course assume that the rules we provide the computer are compatible with the axioms of QM. Now, if it is the case, as you have advocated, that the late time surfaces are disconnected, then it is unavoidable that our computer code will involve specifying a tensor product Hilbert space, and a Hamiltonian operator acting on this space, which therefore has a highly degenerate spectrum. I believe you have now agreed that such a Hilbert space must indeed be introduced in order to compute the inner products necessary to determine probabilities. My point here is that the computer doesn't "care" whether the states in this Hilbert space are "real" or not; but it is simply a fact that we need to introduce such a Hilbert space into our code in order to compute probabilities.

    On the other hand, I claim to have a computer that can predict the outcomes of such measurements, but whose Hilbert space is such that the Hamiltonian is non-degenerate. It further adheres to the axioms of quantum mechanics, and describes gravity in AdS. What is my computer? Of course, it is the dual CFT, or rather a computer programmed to solve the CFT equations. All of the work that has been done on AdS/CFT supports the claim I just made. We conclude that whatever description the CFT gives to black hole formation/evaporation in AdS, it is not one in which there are late time disconnected surfaces.

    By phrasing things in this way we give precise operational meaning to the statement that "the Hilbert space is a tensor product and the Hamiltonian is degenerate", and it allows us to make clear the clash between your AdS picture and the one the CFT provides in a way that is independent of any conceptual issues regarding measurement, wavefunction collapse etc. In my view, this conclusively demonstrates the incompatibility of your scenario with AdS/CFT.

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  172. BHG—

    Thanks for the gift of the supercomputer. Having been so generous, I need no Hilbert space at all. None. Just give me the initial wavefunction for the pre-black-hole situation and a physical specification of the experimental equipment that the experimentalist wants to use, and the absolute square of the wavefunction. We are going to locate that equipment far from the event horizon, setting it up in the initial conditions and calculating probabilities for it to register various possible outcomes in the late period. Given the initial wavefunction and the Hamiltonian (and whatever boundary conditions are needed) the computer computes the time evolution of the wavefunction. That goes into the guidance equation. Next, use the square of the initial wavefunction as a probability measure over the initial configuration space, plug the wavefunction into the guidance equation, and calculate what the final outcome for the experiment will be for each possible initial state of the additional variables (which will be both the exact space-time geometry and the exact distribution of local beables in the space-time).Use the initial probability measure to get probabilities of the various possible outcomes. Done and done.

    No Hermitian operators. No POVMs. No other wavefunctions beside the single actual wavefunction that is the solution for the given initial state. No measurement problem. Everything done in a completely precise and principled way. No need for any Hilbert space, tensor product or otherwise.

    I look forward to understanding how you will provide such a clean, precise, unambiguous calculation.

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  173. Tim,

    you write "I am not quite sure what you have in mind by the "textbook rules" of QM. Why do you say that they can't be used on my final state? I am completely puzzled by that comment."

    I mean the standard Copenhagen rules in the setup I referred to in my first comment. Specifically, the rule that the state after measurement is projected onto an eigenstate of (in this case) position of the external particles. The resulting projected state does not exist according to you, so I cannot follow this rule.

    That's a radical modification of QM by my lights. To me, it's not very interesting to "solve" the info paradox by changing the rules.

    For unitary measurements, the "danger" I referred to is that it's not obvious to me what properties the space of states you like actually has, or that unitary operators like the one describing an external measurement preserve it.

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  174. Mainly for my own benefit I will summarize my thoughts, with no obligation for anyone to respond to or correct them, on this discussion --from the viewpoint of a layman who admittedly does not understand QM or GR on a technical level.

    The hammer-nail analogy I am using to try to comprehend the discussion is probability/statistics math. As I conceptualize it, scientists currently try to predict (and with great success) outcomes of QM events by applying a probability measure over a "physical space" of possible outcomes which satisfy a set of constraints. That is, the total probability for all events within that space is 1. Applying this method to BH evaporation with a remnant solution implies a degenerate set of solutions (which contradicts ADS/CFT).

    The philosopher claims that the phsyical space should be further reduced to a set of specific outcomes which an omniscent observer with an omnipotent supercomputer could calculate for the BH evaportation, using the as-yet unknown actual equations of quantum gravity (that is, an exact simulation of reality) (with a total probability of 1 over only those outcomes); and that the resulting reduced space might not (or might) contain degeneracies. This then could allow a remnant solution to BH evaporation, as claimed by the philosopher.

    Philosopher: problem solved! There is some unknown chance I could be right!

    Scientists: but we can't predict what happens in BH evaporation, except by estimating with our current method. For us, the problem remains! We want to know what reality is, not just what it might be.

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  175. Tim,

    Your introduction of a pilot wave theory, which you say involves no HIlbert space at all, is just getting us off topic. In general, I am highly skeptical that understanding the physics of black hole evaporation hinges on interpretational subtleties of QM, or questions about measurement. Regardless, let's recall the question at issue, which is: is black hole evaporation consistent with the standard axioms of QM, and if so what is the basic structure of this QM theory? For the vast, vast, majority of practitioners of QM, its axioms are understood to be as I have stated them previously, in terms of Hilbert spaces, Hermitian operators, and inner products. So let's try to focus on this question; maybe at the end of the day you will conclude that only a pilot wave theory can describe the physics of black holes. If correct, that would of course be a revolutionary discovery, but let's be more modest for the time being and just ask about the compatibility with standard QM, as usually understood, and not get into questions about (non)collapse, but rather focus on the practical rules for computing probabilities.

