## July 22, 2004

### To loose or not to loose information

#### Posted by Urs Schreiber

Over here and here a link to a transcript of Stephen Hawking’s talk on the apparent black hole information loss problem can be found.

The key argument in, well, a nutshell, is that the Euclidean path integral for gravity over topologically trivial manifolds gives an invertible mapping from initial to final configuration, while that over topologically non-trivial manifolds does not.

Hawking concludes from that that the total path integral will be unitary.

I am looking forward for seeing this detailed in a paper, because I am not sure what to make of it. Something unitary plus something non-unitary certainly does not give us something unitary, so this cannot be what is meant. I would understand the final claim if we had restricted ourselfs to the path integral over trivial topologies, but is this what is meant?

[Almost immediate update: Ah, now I get it, the idea is that the contribution from the nontrivial topologies completely factors to a constant and can be devided out. ]

In the talk, the AdS/CFT correspondence is mentioned frequently. Right at the beginning it seems like Hawking is crediting AdS/CFT and hence Maldacena for giving the solution to the information paradox and that his talk is merely supposed to elucidate how this happens in detail on the gravity side of the duality.

What I find puzzling is that AdS/CFT makes the ‘gravitational’ path integral well defined by giving it a UV-completion, namely string theory. Hawking on the other hand argues purely from the Euclidea path integral for Einstein-Hilbert gravity as well as its canonical quantization. But as far as I know the Euclidean path integral is only gradually better behaved than the Lorentzian one, Wick rotation in a scenario where no background metric and much less timelike isometries are present is a mystery, really, and finally nobody knows how and even if that canonical Hamiltonian operator of pure gravity can be defined, which Stephen Hawking argues to generate the unitary time evolution.

Some work by Maldacena on 3-dimensional AdS gravity is mentioned which seems to support the main claim that information loss and non-unitarity is related to nontrivial topologies, but I don’t know about the details here.

The last but one part of the talk is concerned with a rough (looks hand-waving, indeed, but it is not clear to me which omissions are due to the nature of the talk or actually due to unsolved problems) argument how one could go about actually calculating a solution which shows the unitary formation and evaporation of something that would be a black hole for practical purposes.

The very last part of the talk is about merchandising in theoretical physics. :-)

Posted at July 22, 2004 11:21 AM UTC

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### Re: To loose or not to loose information

Ok, theres a few things that bother me technically, maybe someone can help me here.

One… Why is the Euclidean path integral being used in a semiclassical regime, and only to 2 orders it seems. Usually, its the late orders of perturbation series that reveal the semiclassical limit of the theory (take the anharmonic oscillator as an example)

Two.. I don’t follow the argument that the nontrivial topologies factor to a constant, or alternately that the nonunitarity goes away asymptotically. Why exactly? He mentions that there is no conserved quantities (which strikes me as more a problem with the way time is defined here, eg a mathematical problem, not necessarily a physical one)

I don’t have a problem with taking the values at the boundary, but I don’t get the relevance of ADS/CFT either in this situation.

Posted by: Haelfix on July 22, 2004 2:21 PM | Permalink | Reply to this

### Re: To loose or not to loose information

Hi Haelfix -

you wrote:

Why is the Euclidean path integral being used in a semiclassical regime?

I wouldn’t put it this way. The Euclidean path integral is supposed by some (few) people to be the correct definition of full quantum gravity, not just the semiclassical limit.

The resoning is that the Euclidean path integral is better defined than the Lorentzian one, and that one hopes that the transition amplitudes it yields are those of Minkowski-signature quantum gravity when analytically continued back to ‘real time’. The problems are

- that not every Riemannian metric has purely-Riemannian sections after complexified to a pseudo-Riemannian metric, so that there is no direct map between the ‘states’ that the Euclidean path integral sums over and the states that we are really interested in.

- that it is not at all clear what Wick rotation is supposed to mean when there is no notion of time. Just consider the case where spacetime topology is nontrivial, which is quite essential for Hawking’s considerations. Then what does it mean to do a Wick-rotation?

Usually, its the late orders of perturbation series that reveal the semiclassical limit

Right now I don’t know what you have in mind here.

