# Shores of the Dirac Sea

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## Maldacena’s information paradox

January 15, 2009 by Moshe

Back in 2001, in a truly beautiful paper, Juan Maldacena formulated a version of Hawking’s information paradox, which has the added advantage that it could be discussed and analyzed in the context of a complete background independent theory of quantum gravity, namely that of the AdS/CFT correspondence.

This variant is similar to the original paradox, formulated for black holes surrounded by flat space, in that it displays a sharp conflict between properties of black holes in classical General Relativity, and basic postulates of quantum mechanics. Alas, it is also different in many crucial ways from the original paradox. Despite that, Juan’s proposed resolution to his paradox seems to have led to Hawking’s arguments, who managed to convince himself (though I think it is fair to say not too many others, unless they were already convinced) that information is not lost after all in the process of black hole formation and evaporation.

To present the paradox, first I’ll have to tell you just one fact about AdS spaces. AdS space has a center and a boundary, and a gravitational potential which makes it difficult to approach the boundary. Anything you throw out from the center towards the boundary of AdS space will eventually come back and hit you. It is as if you are surrounded by reflecting mirrors. This fact is a useful in many ways, in particular in stabilizing black holes.

For certain black holes (the so-called large ones, which are the ones we’ll discuss) this  means that the Hawking radiation emitted by the black hole is constantly reflected back towards the black hole. Soon enough the system will achieve an equilibrium state, where the black hole exists is in detailed balance with its Hawking radiation, it absorbs and emits the same amount of radiation.

Which means that the black hole never disappears, unlike its cousin in asymptotically flat space, it is eternal. Therefore, the original information loss paradox (roughly, what happens to all the information stored in the BH after it completely evaporates?) cannot be formulated in this context (at least using the black holes I am describing). Tough luck.

So, instead of looking at the complicated process of black hole formation and evaporation, let us start with one of those stable black hole and perturb it, add to it one bit of information, poke it just a little bit, you get the picture. Then we can try and follow the history of the perturbation and ask: after some time has passed, can we trace back the nature of the perturbation? can we retrieve the information we sent towards the black hole from the set of measurements we can do later in time?

Turns out, this process is puzzling in the same way the original paradox was, but the advantage is that the new puzzle can be formulated in a complete microscopic theory of quantum gravity, namely in the gauge theory variables of the gauge-gravity duality. Let me describe the process and show where the problem lies.

The original black hole, formulated in gauge theory variables, is simply a thermal state, it is a thermal equilibrium of quarks and gluons at a temperature which equals the black hole Hawking temperature. Now, imagine shooting a very energetic particle into that thermal gas. What will happen?  it will start colliding with all the gas particles, thereby slowing down while speeding up the average speed of the other particles. Wait long enough and you will find a gas of particles in a new thermal equilibrium state, in a slightly higher temperature.

This is exactly what happens in the gauge theory, and it has a simple interpretation in the gravity language: you send some bundle of energy towards the black hole, and it falls behind the horizon. The black hole “rings” for a little bit and then settles back to an approximate equilibrium state, which is a slightly larger black hole. The perturbation decays exponentially as function of time (the exponents are  known as quasi-normal modes), after a little while you forget the black hole was perturbed, the information has disappeared behind the horizon.

But now comes the paradoxical part:  in quantum mechanics this description can only be an approximation to the complete situation, because information is never lost. In principle you can always perform precise enough measurements on that seemingly thermal state to retrieve the information about the original perturbation. This is the principle of quantum mechanical unitarity, evolution in quantum mechanics is reversible.

The clearest way to see what is going on is looking at the behavior of the perturbation after a very long time. In the story we told, the effect of the perturbation dies off exponentially with time.  In the complete quantum mechanical system that is what happens initially: for a very long time the effect dies off, but it cannot go all the way to zero, lest the information really be lost. Instead, after very long time the system will undergo Poincare recurrences: it will come back arbitrarily close to its initial configuration. So, looking at the behavior of the perturbation after extremely long times, of the order of the so-called Poincare recurrence time, is one clear way of deciding whether information is lost or not.