    So, getting back on topic, here is where we are. Suppose I have a computer which has been coded to know QM, in the sense that if you input a Hilbert space, complete with a rule for computing inner products, and various Hermitian operators including the Hamiltonian, the computer will spit out probabilities for measurement outcomes associated with the eigenvalues of the Hermitian operators. The claim, which I believe is now well established, is that if you want this computer to supply you with probabilities for late time external region measurements, then you will need to input a Hilbert space which has as a subspace a tensor product space on which the Hamiltonian is hugely degenerate. Or rather, this is the case if your disconnected surface scenario is realized. On the other hand, by inputting the Hilbert space and operators of the CFT I can explicitly get around this, so there is a clash. Perhaps we can at least agree about this specific claim.

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  176. physphil,

    As I said, if you are using a collapse theory where you collapse onto eigenstates then there is massive and irretrievable information loss all the time in QM. So the idea that there could be any "paradox" associated with evaporating black holes is a non-starter. Furthermore, theories with collapses, when they are well constructed so as not to have any measurement problem, do not project onto eigenstates of the "observable". If you do a position measurement, you better not project on to the position eigenstates because there aren't any in the standard Hilbert space. And even if there were, the breakdown of conservation of energy would be immediately noticed.

    The "Rules" of standard QM, if you try to reduce it to rules, are both self-contradictory and empirically refuted. That's why "standard QM suffers a measurement problem. There are rigorous approaches to understanding how an underlying exact physics could produce the same predictions as QM, and among them both the pilot wave and Many Worlds theories have no collapse of the universal wavefunction. So in these theories information is not lost, and there would be a paradox if it were lost in black hole evaporation. So no one in this discussion has been using an objective collapse theory. Your notion that abandoning collapses is abandoning QM is dead wrong.

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  177. JimV,

    I can't understand how you could have been paying any attention to this exchange and then produced this account. After having dealt with various technical objections to the solution, BHG decided to start asking for practical advice for experimentalist. He added to this request the demand that only wavefunctions that represent physical states be used in giving the advice. So in the first place, I rejected this arbitrary constraint. Then BHG decided to take away the issue of practicality and made me the gift of a supercomputer so long as I stayed within physical states in making the prediction. That seemed reasonable to me, so I did that in a complete and exact way when discussing .

    Once again, making this a dispute between "Scientists" and "Philosophers" completely misses the point. At the moment, it is a dispute between a regular physicist who has not studied foundations and an expert in foundations, and the measurement problem has taken a central role. Your rather snide remarks about discussing foundational problems are completely unjustified by the preceding dialectic.

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  178. Tim,

    it doesn't seem that you are making any effort to respond to what I'm saying, because your latest response simply ignores what I wrote. Repeating myself, you are still conflating two separate issues - do bh states evolve unitarily, and how to describe QM measurements. These are clearly logically distinct issues (unless you want to claim bhs are making measurements, which would be radical).

    One other comment. I think you've said several times the external state is entangled with the internal state, and the external state is mixed when you trace out the interior. But to do such a trace you have to sum over states that do not exist (if they do exist the spectrum of H is very degenerate). Can you write a formula for that partial trace that is consistent with your claims, that doesn't include states that do not exist according to you? I don't think you can, which is another way of illustrating that your idea implies massive degeneracy (or is logically incoherent).

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  179. BHG—

    Why is this "off-topic"? If the topic is "Give me a mathematical technique for making predictions along the lines of the standard techniques", then there is no issue: I can use whatever mathematics I like, just as the standard theory derives probabilities for "position measurements" even though their are no position eigenstates in the Hilbert space. The demand that the result of a projector operating on a state be a physical state is not even respected in non-Relativistic QM. So if you want a technique akin to the standard QM techniques, no problem. Which mathematical objects are used in the rule is irrelevant to the question os what is physical.

    If the topic is "Give a measurement theory, but you are required to treat the results of operating on a physical wavefunction with projectors as yielding physical states because there is a physical collapse on measurement", then this whole discussion from the beginning has been crazy. If there are physical collapses then there is irreversible loss of information all the time, endemic to quantum theory, and there was never a paradox in the first place.

    If the topic is "Give a physical resolution of the measurement problem consistent with no collapse of the wavefunction, and implement it so as to produce a physical account of how measurement outcomes actually occur, deriving the standard probabilities from that account by reference only to physical items", then I am more than happy to do that. In keeping with the no-collapse demand, and rejecting a Many Worlds approach, I *have* to use a pilot-wave picture. This is the only real *physical* problem to be solved here, and I will go into lots of detail about how, in principle, it is solved. I would be really, really curious to know how you would approach this physical problem.

