## Wednesday, August 26, 2015 ... /////

### Stephen Hawking "solves" the information loss paradox again

Value not clear but he builds on some very interesting recent research

Stephen Hawking has visited Stockholm where he announced a paper that will be released in 30 days or so. You may watch the video of his 9-minute talk at a Swedish page or read a short paper that Hawking will submit one week from now. And you may check reports from Sabine Hossenfelder and the Nude Socialist.

Recall that in mid 1970s, Hawking backed Bekenstein's idea that black holes have a nonzero entropy because the black holes also have a nonzero temperature. They radiate just like the black bodies. The process that makes this radiation is possible is the Hawking evaporation. However, it seemed inevitable that the resulting Hawking radiation was exactly thermal and basically uncorrelated to the initial state of the star that has collapsed to the black hole.

The Hawking radiation escapes outside – but the black hole exterior is causally inaccessible from the interior, so the interior information can't be imprinted in the Hawking radiation. This conclusion inevitably follows from quantum field theory on a curved background – and it holds at least to all orders in perturbation theory.

It seemed that the pure states of the star evolved to mixed states of the Hawking radiation when the black hole is completely evaporated. This pure-to-mixed, specific-to-universal evolution contradicts the general rules of quantum mechanics and is incompatible with the description of the scattering processes in terms of a unitary matrix. So we talk about the "non-unitarity".

In the 1990s, string/M-theory finally made it obvious that the actual evolution is perfectly unitary and the information doesn't get lost. The "causality" problem is overcome by some degree of nonlocality which has to exist – because we know that the radiation is unitary and which is probably allowed because the geometric background obeys the laws of quantum mechanics as well, a "detail" that was ignored by QFT on curved spaces – but this nonlocality has never been described too clearly.

The reason is that the stringy descriptions make it obvious that there is a well-defined Hilbert space of microstates. They make it clear that there is a Hamiltonian that acts on this space. And the evolution is therefore unitary – which is very manifest in Matrix theory and AdS/CFT. But these stringy descriptions don't exactly tell us "where" the information is located and how and why it gets out. These descriptions are not manifestly local – they are not necessarily in terms of "objects and fields living on a fixed geometry".

After the Duality Revolution in string/M-theory, Hawking admitted that the information gets out. It meant that he surrendered in a bet against John Preskill who had correctly argued that the information had to be saved. Kip Thorne who has shared Hawking's incorrect "information is lost" opinion in the bet hasn't admitted his defeat yet. And it makes pretty clear that he never will.

Hawking continued to be heavily interested in this topic. In 2005, he announced that he had explained why the information wasn't lost, after all. The key idea of his argument was that when we describe the black hole formation and evaporation by the Feynman path integral, we should only sum over "topologically trivial" spacetimes and not the spacetimes of a different topology, those with the black hole.

Well, that's perfectly sensible at a verbal level. But it doesn't eliminate the whole mystery because it doesn't explain why the spacetime macroscopically looks like a topologically nontrivial one, and if it allows this fact, it doesn't explain the map between the topologically trivial description and the topologically nontrivial description.

Back to 2015

Hawking tells us that he had a wonderful idea after he heard Andy Strominger's talk at Strings 2015 and his and his collaborators' recent work on the BMS group, memories, asymptotic symmetries, and soft theorems which I discussed 2 months ago. Hawking also uses some recent results by Polchinski et al.

He informed Andy Strominger and Malcolm Perry about the great idea and it seems that they haven't quite convinced him that it's wrong or vacuous, so he decided to make the big announcement.

I haven't quite and fully understood or swallowed the 9-minute announcement yet. But Hawking claims, among other equivalent or related things, that the information is stored in the supertranslations at the event horizon that the infalling particles caused; and the supertranslations imprint the information through subtle delays in the emission of the individual Hawking particles. As a reader observed, this basically looks like just another way of saying that the event horizon is a hologram of the interior, producing a very-hard-to-decode code translation between the initial and final matter, an idea of the kind that has been discussed for more than 20 years.

