Friday, October 09, 2015

A thesis on bulk locality in AdS/CFT

RAF III is clearly following the papers on the arXiv in some detail so he was able to see the PhD thesis by Jennifer Lin,
Bulk Locality from Entanglement in Gauge/Gravity Duality,
who is getting her PhD under David Kutasov at the University of Chicago. She has written almost a dozen of papers and cooperated with folks like Klebanov etc. at Princeton and Ooguri etc. at Caltech, not to mention interactions with Jeff Harvey in Chicago and others.

Also, RAF III was able to notice that the thesis refers to a 2013 TRF blog post. I assure you, it's not usual to cite blog posts and the technical author needs some balls to do a thing like that in an otherwise formally flawless technical piece of work. Thanks for that, Jennifer!

Jennifer cites my blog post as the first source of the observation that the "existence or non-existence of a [non-traversable] wormhole in quantum gravity" is not observable by a well-defined apparatus or a procedure; and, correspondingly, it is not an observable in the theoretical sense (i.e. a linear Hermitian operator). This negative statement has to hold because the "existence of entanglement", which is dual to it, isn't an observable, either.

I believe that the key reason why the "existence of a wormhole" isn't a strict observable is that (for the pair of infalling observers) to demonstrate the "gluing" connecting two black hole interiors, they have to verify that the observations in both interiors are the same (they are looking at the same objects in one throat). But these features may always be said to coincide "by chance" and be two copies of the same thing rather than "one connected throat". There's no safe way to count the number of copies of objects inside the black holes if causality (of two black holes' geometry) prevents you from observing both/all copies at the same moment.

The insight that the "existence of a wormhole" cannot be measured with certainty in quantum gravity is a consistency check on ER=EPR and related ideas and I believe that the authors of those ideas are aware of this consistency check. On the other hand, it's probably true that a blog post of mine was the first place where the claim and its justification was articulated explicitly. A paper by Bao et al. that is three weeks old (and discussed on TRF) is another source that Jennifer cites.

More generally, her thesis is dedicated to a fascinating topic of the "emergence of spacetime". Many recent developments in quantum gravity in general and string theory in particular have made it increasingly clear that the spacetime as a concept is "absent" in some fundamental formulation of the laws of physics and it only emerges in certain situations.

It's a seemingly metaphysical slogan that has been said for decades but its importance was fluctuating and the specific technical ideas that make it more tangible were changing over the years. What people found within string theory and especially AdS/CFT – with the equivalence between the quantum entanglement and the geometric glue – arguably makes much more sense than many previous attempts to make the "emergent nature of the spacetime" more rigorous and material.

Maybe I should mention that in the professionals' quantum gravity, the spacetime emerges much more fantastically from something "really different". In the crackpots' childish and naive caricatures of quantum gravity such as loop quantum gravity, one shouldn't even say that the spacetime emerges. The spacetime arises from "small pieces of LEGO". But those pieces may be said to be pieces of the spacetime. So you have a spacetime – although divided to small pieces – to start with. You put it in and you shouldn't say that the spacetime is emergent. On the contrary, in the actual beautiful adults' quantum gravity, the very locality in the spacetime is emergent. There is nothing such as a "locus of an object" to start with. Holography is a paradigm that makes this "really unexpected" emergence of the spacetime clear.

In late 1997, Juan Maldacena proposed his AdS/CFT correspondence. Already in his original paper, the amount of evidence was extremely strong and by the end of 1997, the top physicists had no doubt that he was right. (I do admit to have been skeptical as recently as in December 1997.) It just seemed to work very well. Some paradoxical questions that were previously raised were suddenly resolved. Additional papers by other authors verified some details about the correlation functions and wrapped branes and later excited strings and so on which is why I don't know a competent quantum gravity theorist who would seriously doubt that the AdS/CFT correspondence is an exact equivalence, at least in the cases of the most carefully verified examples.

The AdS/CFT correspondence says that a non-gravitational theory in \(d\) dimensions, \(CFT_d\), is exactly equivalent to a quantum gravitational (i.e. string/M-theory-based) theory, \(AdS_{d+1}\), in an anti de Sitter space. The numbers of large dimensions disagree: the quantum gravitational "bulk" theory has one extra dimension, the "radial" or "holographic" dimension, aside from the dimensions that also exist in \(CFT_d\). This dimension of the gravitational theory that can be "projected out" is of course the defining sign of the holographic principle. But how does it work? How can a \(d\)-dimensional conformal field theory secretly have a higher number of dimensions?