    But the "topic" you seem to have in mind is an incoherent mish-mash of these three precise topics, with AdS/CFT thrown in for good measure. Suppose I program the computer with the kinematic Hilbert space, which is a tensor product space, and proceed as usual. I get predictions for all experiments. So what's the problem? Is the problem that I didn't confine myself to only "physical" states? Neither does standard QM, as I have said. If it is that the technique is not isomorphic for the CFT? Well, please begin by showing how the Hilbert space for the CFT is isomorphic to the Hilbert space for AdS: they are associated with space-times of different dimensionality! This whole AdS/CFT conjectured "duality", as far as I can tell, is not based in any rigorous demonstration of any isomorphism of structure at all. But now you want that from me? This just isn't a vaguely serious demand, unless you can show me precisely how you can do the same thing. If all you have as evidence for the duality is that some computation done some way for gravity in AdS gives the same result as another computation done another way for CFT (which was the whole point of the duality: if you have to do the same computation in both theories then you aren't gaining anything), then on what basis are you complaining here?

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  180. physphil,

    Perhaps you can now appreciate why I said I didn't want to bring up collapse. But of course you are correct. Tim is grabbing onto the weakest possible interpretation of your remarks and running with it. He is incorrect in asserting that you can't meaningfully talk about information preservation and unitary evolution within a collapse interpretation of QM. This is obviously incorrect, since experimental physicists all the time are testing unitary evolution, and pretty much all of them use the rules of a collapse theory. As I am sure you know, this is possible because we carry out not a single measurement but repeated measurements with identical initial conditions. That way we can verify or refute the hypothesis of unitary evolution within a collapse theory, and indeed this is how science actually works. So I agree with you and find's Tim's objection silly.

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  181. And I don't understand how you could think that BHG was asking you for technical advice on how to compute probabilities. (I think he had a different motive and objective). I could be wrong, but it seems to me that falling back on blanket assertions as you apparently like to do is not how scientists debate, or should debate.

    As for foundations, I see the progress of science as blazing a trail through a jungle, not constructing a building with a blueprint and foundation. We see the world very differently. How I see it is how I called it. Which is of interest, if at all, only as one piece of random external feedback as to how the thread has been perceived.

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  182. Tim,

    It appears to me that you are now trying to throw up lots of dust to avoid conceding. What's the purpose of bringing up issues regarding position eigenstates -- this is just sophistry that has nothing to do with anything we've been discussing, and of course can be refuted by simply using wavepackets, as everyone knows. Please stick to the topic.

    As far as I can tell you have now conceded the main issue we have been debating. We have established that in your scenario if you want to obey the axioms of QM as usually understood then you are required to use a tensor product Hilbert space with a degenerate Hamiltonian. This is a precise mathematical statement, and you have no counterargument. And this conflicts with AdS/CFT. If you would simply agree that is the case, then we can move on.

    I am not impressed by pilot wave theory, and yes I have studied this before.

    I am potentially open to explaining more about AdS/CFT, but I am not going to exert the effort unless it is worthwhile. Namely, it is clear from our discussions that you are not well versed in the technical details of gravity or quantum field theory, whereas I am. I am not interested in endlessly debating what to me are trivial technical issues. As one representative example, look at your messages from around Jan. 18, where you demonstrate that you don't understand basic issues of the canonical formalism, such as the phase space variational principle. Properly understanding AdS/CFT requires a much higher technical background than you possess. I can explain more, but only if you are willing to accept this fact and respond accordingly.

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  183. physphil, BHG, and JimV—

    As I predicted, and completely unsurprisingly, once we touch on actual foundational issues (such as how to solve the measurement problem) all of you are at sea. Since you have never studied these questions, that is not much of a surprise. So one more time, and please try to keep any responses actually relevant to these points.

    As far as actually computing a density matrix for all measurements that can be carried out in the external disconnected piece, Gleason's theorem already assures us that so long as you are following the usual pragmatic rules, such a density matrix exists. To ask about how to calculate it using only some restricted class of wavefunctions is just pointless. As I said, it is like demanding that someone not use Fourier analysis in non-relativistic QM if the Hilbert space is the space of square-integrable functions. You are all jumping on this rather ridiculous restriction on the mathematics that is allowable for making a computation without even pretending to offer a justification for it.

    If you demand that only mathematical representations of physically possible states be used in making calculations, the only possible justification is because you want the calculation itself to directly reflect what is going on physically. I am actually all for that demand, which obviously brings with it a demand for a physical solution to the measurement problem. I have planted my flag about how to solve that problem, and will happily explain how to do the mathematics that directly reflects the physics. But all of the usual stuff about Hermitian operators and projectors and so forth gets chucked out the window, since it plays no role at all in specifying the physics. If you have some other preferred method to solve the measurement problem and want to use the mathematics appropriate to it to do your calculations, then once more: Bravo! Put your solution on the table. But the way you are proceeding now, as if collapses are somehow essential to the mathematical formalism without committing to any real physical collapses is just completely unprincipled and actually silly. The "standard formalism" is an incoherent mess. You can't even articulate it without reference to "measurement" and, as Bell insisted, the term "measurement" is unprofessionally vague for any serious physics that aspires to be exact. If any of you three want to dispute that, I'm raring to to go.

    BHG's comments about how there is obviously unitary evolution with collapses are so incoherent that I won't even go into them unless he insists. But here is a clue: to distinguish a collapse from a non-collapse interpretation by empirical means is very. very difficult. There is a large, detailed literature on this regarding tests of the GRW theory. This also shows how ridiculous the paper of Peskin, Banks and Susskind (that is often cited) is. I am completely ready for a detailed discussion of that paper, if any of the three of you want to try to defend it.