The papers by Andy Strominger make some of the relevant mathematics more explicit but if you ask me how Stephen Hawking goes beyond both the holographic principle and the recent work by Strominger et al., then I must admit that right now, I don't understand this added value at all. Maybe, Hawking does mean or admit that he has discovered the papers by Strominger et al. and this is his discovery. ;-) But I find it plausible that the added value actually does exist and we will learn about it at the end of September.

Sabine Hossenfelder tries to argue that the "information is there twice to start with". This would obviously be wrong. The information may only exist in two, equivalent – mutually dual – descriptions but we can't believe both of them at the same moment because quantum mechanics prohibits xeroxing due to its linearity. So if the supertranslations or some other horizon-based degrees of freedom remember exactly the information that is also carried by the infalling matter, the map between these two ways to describe the information must be a duality map – not a doubling of information.

According to Hossenfelder, she, Gerard 't Hooft, Paul Davies, and Carlo Rovelli participated in the debate with the speakers such as Stephen Hawking and Malcolm Perry. Especially at this level when the exact proposal isn't clearly written down, it must be a rather frustrating interaction for Hawking, especially because most of the criticisms of some general points that are probably involved seem fundamentally wrong to me.

For example, Hossenfelder writes:

Ok, so Hawking is saying in reply to Rovelli that it's an effect caused by the classical gravitational field. Now I am confused because the gravitational field doesn't uniquely encode quantum states. It's something I myself have tried to use before.
If Ms Hossenfelder has tried something and failed, it doesn't mean that it's wrong. Ms Hossenfelder is likely to fail whenever she tries to do something that makes sense.

In the description at low energies, the gravitational field is independent of the other fields, so they carry different pieces of the information. But this separation between "gravity" and "non-gravity" is bound to be ill-defined in the most general description of the quantum reality – which is not just an effective quantum field theory.

In particular, string theory unifies gravity with other forces and matter species and it allows us to say that all forces and particles are manifestations or excitations of a generalized stringy geometry. A trivial old example is the Kaluza-Klein gauge bosons coming from extra dimensions. Gauge bosons are polarizations of gravitons. But even in the case of all the other vacua and all the particle species upon then, it is morally true that all particle species are polarizations of a single giant "string multiplet" (perturbative, it is the "string field"). And this giant "string multiplet" may be organized in many different ways.

Hossenfelder continues with a text that gets even more wrong:
The gravitational field of the ingoing particles does always affect the outgoing radation, in principle. The effect is exceedingly weak of course, but it's there. If the gravitational field of the ingoing particles could encode all the information about the ingoing radiation then this alone would do away with the information loss problem. But it doesn't work. You can have two bosons of energy $E$ on top of each other and arrange it so they have the same classical gravitational field as one of twice this energy.
If the gravitational field is described classically, it may encode some "limiting slice" of the full quantum information. But even though Sabine Hossenfelder is among those who love to deny that the gravitational field must obey the laws of quantum mechanics as well if other things do, the gravitational field is ultimately not purely classical and it's totally plausible that it remembers everything at the quantum level.

In particular, it's simply not true that the gravitational field of two overlapping bosons of energy $E$ per particle is the same as the gravitational field of one boson of energy $2E$. If you have two bosons in a wave function $\ket\psi$, i.e. $\ket\psi\otimes \ket\psi$, this tensor product wave function describes the superposition of different point-localized states of two bosons. For example, if $\ket\psi$ is uniformly spread over the Earth's surface, there are contributions to $\ket\psi\otimes \ket\psi$ that contain one boson in Stockholm and one boson in Boston (a generic contribution).

The gravitational potential of that configuration has the schematic form $\phi(\vec R) = -GM\zav { \frac{1}{ |\vec R - \vec r_{\rm Boston} | } + \frac{1}{ | \vec R - \vec r_{\rm Stockholm} | } }$ And the full gravitational field induced by the state $\ket\psi\otimes \ket\psi$ is a linear superposition (in the Schrödinger cat sense, if you wish) of gravitational field with profiles above. On the other hand, the gravitational potential of one doubly energetic object in spread according to $\ket\psi$ is of the schematic form$-\frac{2GM}{ | \vec R - \vec r_{\rm Seattle} | }$ It only contains one source, so one singularity. You know, the functions$\frac{1}{x-7} + \frac{1}{x-9}$ and $\frac{2}{x-8}$ are different. Qualitatively different. The first function has two singularities while the second function only has one singularity. If you consider a superposition of gravitational fields of the first type, they will still have two singularities. You will only be uncertain about the locations of these singularities. Be careful: When talking about quantum superpositions, we are not adding the potentials themselves; we are adding the quantum amplitudes for one shape of the potential or another! The second type of the potential has one singularity, even in the superposition. The averaged or integrated values of the potential etc. (either over the quantum states or over the space) may be the same but the detailed state of the gravitational field ignited by the two configurations is simply not the same.