This is very much “in theory” type of situation, the common (unfortunate) analogy seems to be trying to retrieve the contents of a burnt book from its ashes. Nevertheless, this is a question of principle: is there a set of measurements we can do on that perturbed black hole which can be used to retrieve the complete information in the original perturbation?

Looking at the black hole solution of  General relativity, if we take this seriously as the complete description of the situation, the precise answer is that the effect of perturbation dies of exponentially, for all times. Wait long enough and the system is arbitrarily close to an equilibrium state. In that case, you can never retrieve the information, it is lost.  You can say words like “it continue to live behind the horizon”, but for you, the observer staying in the safety of the outer region, that information is lost.

On the other hand, the system has a dual which is a more complete and microscopic description of the theory. When looking at the process in the complete theory you can see that information is never really lost. In fact, you can do more – you can also truncate the complete theory, taking into account classical gravity only, and observe that in fact with this truncation information really is lost. In other words, you can see the precise details of how the illusion of information loss comes about.

So, what is the current state of affairs? in the dual language, it is clear that the system is unitary, but it also became clear that in the limit corresponding to classical gravity, there is an apparent information loss. In fact, this information loss looks like a generic phenomena of that (so-called large N) limit in a whole slew of models, even those not dual to any known gravity theories. So, it looks like this paradox is pretty much solved, at least no mystery as to whether information is really lost or not.

What is missing at the moment is a phrasing of the resolution in the original, gravitational variables. Since we have a complete description of the story, it is known in detail that information is not lost, and why it seems to be lost as an artifact of the specific approximation we are making.  What we are missing is the always elusive question of the “mecahnism” of retoring unitarity. What are the essential ingredients of the complete story that are needed in order to restore unitarity. Can we phrase the solution in terms of some coherent narrative phrased in the gravitational language? would be nice if we could, maybe we are not that far off, stay tuned.

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Posted in high energy physics, Quantum Gravity, string theory, thermodynamics | 55 Comments

### 55 Responses

1. I don’t see where information would be lost, even if the perturbetions die off exponentially. thy are still there after any finite amount of time, or not?

As a simple example, take a lightily damped classical harmonic oszillator. it’s amplitude falls of exponentially, but it never comes to a complete halt, if you are looking close enough.

2. The automatically generated possibly related posts make quite an interesting match.

Question: Does that mean you can only trace the fate of information that was ‘thrown in’ from the boundary? What can you say about states that were not formed this way? Could you maybe comment on this paper?.

3. Matthias, damped system are exactly those where unitarity is violated, they are not time reversible. Wait long enough and all that distinguishes different states gets damped.

Bee, I am only saying that we can form a clean puzzle for those set of states. We can also ask more general questions, and those would be more difficult to answer.

The possibly related posts are, on this rare occasion, possibly related. At least some of them are. It’s a miracle!

4. on January 16, 2009 at 12:46 am Just Learning

Incidently, there is an article in the January issue of Physics today by Klebanov and Maldacena called

Solving quantum field theories via curved spacetimes

5. I ‘ve always been curious, where do gravitons come into the picture of AdS/CFT? One always hears about the dual picture; the gauge field and the gravitational field. Is the graviton a complication here?

6. Mark, precise details depend on context but very roughly speaking the graviton is a complicated state when written in terms of the gauge theory variables. It’s the no free lunch principle.

7. mark a. thomas:
The gravitational field is a classical description of the dual to the boundary gauge quantum field theory. If you are in the large N limit, large lambda limit, then the first limit tells you that quantum corrections (handles in your stringy diagrams, if you wish) are suppressed, while the second limit tells you that while your gauge theory is strongly coupled, the dual bulk theory is weakly curved. As far as bulk calculations go, this is great, which is why people work in these limits, but the correspondence is conjectured to work away from this special point.

Back to your question, gravitons are a quantum mechanical description of weak-field gravity, so unless you want quantum answers to quantum questions, you don’t need to consider gravitons.

The big question is how is the information problem resolved? Is it because gravity is only a true description at the strictly infinite N limit, and somehow finite N resolves it (stringy black holes).