    The three of you can't even agree among yourselves. physphil want to model a measurement as a von Neumann perfect measurement with a collapsed final state. There is no precisely defined, empirically adequate theory that uses such a rule. The nearest thing are the spontaneous collapse theories of GRW and Perle, and they just don't work that way. But physphil also says that the measurement problem is different from the question of unitary evolution. Not in any serious sense: no collapse theory that has a prayer to work could get from the pre-formation to the post-evaporation states of a black hole without many, many collapses along the way. In which case, as I said, the whole Information Loss issue evaporates completely.

    So here's a simple request. Clearly state what you are demanding here and why. Justify and restrictions on the mathematics used in a calculation. And be prepared to show how you can meet the same standards as you are demanding of me.

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  184. Tim,

    you write "Not in any serious sense: no collapse theory that has a prayer to work could get from the pre-formation to the post-evaporation states of a black hole without many, many collapses along the way. "

    That's either obfuscation or a basic misunderstanding. A small black hole can form and evaporate in a tiny fraction of a second. People do experiments all the time that test unitary evolution (like any process in a quantum computer) that last longer than that.

    I agree with bhg. You've conceded your proposal requires a tensor product hilbert space. That means it is ruled out by ads/cft, and you're just throwing dust in all directions.

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  185. BHG—

    So it "appears to you" that I am "trying to throw up lots of dust to avoid conceding"? Well, I can do a bit better that report on how things "appear to me": I can actually cite chapter and verse about how we got to where we are. I will do this in complete detail because my guess is that this is the final post I will write, so it is a good time to sum things up.

    For months, you have been pursuing a line of argument that was supposed to show—if I granted the truth of AdS/CFT which 1) you have never even been able to precisely define ("It is a work in progress") 2) is restricted to AdS which is neither physically acceptable as a description the actual world nor a spacetime mentioned in Hawking’s or my papers and 3) is an unproved conjecture that has severe conceptual issues—that the solution I advocate is incompatible with some fundamental feature of quantum theory. That fundamental feature is supposed to be the unitarity of the evolution of the universal wavefunction. I accepted unitarity as a fundamental feature of quantum theory because it is a critical premise of the "Information Loss Paradox" in the first place. Anyone even casually acquainted with the "paradox" could tell you that: if you adopt a collapse theory, then there is no paradox to begin with because unitarity is not a fundamental postulate of such a theory.

    Indeed, as I took pains to point out in my paper, unitarity is a stricter condition than the condition needed to construct the paradox. The "no information loss" condition that serves as one horn of the supposed dilemma is really code for having a bi-deterministic theory—a theory in which the fundamental dynamics is both deterministic in the forward direction of time and retrodictable in the backward direction—which follows from unitarity but is not equivalent to it. Any fundamentally indeterministic theory, such as a collapse theory, is not subject to the “paradox” at all. And what my paper spells out, following Wald, is that in GR one only has a right to expect determinism and retrodictability for evolution from the state on one Cauchy surface to the state on another, whereas the evolution alluded to in the "paradox"—from Sigma_1 to Sigma_2out—is from the state on a Cauchy surface to the state on a non-Cauchy surface. So from the get-go, no one ever had any right to expect information to be preserved in that evolution, i.e. that the evolution be retrodictable.
    Hawking appears to have come to the mistaken belief that this constitutes a paradox from his sloppy usage of the term "no longer exists". He mistakenly concluded from the fact that the event horizon no longer exists relative to an observer in the future light-cone of the evaporation event that the interior of the light-cone also does not exist relative to such an observer. That is a demonstrable error once you properly define what "no longer exists" means in the context of Relativity. And having shown that the interior of the event horizon still exists after the evaporation, I locate the place where the "lost information" resides: in the interior. The only surprise in this solution is the realization that while the initial Cauchy surface is connected, the post-evaporation Cauchy surface is disconnected. I explain how that could come about due to the failure of a standard presupposition of GR (not of QM), namely that the space-time forms a manifold. That condition need only fail at the Evaporation Event to disconnect the Cauchy surfaces.

    Con't

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  186. That is the argument of my paper in a nutshell. You have never come close to laying a glove on it. You have not shown that there is any error in the argument, and haven't even been trying to for months and months. What you have done is to try to shift the argument to the context of AdS—which is not a globally hyperbolic spacetime and therefore admits of no Cauchy surfaces—so you could bring to bear the unproven and unformulated AdS/CFT conjecture in a bid to show some inadequacy of my argument in that setting. That gambit has not gone well.

    In the first place, since the "information loss paradox" has always been formulated in the setting of a globally hyperbolic space-time—absolutely everyone refers to Sigma_1 as a Cauchy surface when explicating the paradox—it is not clear what the paradox is even supposed to be in the setting of AdS. In order to get any sort of determinism there, something must be done about the timelike boundary of AdS. The technical fix for this involves adding a surface term to the Hamiltonian and some boundary conditions for the boundary. These technical details, spelled out using the formulation of Regge and Teitelboim, are of zero fundamental significance for my solution since all of the important physical action takes place deep in the bulk and not at the boundary.

    So what you have been attempting—and repeatedly failing—to do is to argue that my solution must fail for evaporating black holes in AdS. Sometimes the argument is that the pre-evaporation physical state space (Hilbert space) in not a tensor product space while the post-evaporation physical state space is a tensor product space. If the evolution from the pre-evaporation condition to the post-evaporation condition were unitary this could not occur, because the structure of the Hilbert space would be preserved under a unitary evolution. The problem is that once one defines the “post-evaporation Hilbert space” properly, viz. as the space of states that could arise from the evaporation of a black hole, this argument goes up in smoke.