You probably know that, but just to make myself clear let me say that the semicalssical limit is usullay defined as the limite where $h\to 0$ asymptotically. This means that in the path integral

(1)$\int D\left[\varphi \right]\mathrm{exp}\left(S\left[\varphi \right]/h\right)$

contributions from the action $S\left[\varphi \right]$ which vary rapidly with the field configuration $\left[\varphi \right]$ on scales of $\hslash$ tend to cancel each other, due to their quasi-random phases. Hence when expanding $S\left[\varphi \right]$ about a classical solution ${\varphi }_{0}$ one can write

(2)$S\left[\varphi \right]=S\left[{\varphi }_{0}\right]+h{S}_{1}\left[\varphi \right]+{h}^{2}{S}_{2}\left[\varphi \right]+𝒪\left({h}^{3}\right)\phantom{\rule{thinmathspace}{0ex}},$

i.e. expand the action in powers of $h$ and only keep some low orders.

I don’t follow the argument that the nontrivial topologies factor to a constant, or alternately that the nonunitarity goes away asymptotically. Why exactly?

I think that’s the part that would definitely need a real proof. What Hawking does is mentioning a plausibility argument, which uses the fact that, he claims, whenever the topology is nontrivial ‘the fields’ will decay exponentially in the asymptotic future.

I don’t know the details of that, but I got the impression that for the case of ${\mathrm{AdS}}_{3}$ this is a result by Maldacena, which he briefly mentions somewhere in the second half of the talk.

I don’t get the relevance of ADS/CFT either in this situation.

To my mind AdS/CFT might be the only relevant thing here, to put it bluntly!

If AdS/CFT is right (so far it has been checked only to second order (first stringy contributions) plus maybe a little bit more) then it follows that the stringy UV-completion of supergravity for asymptotically AdS configurations can be mapped in a weird but 1-1 way to a non-gravitational quantum field theory on the boundary of AdS. The important point is that this quantum field theory can be much better understood than the theory in the bulk. In particular we know that the field theory is nicely behaved in that it is unitary. This implies (at least naively) that also the gravitational theory in the bulk is. And this again is precisely the result that Hawking would like to obtain.

Posted by: Urs Schreiber on July 22, 2004 3:16 PM | Permalink | Reply to this

### Re: To loose or not to loose information

Ok, real quick as im pressed for time, I’ll try to flush out what I mean later with some math.

”- that not every Riemannian metric has purely-Riemannian sections after complexified to a pseudo-Riemannian metric, so that there is no direct map between the ‘states’ that the Euclidean path integral sums over and the states that we are really interested in.”

Thats right, so its not clear at all how to analytically continue, and the full integral becomes pathologically ill defined. One could probably restrict the sum to geometries with nice symmetries (time like killing vectors and so on), but then its not evident to me at all what that *is*.. Certainly not the full picture.

“This implies (at least naively) that also the gravitational theory in the bulk is.”

Yea I don’t see why that is so, or even why we should expect it to be.

In general, as you mentioned, the euclidean path integral is a nicely behaved mathematical object, as we dont have oscillating phases going berserk. However, I don’t see why String theory should have relevance to this interpretation of QG naively, as many fields of interests are being completely neglected. Again, ADS/CFT comes to rescue, but one would have to prove it to me that indeed this particular case is applicable. Not to mention there is the usual argument, that we are not interested in ADS/CFT, but rather DS/CFT.

The bit about his talk on the counterterms scares me a little bit, it looks like he’s going to start truncating things. Which is kinda a cheat, since much of the difficulty is in precisely summing the late orders (which naively are going to be problematic if as I expect they encode the semiclassical limit.. I’m thinking of analytic analogies with Condensed matter, where Borel resummation often shows this)

Posted by: Haelfix on July 22, 2004 5:46 PM | Permalink | Reply to this

### Re: To loose or not to loose information

Yea I don’t see why that is so, or even why we should expect it to be.

Because the two theories are equivalent. Every state of one theory maps to a state of the other theory and vice versa. So if you know that the evolution on one side is time-invertible and does not loose information about initial states, then the same must be true for the dual theory.