In non-gravitational local theories, the number of dimensions may be defined pretty much rigorously. You find local operators that commute at spacelike separations and reconstruct the spacetime dimension from ensembles of such local fields. In a gravitational theory, the spacetime dimensionality is much more subtle because the locality doesn't hold "exactly", at least not "exactly and in all respects". The ability of the information to get out of the black hole through the Hawking radiation is an example of a "small violation of the locality".

What I mean is that the locality means that the operators commute or anticommute at spacelike separation but the separation may only be determined relatively to a background metric but the metric tensor is a dynamical variable in quantum gravity so whether two "points" are spacelike-separated depends on the state (of the metric tensor field). The local operators and "points" are simply not as easily definable as in non-gravitational theories.

But in some limits, even the quantum gravitational theories look similar to the local quantum field theories and the metric tensor is nearly constant and "classical". How is it possible that AdS/CFT produces an extra dimension? What is being produced is really remarkable. The extra radial dimension means that the operators \(\phi(x,y,z,t)\) in \(CFT_4\) may be given an extra continuous label, \(r\), and be reorganized as \(\phi(x,y,z,r,t)\), which commute or anticommute according to the separation measured in a five-dimensional spacetime.

Most obviously, two such operators \(\phi(x,y,z,r_1,t)\) and \(\phi(x,y,z,r_2,t)\) supercommute with each other because they become spacelike-separated. In some sense, they have to be constructed from field operators in the CFT "located" at the same point in four dimensions, \((x,y,z,t)\), but there are hidden degrees of freedom that allow us to extract "many more operators" that commute with each other. It's the (approximate) locality in the extra holographic dimension that is the root of the holographic miracle.

Since Maldacena's paper, people were trying to prove the approximate locality and they were using various methods to do so. Those proposals were interesting but none of them has led to a clear rigorous proof that would teach us a new conceptual lesson at the same moment, at least not one that made perfect sense. People knew that the "location in the radial dimension" was something like the "energy scale" in the CFT. And the separation of the different regions along the radial dimensions was somewhat related to the "independence of individual scales in quantum field theories", as manifested by the renormalization group. But the renormalization group doesn't really say that one scale is independent from all other scales. It says that the low-energy physics may be independent of the detailed high-energy physics – but not the other way around! So there were various puzzling questions.

In the recent 7 years or so, the problem to "prove the bulk locality" has been attacked from a shockingly new perspective – the geometry in any quantum gravitational theory (the very "continuity" of any gravitating spacetime) began to be "visualized" as quantum entanglement. Mark van Raamsdonk has made insights of this sort and Ryu and Takayanagi (I also know the latter guy, Tadashi, from Harvard, a nice guy) presented their formula which is "quantitatively" an emulation of the Bekenstein-Hawking formula for the entropy and area but has a new interpretation. The entanglement entropy between a spatial region in a CFT and the rest of the world volume is related to the minimum surface that may connect the two sides in the quantum gravitational description (over four times Newton's constant).

So there's some sense in which the areas of surfaces in the AdS bulk, regardless of their "direction", are encoded in the entanglement entropy of some degrees of freedom that may be seen in the CFT, too. There are many spatial situations – with or without black holes in various qualitatively different geometries etc. – that can be used as a playground to test this general paradigm and turn it into very specific insights and equations that depend on the situation.

Jennifer may have been influenced by all the collaborators but at any rate, I think it must be interesting to read what someone who has become an expert but who has only begun to think about the issues "rather recently" has to say about all these topics.

Because physicists may reduce many – or all – aspects about the geometric relationships in the spacetime to the quantum entanglement which is basically a "research of information", they are finding new proofs of John Wheeler's prophesy "everything is information". Well, I would surely not argue that Wheeler has co-discovered any of the cool, more specific insights in quantum gravity of the recent years. It's easy to say general metaphysical things that will be "shown true" in some way in the future. What Wheeler may have been imagining was probably much more primitive. But it's still funny to conclude that the qualitative difference between the spacetime and the information is being "blurred away".

We are being switched to a new way of thinking in which the number of independent concepts is even smaller than it was after the previous revolutions and unifications. Every time a paradigm shift like that takes place, we feel closer to the ultimate architecture underlying Nature. Maxwell has unified electricity, magnetism and light. Einstein has unified space and time. Quantum mechanics has unified waves and particles. And there are a dozen of similarly important unifications in the history of physics. But I think that the unification of the quantum entanglement and continuity in the spacetime geometry is a comparably important advance.

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