At the end, Hossenfelder's counterargument is either assuming that the physical theory has to be totally classical forever which would be very bad; or it at least denies the fact that the "classical limit" of a quantum theory may mean various things. There may be many inequivalent classical limits of a quantum theory.

AdS/CFT is a trivial example. (Not just "an" example; morally speaking, this holographic map is almost the "very same" example.) It is the exact equivalence between two fully quantum descriptions with Hilbert spaces and operators. One of them is a gravitational, stringy theory in the AdS bulk; the other one is a non-gravitational conformal field theory on the boundary. Both of these theories have certain classical limits. But the classical limits are simply not equivalent to each other. The straightforward classical limits of the conformal field theory are non-gravitational theories on the boundary; the classical limit of the bulk description is some classical GR in the anti de Sitter space coupled to matter. The duality works because it maps the classical limit of one of the equivalent theories to a hugely quantum regime of the opposite theory!

But even if we stick to the classical limit, and even if Strominger and/or Hawking or their pals were talking about the mapping of classical limits themselves, it has to be done right. As the formulae above were supposed to explain, the gravitational field only has a "sharp classical profile" if the profile of the energy density is classical. And that's only true for the states with perfectly localized elementary particles (to points). At finite momentum (and energy), particles are never quite localized which is why it's incorrect to say that the two overlapping bosons in identical wave packets have a sharp classical profile of the gravitational field. They don't. They produce a "Schrödinger's cat" state of the gravitational field.

(The perfect localization isn't possible in quantum gravity, not even in principle. If you localize the particle to a region smaller than the Planck length, the uncertainty principle implies that the average energy is greater than the Planck energy and the object will start to be a black hole – the size, another source of delocalization, grows if the desired localization gets more accurate. So no localization of a particle in quantum gravity can be better than the Planck length which also means that no quantum state of matter may produce a gravitational field that would be "perfectly" classical.)

So it is obvious that Hossenfelder's – and, as I understood it, also Rovelli's – criticism is just bullšit. They haven't found any genuine problem. All the information about the state of the infalling matter might be encoded in some purely gravitational information when the space of possibilities is parameterized appropriately.

Aside from holography, it reminds me of the fuzzballs. There are lots of controversial things that Samir Mathur (and collaborators) say about the information loss in the context of the fuzzballs. But what they do have is an infinite-dimensional family of BPS solutions that are "pure geometric" but they still manage to carry the same entropy as "the states of the stringy and braney matter" that may combine to create the same black hole. It's another (or closely related?) example of a seemingly viable map that stores all the information about the matter into some purely gravitational degrees of freedom. No valid proof of a contradiction exists.

The quantum black hole puzzles combine very technical and well-defined aspects and calculations with some body of "philosophical interpretations and comments". There are lots of tensions if someone doesn't share the philosophical attitudes with you; and things are even worse if she doesn't understand the actual technology rooted in mathematics. (Note that to be politically correct, I have used "she".)

So even though I tend to bet that Hawking's picture won't be absolutely clear and they won't settle all the mysteries, like his 2005 comments on the same issue didn't (and I will love to be proven wrong), I think that much of the criticism is meant to "shoot down" many important insights that are pretty much demonstrably correct and don't suffer from any flaws.

The number of people talking about these matters who are really confused about rather basic things – and, in some cases (including Hossenfelder and Rovelli), don't even understand the basics of string theory which are absolutely critical for a comprehensive understanding of all the black hole information issues – is so high that making fundamental and important contributions in that field is an example of casting pearls before swines. I feel sort of relieved that I no longer have to sell important findings to people who aren't capable of appreciating them.