In answering your question, I have managed to confuse myself, wondering if unitarity is what we’re really violating here (in the classical limit, presumably one needs a different word for unitarity). In any case, I think even the classical field theory should exhibit recurrence.

8. Lionel, I’d say that Liouville thm., about preservation of the phase space volume, is the classical counterpart of QM unitarity. But, classical in the bulk means infinite N on the bdy, the field theory is still QM (which has to do more with lambda, so classical bdy field theory is lambda=0).

9. Dear Lionel:

Classical fields have infinite numbers of variables and don’t exhibit recurrence (except in integrable models). The main reason for this is that a ball of size one half in an infinite dimensional phase space can fit in a ball of size one an infinite number of times: hence the usual proof of Liouville’s theorem doesn’t do anything. In the quantum theory the high frequency modes get regulated away (they are too energetic because of Planck’s energy cost to excite a frequency $E=h \nu$ ).

In classical fields one would expect that everything would dissipate into high frequency noise in the end. You can call this heat if you want to.

10. In this context though we are discussing field theory on a finite space (3-sphere), so I think both the classical and quantum field theory will have recurrences at finite N, and dissipation at infinite N. Unless of course I am missing something.

11. Great post. Thanks for taking the time to write it.

Although I’ve personally always believed that the information is not destroyed, I can see why a true skeptic would not be moved by Maldacena’s argument. If we can’t actually tell an internally consistent story on the gravity side that explains how the outgoing radiation encodes the initial state, there will always be wriggle room. But hopefully, as you say, we are getting there!

12. Sean, I don’t believe it to be the case, but the logical possibility exists that the gravitational degrees of freedom are just the wrong set of variables to give you the correct answer. I find the case in AdS to be 100% fool proof, even if there is no story to tell in the (fundamentally incomplete) gravitational variables. But again, I do believe there is such a story, it is probably interesting and we might even learn something about the original information paradox.

(Incidentally, Juan did attempt in that paper to resolve the paradox in the gravitational language, all the stuff with the sub-leading saddle points which Hawking took note of and tried to apply to his context, but that solution is incomplete for various reasons.)

13. on January 16, 2009 at 12:03 pm Pope Maledict XVI

I agree that the case in AdS is “100% foolproof”. The problem is to turn this into an understanding of what happens to unitarity in the case of realistic Schwarzschild black holes, which are so very different.

14. on January 16, 2009 at 12:05 pm Just Learning

“So, looking at the behavior of the perturbation after extremely long times, of the order of the so-called Poincare recurrence time, is one clear way of deciding whether information is lost or not.”

Would it also be correct to assert that looking at a large number of similarly perturbed systems one would also be able to see if information is lost or not? Is the poincare recurrence time an expected value or not? Although I imagine it would be difficult to find a large collection of black holes.

15. Hi Moshe:

The way I think of it is in terms of the UV catastrophe. The classical system will equipartition finite energy in an infinite number of modes, so all the energy ends up in the high frequency noise. There is an infinitude of frequencies there, and the Poincare recurrence time will become infinite.

The UV catastrophe is what quantum mechanics cures.

Infinite N is a different classical limit. There, the spectrum of the quantum system becomes continuous at high energies only in the strict infinite N value (it’s akin to going to an infinite box). At finite N the systems has a finite number of states below any energy.

Dear Pope:

I made a recent attempt with one of my students to explain how small black holes in AdS look like in the dual theory. These small black holes are going to be very similar to Schwarzschild. Maybe you might be interested. Here is the link
arxiv:0809.0712. It turns out that quite a bit of the microscopic description is not all that different from the big black holes in AdS. The main difference is that the system is not in equilibirum, but the internal degrees of freedom of the black hole are in some type of dynamical thermos that makes it possible for them to have a different temperature than the ambient space.

16. Dear Moshe,

I agree with David that the Poincare recurrence time is infinite in classical field theory. There are infinitely many Fourier modes and each of them can carry some, arbitrarily low, energy. So the energy will dissipate into them – mostly the infinite-frequency ones.