    You have repeatedly tried to muddy the waters by defining the post-evaporation “physical” Hilbert space in an inaccurate way, as the space of states that satisfy some but not all of the conditions that a post-evaporation state must satisfy (in particular, dropping the condition that it have arisen from the evaporation of a black hole!). The led to an interminable squabble about the conditions for being a “physical” state, which must be the condition I just gave if any observations about the structure of that Hibert space are to have any bearing on the topic at hand. The extent to which you have tried to deflect from actual, concrete physical questions to completely uninteresting purely semantic questions (how to use the term “physical’) cannot be overstated, because your arguments, instead of being physical arguments, arise from logical fallacies of equivocation.

    The other gambit that you have tried and failed at uses the CFT part of the AdS/CFT conjecture to raise a problem for my solution. There the idea is that the spectrum of the Hamiltonian of the CFT is discrete and (up to symmetry) non-degenerate, therefore by the conjecture the spectrum for the Hamiltonian of the AdS gravity theory must be discrete and (up to symmetry) non-degenerate, whereas the Hamiltonian of what you define as the “physical Hilbert space” in the post-evaporation regime according to my solution must be massively degenerate. But here you are caught in a hopeless dilemma. Either by the “post-evaporation (“late”) physical Hilbert space you mean the space of physical states that could arise from the evaporation of a black hole, in which case the isomorphism of the spectrum is guaranteed by unitarity, or you mean the spectrum of the Hamiltonian on some other space of states, in which case who cares?

    Con't.

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  187. Indeed, the CFT part of the conjecture seems to be a complete red herring since you claim to be able to motivate the non-degeneracy of the Hamiltonian by considerations to do only with AdS and gravity. If you can, then bringing up AdS/CFT in the first place was a waste of time. If you can’t, then that is prima facie evidence against AdS/CFT. In short, the whole AdS/CFT gambit has been a bust.

    Realizing that you were getting nowhere, on March 26, one week ago, you came up with this brand new objection: “The problem is that you haven't addressed the obvious issues of how to make sense of measurement theory in your setup. Here I am just asking for a concrete set of rules that could be handed to a practically minded experimentalist -- I am not asking about "deep" philosophical issues’. And that rather unclear demand has led us here.

    “Making sense of measurement theory” could mean either of two quite different things. One is providing a practical calculational technique for deriving predictions about the results of “measurements”, without ever rigorously defining just what a “measurement” is. That is how standard QM is set up, and seems to be what is indicated by “practically minded experimentalist”. Since all that is asked for here is a tractable mathematical technique for making (probabilistic or statistical) predictions, there is no objection to following standard QM here and using whatever mathematics is convenient. My first attempt to answer this question took that tack.

    But then, upset that I was actually answering the question you asked, you tried to put more constraints: the mathematical technique had to be restricted to using only “physical” states, and since the “standard” theory uses projectors the projected states must be considered “physical”. But the only conceivable way to make sense of this constraint is that what is wanted is more than is given by the standard “axioms” of QM, you actually wants a theory that solves the measurement problem in a principled way making reference only to physical facts.

    This is a request I relished, because this is the sort of thing I know quite well. So I offered a principled physical solution of the measurement problem that does not postulate any collapse of the wavefunction but only deterministic unitary evolution as the paradox requires. And since I cannot defend any Many Worlds approach, I had to explicate a pilot wave theory. But now, because I am answering the second possible reading of the question you asked, you are accusing me of “throwing up dust”.

    All the dust is coming, and has been coming for months, from your side. As I have said, if you want to get into the foundational issues, I am more than happy to do that. But if you are going to complain that I am not respecting “the axioms of QM” (whatever it is you take those to be) because I am offering something more clear and exact and physically principled than the “standard approach” then this is completely inane. The “standard approach” to QM is conceptually incoherent and unprofessionally vague, as Bell insisted. Professional physics ought to be able to do better. If you are going to complain because I am doing better (and hence rejecting the standard axioms) then this has reached a level of terminal pointlessness.

    Con't.

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  188. So, according to me, is where we stand and how we got here. I have detailed my solution to the paradox and defended it. I have gone along with your allegation that the solution is incompatible with what anyone working in quantum gravity “knows” long enough to see that there is not a speck of truth in it. I learned a bit about surface terms from Regge and Teitelboim but that turned out to be completely orthogonal to the issues at hand and not very remarkable so long as you keep in mind that the dynamics of the theory is implemented via both the Hamiltonian constraint and through a Hamiltonian in a Schrödinger equation for the boundary term. And I have learned that the people who claim to understand all this have no limit to the extent that they will congratulate each other on their sharp insight when posting anonymously on the internet, as you and physphil and JimV have taken to again. And so I am putting my foot down for the last time. Above you have my summary of where we are, what I argued, what you tried to argue, and why your arguments failed. I will defend every syllable. I will proceed on a discussion of the options for solving the measurement problem and why the “standard axioms of QM” fail to solve that problem so holding up those axioms as something that must be respected is positively idiotic. I will go into more detail about the shortcomings of the Banks, Susskind and Peskin paper, which is relevant to a discussion of foundational issues and “measurement theory”. But I will not continue any more to interact with people so scared of taking responsibility for what they post that they remain anonymous. If you are so positive that you know what you are talking about, and that anyone reading this will recognize your sage wisdom over my confusion, then come out from behind your mask. And the same goes for physphil and JimV and dark star and whoever else there is lurking after all this time. I am willing to engage on even terms, where everybody puts their actual reputation on the line, but I will not respond to anonymous comments any more, including to BHG. If you have the courage of your convictions, show it. If not, enough is enough.