Not to mention there is the usual argument, that we are not interested in ADS/CFT, but rather DS/CFT.

True. But AdS is the case that can currently be understood. And - not unimportant for the current disucssion - it is also the case considered in Hawking’s talk. Be sure to have a look at what Jacques Distler just wrote about that.

Posted by: Urs Schreiber on July 22, 2004 6:19 PM | Permalink | PGP Sig | Reply to this

### Re: To loose or not to loose information

Hi Urs, theres one last thing i’ve been puzzling about.

Wick rotation in general is hard to do outside of flat space, as there is no easy notion of time.

Well, one this is not quite satisfying mathematically, and two its not clear how this generalizes at all.

Now, i’ve actually seen some attempts at solving this before. One proposal is to wick rotate on the space of guage fixed metrics (using the proper time guage), as this will not involve a complexification procedure. The drawback is the topology must be fixed, and we are not looking at saddle points of the path integral like Hawkings claims.

But still, its not clear to me how applicable this guage is, as its not covariant. Nor if its relevant to the talk.

In any case, I’d still like to see a sensible argument given to how we can use wick rotation without full time like killing vectors.

Posted by: haelfix on July 27, 2004 7:48 AM | Permalink | Reply to this

### Re: To loose or not to loose information

I don’t understand this in general, either, and I doubt that anyone really does. It is known that the Euclidean version of some standard metrics, like AdS in standard coordinates for instance, ‘makes sense’.

On the other hand it is also well known that a general Euclidean metric won’t allow a complixification which admits purely spacelike slices. This means that when you perform the Euclidea path integral over the Einstein-Hilbert action most of the metrics that you sum over in principle cannot have an interpretation as a metric on some physical spacetime. They must be regarded as purely auxiliary mathematical objects.

This is, I think, a problem over and above that of interpreting single terms in a Feynman diagram expansion.

To me the use of the Euclidean path integral in Hawking’s talk is a red herring, to some extent. It is introduced boldly as the ‘only sane way to do non-perturbatibe QG’ (which by itself is - hm - questionable) but apparently what Hawking really has in mind is again a perturbative expansion about classical solutions. When these classical solutions admit a sensible Wick rotation I guess one can ignore any potential problems with that method.

Posted by: Urs Schreiber on July 27, 2004 8:32 AM | Permalink | Reply to this
Read the post No Information Lost Here!
Weblog: Musings
Excerpt: Hawking creates a tempest in a teapot.
Tracked: July 22, 2004 5:36 PM

### Re: To loose or not to loose information

“Because the two theories are equivalent. Every state of one theory maps to a state of the other theory and vice versa. So if you know that the evolution on one side is time-invertible and does not loose information about initial states, then the same must be true for the dual theory.”

Maybe im not being clear. I understand asymptotic AdS configurations are mapped back and forth to some guage theory. I just don’t see why that should necessarily encompass the bare gravitational theory without anything else, or has that already been shown in the past? I’ll read up on the subject, I admit to little working knowledge of this.

Distler’s comments incidentally is what I was trying to get at in the last comment. But whatever, it makes sense to consider the working case for the time being, without wondering about setting the CC to zero or something small and positive.

I’ll refrain about the other issues until I see the paper, so as not to go off on hypothetical tangents.

Posted by: Haelfix on July 22, 2004 8:45 PM | Permalink | Reply to this

### Re: To loose or not to loose information

Hi Haelfix -

you wrote:

I just don’t see why that should necessarily encompass the bare gravitational theory without anything else, or has that already been shown in the past?

Oh, yes, right. This is precisely a point which I tried to make before. I was surprised that Hawking adopted AdS/CFT reasoning, because one cannot do that without accepting stringy degrees of freedom and all that. But if we do that - what’s the point of arguing non-perturbatively with the Euclidean path integral of pure gravity?! That’s just self-inconsistent reasoning to me…

Posted by: Urs Schreiber on July 23, 2004 11:11 AM | Permalink | PGP Sig | Reply to this

### Re: To loose or not to loose information

Decoherence would cause a non-unitary step in the evolution of the system, and a black hole will have to either be formed or not. This is irrespective of the fact that there may be an observer at infinity. The experiment with hot fullerene diffraction shows that a system can cause decoherence with itself if it emits a particle that contains enough information to betray a specific state of the source.