In quantum field theory, one photon is a minimum, and that’s why you can only use the modes up to reasonable frequencies linked to the temperature. That’s why the finiteness of the Poincare recurrence time is only dictated by the IR physics, but this statement is only true at the quantum level because the hbar=0 limit is where all frequencies can participate (hf=0, too) and even a finite space contains infinitely many usable modes.

Also, I agree with him that the spectrum of dimensions of operators in the N=4 is discrete for any finite N_colors because it is always a quantum theory on a compact S^3 times time. When one looks at the spectrum of operators and gives them no geometric interpretation, there is really no separate hbar because of all of them are just “some” operators.

One might think that when one interprets the states geometrically, it is possible to be sending hbar to zero independently from N to infinity, effectively by considering large, slow, heavy objects and processes. But that’s an illusion because finite g, finite N really means that the volume and curvatures are close to the Planckian ones, and there are no large classical objects in a Planckian seed of space. One can be sending g and hbar to zero simultaneously to get the planar limit, too. I think it’s fair to say that it’s classical field theory, although equivalently described by the 1st quantized quantum theory of one string (the planar limit).

David’s paper that I had missed is very provocative – approximately thermal states with respect to a dynamically generated subgroup of the gauge group – wow. It must clearly be possible because it’s about the localization on the sphere. For QCD, such a segregation of an SU(2) inside would probably not be possible, would it?

Some comments about the comment #1 whether the information is getting lost if the perturbations “only” decrease exponentially:

http://motls.blogspot.com/2009/01/killing-information-softly.html

Best wishes
Lubos

17. If a black hole is not to accrete faster than it radiates it must have a present temperature greater than 2.725 (+/-)0.0004 K cosmic background radiation. This requires it be less than ~4.7×10^22 kg (~0.79% of Earth’s mass). There are no obtainable circumstances in which a black hole can form in a universe cooler than its own temperature – with one exception: CERN’s LHC.

GARDYLOO!

18. Sorry for the intrusion here from the “three dimensional world and time, into your higher abstract perspectives

Again from a layman’s perspective it is extremely hard for the ordinary person to image what those strange worlds are, that mathematician and physics theorists are talking about.

Now Imagine your going to turn my story of the Cave around?

“Whereas Plato envisioned common perceptions as revealing a mere shadow of reality, the holographic principle concurs, but turns the metaphor on it’s head. The Shadows-then things that are flattened out and hence live on a lower-dimensional surface-are real, while what seemed to be more richly structured, higher dimensional entities (us; the world around us) are evanescent projections of the shadows.” Brian Greene-The Fabric of the Cosmos, pg. 482

He puts an asterisk beside that quote to show below on that support page, to provide the meaning of this.

While this seems like a turning of the “thoughts about of Plato” I really don’t see it that way.

For instance Greene goes on to write, “For instance, in Maldacena’s work, the bulk description and the boundary description are on an absolutely equal footing.” Pg 484.

I think my illustration of the sun and the opening of the cave embeds the realization that in a “reductionistic view,” energy is well considered as we see the entropic values of the universe as descriptively as it is. Not to rewrite the work branes may have enforced from an “abstract and mathematical process” but to concur that the three dimensionality reality is “solidified” to the world around us, from that higher dimensional state.

Geometry is a branch of mathematics that deals with points, lines, angles, surfaces and solids. One of Coxeter’s major contributions to geometry was in the area of dimensional analogy, the process of stretching geometrical shapes into higher dimensions. He is also famous for “Coxeter groups,” the inversive distance between two disjoint circles (or spheres).

Now if one retains this understanding about Plato you might be moved to consider Coxeter views to those abstract objects.

I do not believe that this is inconsistent with what is continued to be expressed in the Tomato soup analogy in recognition of Maldacena’s work in the five dimensional spacetime.

Best,

19. David and Lubos, thanks, that makes perfect sense. I agree that David’s paper is interesting and provocative, blog post maybe?

20. on January 17, 2009 at 1:14 am Just Learning

So the day’s discussion has made it clear:

Information is not lost, only practically undecipherable.

This is a similar statement to that of entropy, where energy is not lost, only made unavailable to do work.

Utility is then maximized with the optimal pareto efficiency.

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