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  189. physphill (for the last time to you this internet mask),

    If you believe that small black holes are forming an evaporating all the time then you are confused. There was a rather extensive physical discussion about whether the LHC could possibly create black holes on the (mistaken) assumption that such a black hole could devour the earth. There was at least some point to asking that question because the high energies of the LHC are capable of producing entities that do not occur in everyday circumstances, e.g. Higgs bosons. But if you are under the impression that small black holes are constantly forming and evaporating in the course of very delicate quantum computation experiments then any High-energy physicist can straighten you out and explain why that is not the case. Indeed, if small black holes were constantly forming and evaporating all the time the reality of Hawking radiation would be more than manifest and Hawking would have gotten the Nobel prize long ago.

    Your confusion is a common one: mistaking "virtual events" that are mathematical fictions for real events, and "virtual particles" for real particles. We can go into why this is a mistake, but the paragraph above assures you that it is indeed a mistake, so you should reflect on that. And if you want to continue discussing this with me, you will have to post under your real name. See the post to BHG above.

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  190. Tim,

    You are evidently still missing the point, on both technical and conceptual levels.

    Regarding the former, we have established the following "theorem": to realize your scenario within the axioms of QM, as enunciated in essentially every text on the subject ever written, the Hilbert space that appears in the axioms needs to have a tensor product subspace, and a Hamiltonian which is highly degenerate in this subspace. Note that your complaints about being forced to use a specific mathematics, or about whether all the states are "real", have no bearing on this -- use whatever math you want, and call the states whatever you want. The theorem still holds. On the other hand, we know that the CFT describes gravity in AdS with a non-degenerate Hamiltonian. So however AdS/CFT describes black hole evaporation, it is not realizing your scenario. It's that simple, although I have by now given up on expecting you to admit that this is the case (why not surprise me...).

    On a conceptual level, it's puzzling that you don't seem to appreciate the basic issue, which is incredibly simple and intuitive. To say that information is lost to the external observer is to say that for whichever way the outer region can be, there are many ways that the inner region can be. Since energy is a property of the outer region alone, it follows that all these different ways the inner region can be don't contribute to the energy, and hence the energy spectrum is hugely degenerate. Now, the way I have stated this is of course slightly imprecise, which is why we turn to the precise version above, but it gets the basic point across.

    It's clear that you are hoping that issues regarding foundations and interpretations of QM will be needed in this discussion, but I just don't see it. No one is denying that there are various unresolved issues and confusions regarding QM and in particular measurement, but there is no indication that these issues are any way tied to black hole evaporation, and trying to force them into the discussion appears designed to obfuscate rather than clarify.

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  191. A couple of points of logic:

    Even if we grant all TM's premises, which I implicitly did in my previous summation, although I do not accept them all in fact, by the same token that BHG cannot prove that all states which can be traced through a BH formation and evaporation have the form of a tensor product, TM cannot prove they do not. Thus, as I said (trying to be charitable) at best there is some unknown chance that a remnant solution could work under ADS/CFT. (To evaluate that chance we would have to do the full calculation of all possible such states, which we lack the equations and technology to do.)


    physphill said... "A small black hole can form and evaporate in a tiny fraction of a second."

    Tim Maudlin said... "If you believe that small black holes are forming an evaporating all the time then you are confused."

    The second statement seems to be a non sequitur.

    And the part of TM's argument that my mathematical intuition, such as it is, rejects is that these two statements are compatible:

    a) the only physical states that result from a BH evaporation are those that could be mathematically shown to have evolved through that process, and only these should be considered to determine whether degeneracies exist.
    b) all standard physical states (not just the above) can be assumed in the QM process for predicting statistics of experimental results involving the thermal radiation from the BH.

    This seems analogous to me, for example, to applying a Gaussian or normal distribution to the results of a sampling from a population that is bounded between 0 and 1.

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  192. For me the most interesting thing about this discussion has the claim that conservation of information depends on our interpretation of quantum theory. It’s been stated here that if we espouse a collapse theory then information is lost all the time (measurements are irreversible), whereas if we espouse some other interpretation (many worlds, pilot wave, …) then information is conserved. This is interesting because people usually say that viable interpretations of quantum mechanics are empirically indistinguishable. If so, then conservation of information is not an empirically meaningful principle. The alternative is that different “interpretations” actually are different theories with different empirical predictions involving the (non)conservation of information.

    People have mentioned here that tests of unitarity have been performed, but I think those tests just cover the evolution between “measurements”, so they may not address the question at hand. This has always been a problem for trying to apply von Neumann style rules to the whole universe. The very notion of empiricism seems to imply measurement, so are we to conclude that the concept of empiricism just doesn’t apply to the universe as an unmeasured and unmeasurable entirety? If we restrict our attention to actual measurements we make within the universe, is information conserved in a meaningful sense during those measurements? (We might define “meaningful” as excluding any information that may hypothetically exist in other branches of the “universe” that are inaccessible to us.)