However, the two particles that cause Hawking radiation may be entangled across the event horizon and the state of some particle within the event horizon may be teleported to the particle that escapes away from the horizon.

Posted by: Rahul Jain on July 22, 2004 9:50 PM | Permalink | Reply to this

### Re: To loose or not to loose information

I agree. Knowing that the gravitational evolution is unitary alone does not tell us how the information is preserved in detail.

Apart from issues with the measurement problem I think one also has to take into account that AdS/CFT predicts stringy degrees of freedom, which might play a crucial role in how information evolves through a black hole.

This is for instance the point of view proposed by Mathur, e.g.:

Samir Mathur: Where are the states of a black hole? (2004)

Abstract:

We argue that bound states of branes have a size that is of the same order as the horizon radius of the corresponding black hole. Thus the interior of a black hole is not `empty space with a central singularity’, and Hawking radiation can pick up information from the degrees of freedom of the hole.

Posted by: Urs Schreiber on July 23, 2004 11:08 AM | Permalink | PGP Sig | Reply to this

### Re: To loose or not to loose information

“I agree. Knowing that the gravitational evolution is unitary alone does not tell us how the information is preserved in detail.”

That’s not what I was saying, and I don’t think that your analysis is correct here. If the evolution is unitary, it is information preserving. How it is that the wavefunction evolves is orthogonal (no pun intended!). We know that there is information and it is preserved, as the evolution is reversible. However, decoherence is not unitary and actually creates information.

Note that my alternative theory does not require any kind of quantum theory of gravity, although the coherence limits will affect whether the particles can actually manage to get entangled and teleport the state of the captured particle across the event horizon.

Hawking’s argument seems to me to be saying that if a tree might have fallen in the forest and you don’t look at it closely, you can assume that the tree both fell and didn’t fall at the same time. Spontaneous decoherence of hot fullerene atoms has been observed experimentally. We now know (or at least have evidence) that the undefinable “observer” of the Copenhagen interpretation is really not some mystical property of a human being, but rather a property of Heisenberg uncertainty – the decoherence was forced when the emitted photon had a small enough wavelength to fix the path of the fullerene though one of the gaps in the diffraction grating.

Posted by: Rahul Jain on July 23, 2004 4:54 PM | Permalink | Reply to this

### Re: To loose or not to loose information

Decoherence is something that happens to a subsystem of a total system when the rest is ‘traced out’. The total system always evolves unitarily. This has therefore no bearing on the question whether the evolution of the universe as a whole is unitary or not.

Decoherence might play a role when considering local or quasilocal observables in a, say, AdS universe, but that’s not the issue that is meant with ‘information loss paradox’. Decoherence formalism is just a means to talk about subsystems.

As far as you ‘alternative theory’ is concerned, lets not get into overly speculative territory. We’ll leave that to people like ‘t Hooft. ;-)

Posted by: Urs Schreiber on July 23, 2004 5:22 PM | Permalink | PGP Sig | Reply to this
Weblog: BLOG@STEFANGEENS.COM
Excerpt: Stephen Hawking gave his talk yesterday, the media came and went, and now the interpretations are beginning to trickle onto the web. The...
Tracked: July 22, 2004 11:06 PM

### Re: To loose or not to loose information

Hello all,

an extended account for the general audience on why Stephen Hawking lost his bet just appeared in the December issue of the monthly popular science magazine bild der wissenschaft:

Rüdiger Vaas (2004):
Warum Stephen Hawking seine Wette verlor.
bild der wissenschaft, no. 12., pp. 42-49.

Unfortunately it is not online and only available in German yet. The article also deals with the limitations of Hawing’s approach and the string cosmological background, i.e. the AdS/CFT correspondence. Many thanks to Urs for his informative support. On p. 41 there is also a reference to the SCT because this interesting blog was quite helpful.

Best wishes,
Rudy

Posted by: Rüdiger Vaas on December 11, 2004 7:38 PM | Permalink | Reply to this

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