    The pilot wave theory is (as far as I know) non-relativistic, so I don’t think it’s valid. Maybe the “information paradox” really only applies to an unmeasurable many-worlds universal wave function evolving under a unitary law, so the measurement problem doesn’t come into play (since there is no measurement), but if so, the empirical significance of the “paradox” seems questionable (to me).

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  193. BHG—

    Just to be clear, this statement of yours is nonsense: "Regarding the former, we have established the following "theorem": to realize your scenario within the axioms of QM, as enunciated in essentially every text on the subject ever written, the Hilbert space that appears in the axioms needs to have a tensor product subspace, and a Hamiltonian which is highly degenerate in this subspace.". Try this: go pick up half a dozen texts on QM, search through them, and write down the "Axioms of QM" according to each of these texts. See 1) whether you can even find them in each text (good luck) and 2) if you can find them, whether what you find in each text agrees with what you find in every other text (good luck squared). Now if things turn out as you assert they will, you will find the same set of axioms in each text. Next, write down these axioms in explicit form, and investigate to your own satisfaction whether they are consistent. And once you are satisfied that they are consistent, present them here. I would love to know what these commonly held axioms in "essentially every text on the subject ever written" (you certainly are well read!) are. I don't believe for a moment that you can do this.

    In 'Against 'measurement'" John Bell did this exercise for three different texts and found three different sets of "axioms", all inconsistent with each other and all incoherent in themselves. Shankar announces a set, which are essentially those of von Neumann, and which require a sharp distinction between measurement and non-measurements, a distinction that has no physical basis. But despite this, you act as if there are some unproblematic "axioms of QM" that everyone agrees to. I assert that there is no such set, and that trying to hold my account to such a fictitious set of axioms is nothing but a ruse. You just wheel in whatever principle that is convenient and declare it is an axiom that can find in all the texts. Very well then: go find it in all the texts. Or admit that you can't.

    We have gotten into foundational issues not because I have been insisting on them but because you have. If you are confident you really understand these issues and can speak authoritatively on them, feel free. But I certainly won't be either persuaded or bullied by assertions that "in essentially every text on the subject ever written, the Hilbert space that appears in the axioms "needs to have a tensor product subspace, and a Hamiltonian which is highly degenerate in this subspace." I already explained why this is false. If you go through the exercise, at least one of three things will happen. !) you can't even locate "axioms" in most books, and what you do find in different books will be different, 2) when you try to assemble a list of axioms you can't get anything coherent and/of 3) if you focus down on some "axiom" that is supposed to rule out my solution, it will at the same also rule out your account if AdS/CFT. Try it. The outcome will be instructive.

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  194. Tim,


    There is absolutely no reason to expect that the question of how a measurement/collapse occurs in QM has any particular relation to the issues raised by black hole evaporation, and getting into this topic is not helpful at this stage. That's why I have emphasized that I have demonstrated why your scenario clashes with AdS/CFT without wading into these murky waters. The axioms of QM, defined as rules to compute probabilities, are explained in every standard QM textbook that I know of, and are learned by all students who have taken an introductory course in the subject. And the postulates are just as I have stated them: A system is described by a state in a Hilbert space, an observable by a Hermitian operator, and probabilities computed via inner products of states with eigenvectors. Why do all the books say the same thing? Because these rules have been found to accurately describe all known physical phenomena. The CFT in AdS/CFT certainly obeys these rules, so I am hardly making an unreasonable request by demanding that we formulate the gravity side in these terms as well. Within this framework, after much prodding, you eventually conceded that for your scenario to be realized your Hilbert space needs to be a tensor product, which implies that the Hamiltonian is degenerate. This clashes with AdS/CFT. So this theorem definitively rules out your scenario (assuming AdS/CFT) without getting into issues regarding measurement.


    I also provided a simple intuitive version of this argument that is easier to grasp. If you would put forth even a glimmer of a counterargument then we could potentially get somewhere, but I see no indication that you even understand the argument to begin with. All I get back is sneers, insults and chest thumping.


    I also can't help commenting on your response to physphil, who quite accurately noted that ". A small black hole can form and evaporate in a tiny fraction of a second." to which you responded "If you believe that small black holes are forming and evaporating all the time then you are confused.", and then went off on an unrelated tangent about black holes at the LHC. Either you are totally missing his point, or this is just a cheap debating tactic (notice how you twist his/her wording?) -- either way it is the kind of response that is utterly exasperating. If you collide two particles at Planck scale energies then every expectation is that a black hole will indeed form and evaporate in a fraction of a second, which was physphil's point, and one that is highly germane to the topic at hand. You respond by twisting his/her remarks to make it seem like they were referring to black holes being formed under experimentally accessible conditions, i.e at the LHC. This behavior is not conducive to constructive dialog.

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  195. BHG (since you refuse to identify your self, but I am just going to stop responding)

    Your comment about physphil reveals beyond all doubt that either you are not even reading these posts or you are just trolling. Here is what physphil wrote:

    "you write "Not in any serious sense: no collapse theory that has a prayer to work could get from the pre-formation to the post-evaporation states of a black hole without many, many collapses along the way. "

    That's either obfuscation or a basic misunderstanding. A small black hole can form and evaporate in a tiny fraction of a second. People do experiments all the time that test unitary evolution (like any process in a quantum computer) that last longer than that"

    Physphil is clearing asserting that microscopic black holes are forming and evaporating *during the time that people do tests of unitary evolution, such as quantum computation tests*. The functioning of a quantum computer does indeed require unitary evolution of the computer's quantum state: that's why it has to be shielded against interaction with the environment and the consequent decoherence. The notion that anyone has ever tested a quantum computer in an environment in which black holes were forming an evaporating is absurd. This is exactly the sort of thing I am fed up with: if you are going to be so sloppy and inattentive (or flatly dishonest) to portray my perfectly on-target response to physphil as a "cheap debating tactic" then your behavior should come back to you. As it is, you throw around these baseless accusations with impunity.

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  196. No, that has nothing to do with what I was saying.

    My point was that in a (future) experiment, someone could create a microscopic black hole in a lab starting from a controlled initial state, and then observe it evaporate (measure the outgoing radiation). That whole formation/evaporation process could take a small fraction of a second and could be repeated many times. Done carefully enough and repeated enough, that would test whether the (outside) evolution is unitary, just like other controlled quantum experiments do.

    If (as you claimed) a "collapse model" would preclude unitary evolution in that experiment because many collapses would take place during the formation/evaporation process, it would have exactly the same effect on any other QM experiment or on a QM computer. Since QM computers work, no model like that is viable or interesting and I don't know why you would even bring it up. I guess you thought black holes always live a long time?

    Just like bhg says there's nothing special about black hole formation or evaporation that forces us to think about measurement any more than in any other application of QM. My point all along is that there's no reason why I can't use standard Copenhagen/projective measurements, but then I get nonsense if I follow your rules since the state after measurement isn't supposed to exist. It's just one more way of showing that you are radically modifying the rules of QM.

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  197. For the last time to anonymous physphil:

    You write: "Since QM computers work". No one has ever demonstrated that QM computers do work in the presence of evaporating black holes. Period. And if you insist on a collapse interpretation, as you are doing, there will be deviations from the "standard" predictions, as that phrase is usually understood. That's why one can test, and people do test, collapse theories. If you are unfamiliar with this then you are unfamiliar with the basic results in Foundations.

    If you think there are "rules of QM" try to explicitly write them down. There are no such rules, and nothing I have said is a "radical modification" of any actual practice. You keep insisting on collapses during measurement, which puts you out of the mainstream. Indeed, collapse theories are generally considered to be modifications of QM. So this line of "argument" (which actually contains no arguments) is of no use. If you want to talk about Foundations, then first learn about Foundations,

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  198. Tim,

    Your exchange with physphil does in fact illustrate very well the situation. I strongly suspect that anyone reading this will view it the same as I do: you are just missing his/her point entirely, and putting words in his/her mouth that are completely unrelated to the argument being made, which is in fact quite clearly expressed. There was no claim being made about quantum computing in the presence of black holes! The point being made was that the time scales involved in the evaporation of a (gedanken) microscopic black hole is the same time scale over which quantum computers exhibit coherence, not that one is happening in the presence of the other.

    This is more or less the same situation that has occurred over and over in our exchanges.


    Reading this over, I don't think you are willfully obfuscating. Rather I think that you tend to operate under the assumption that your debate partner is a drooling idiot, and so you assign to them the worst conceivable interpretation of their remarks. A great leap forward would occur if you would simply ask for clarification when an argument seems faulty to you, rather than telling the person how "silly" they are and then composing a long response that utterly misses the point.

    So that said, I can now understand how things fell apart in our dialog. For example, I have the strong suspicion that you think I am trying to "disprove" your scenario, whereas I have been trying to emphasize that it clashes with AdS/CFT according to what we know about the latter, and that you should take this seriously since AdS/CFT is by far our most concrete formulation of quantum gravity. You can think of the CFT as a machine that operates in Hilbert space and spits out probabilities according to the quantum rules. It definitely has unitary evolution and a non-degenerate Hamiltonian. Now, we want to translate the story it tells into bulk language. What I have demonstrated to you, is that this bulk story cannot involve a splitting hypersurface, since that leads to a degenerate Hamiltonian. And I also explained this on general grounds: if you want to lose information by sequestering it in a disconnected region of space, then a degeneracy of the Hamiltonian is inevitable, since a disconnected region carries no energy.

    I suspect that if you considered my arguments after dropping the assumption that I am a raving lunatic, you might see that it in fact makes sense, and progress could even be made by engaging seriously with these ideas.

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

    Non-sequitor. I don't think you understand me at all. Or maybe you are throwing more dust.

    "And if you insist on a collapse interpretation, as you are doing, there will be deviations from the "standard" predictions, as that phrase is usually understood."

    It was you who brought up collapse interpretations. I just follow the standard rules of QM.

    "If you think there are "rules of QM" try to explicitly write them down."

    I learned QM from Landau+Lifshitz. The rules are there. They are also in Dirac and every other QM book I know and they all agree. The only rule that matters now is that the state after measurement is an eigenstate of the operator. If that was not true, immediately repeated measurements of the same thing would give different results. But that eigenstate doesn't exist, according to you. That means repeated measurements of the same thing (positions of the Hawking particles) will give very different answers every time. Particles will magically teleport. That's radical. Sorry if you do not like it.

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