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Maldacena's talk at Strings 2016

I could download the slides from all the Strings 2016 talks just fine but the videos that were posted were unusable for me, due to the low bandwidth etc. Sometimes when you need it, China no longer looks like the ultimate 21st century superpower. Fortunately, Alexander Comsa posted 7 first Strings 2016 videos from some of the most famous speakers. Even though only dozens of people in the world appreciate it, I hope that he is not finished yet.

One of the talks that were posted was Juan Maldacena's talk about entanglement, quantum gravity, and tensor networks. I find it rather amazing how close his conclusions about the current state of affairs – and even the proposed or expected next major steps – are close to mine, especially if one compares it with the overwhelming majority of the "people around physics" (and, often, technically "inside" physics) who can't agree even about basic issues that physics settled 90 years ago.

Maldacena has written over a hundred of very important highly technical papers but I think that he has developed so much experience and hindsight that he should be talking about more conceptual things and visions as well and this talk is an example of that.

He spoke in Maldacena's typical low-key, disproportionately modest Argentine English. Princeton should hire a special person who will do just P.R. for Juan, make sure that people actually listen to his ideas, and so on. That would be a good investment. Most of us who don't get isolated from the Internet distractions, including your humble correspondent, are subject to the brainwashing by the media, social networks, and the broader moronized postmodern society and we spend lots of time by reading articles about (and by) crackpots like Sabine Hossenfelder or Lee Smolin, time which should be better spent by reading articles (and listening to talks) by the top physicists like Maldacena and our own research.

OK, he wants to solve all big issues of quantum gravity and everything, we're told, and solve all these things by analyzing the fate of the quantum information. To start with, he defines the relative entropy – a von-Neumann-like formula for a "distance" between two density matrices.

It's positive. This positivity may be proven and because it applies for two density matrices describing half-spaces etc., one may also use the positivity to derive a Bekenstein bound saying that the entropy is smaller than the expectation value of a boost generator (a form of energy).

I find his formulation of some of the well-known ideas highly refined. For example, he rephrases the Unruh effect by saying that the density matrix for a half-space is thermal. The thermal formula derived by focusing on the boost generator which enters the exponent in the density matrix is a clever method to integrate out the other half of the space. This method ultimately makes you see some causal diamonds (Rindler wedges) in the spacetime but that's just an artifact of the method. You may still imagine that you discuss states on a single slice at \(t=0\) and what is thermal is simply the state of one-half of the space.

This clearly, provable assertion is a more correct statement that may replace similar statements that are often vague. While explaining things like the Unruh radiation, people don't make it too clear what states are pure and mixed, why mixed states suddenly enter at all, and so on.

He describes some results in quantum field theory that were derived by similar methods, too. Various somewhat boring inequalities involving the relative entropies etc. The entropy is shape-dependent and the shape dependence may be studied by replacing shape changes by metric changes along with diffeomorphisms (generated by the stress-energy tensor). Lots of people have worked on these issues. Special subtleties arise when the theory has gauge fields; there are edge modes. Maldacena also switches from QFT to semiclassical gravity, corrections to black hole entropy are included in this segment. Classical and quantum focusing theorems are reviewed; Bousso and others have worked on those things.

The following segment of the talk discusses the links between holography and entanglement – minimal areas in the bulk enter etc. Also, the relative entropies in the bulk and in the boundary theory must be equal thanks to the holography. This tautology (resulting from the holographic duality) may have interesting implications. Two states in the boundary are only distinguishable if their bulk counterparts are distinguishable, too.

Following minutes sketch the derivation of Einstein's equations from the entanglement formula etc. Many people would credit Ted Jacobson with this whole industry. For Maldacena, Jacobson is a not fully general preparation. Maldacena emphasizes that one must know the formula for the entropy to derive Einstein's equations.

Wormholes become heroes of the next segment. Negative energy is enough to communicate (Gao-Jafferis-Wall). The negative energy and the communication are two seemingly different types of a violation of the relativistic causality. But they may be shown to be fully equivalent even though the "locations" of the violators are different in the two descriptions. A black hole fully entangled with a quantum computer is used to study these affairs – a highly controllable strategy.

Now, the entanglement entropy isn't the only interesting type of entropy. He shows that by displaying the difference between the causal wedge and the entanglement wedge.

At some moment (38:10), he gets to the AdS-style tensor networks. This picture – which originally was a method to represent some simple wave functions; embodies the renormalization group; and seems analogous to quantum gravity – has some problems when you try to use the method to write the cosmological wave function in quantum gravity (five-index tensors seem right or simplest to discuss scale-invariant wave functions). In the discussion, Andy Strominger complains that the tensor networks don't seem to preserve the number of degrees of freedom i.e. apparently violate unitarity. (Cumrun Vafa and one Verlinde brother also ask questions.) Juan highlights another problem of the tensor networks: they break the Lorentz invariance (in the bulk).

It's obvious but I have never quite appreciated this problem. A solution of this problem – a mathematical structure that generalizes tensor networks – should overcome the problem of the Lorentz violation (and perhaps other problems). But maybe when you overcome this problem, it's enough to find the "healthier refinement" of the tensor networks in quantum gravity.

I feel that this new mysterious framework could generalize the tensor networks in the same way in which multivariate techniques and machine learning generalize simple statistical techniques based on bins while analyzing the LHC data. Maldacena was just slightly less specific about these matters than I am but he said a very similar thing in the last minute or so – which was a challenge for the audience to make breakthroughs.

Most importantly, he considers all these tensor networks etc. to be heuristic hints that should lead someone to find a well-defined example of these networks or their generalization that may actually describe some well-established (stringy or M etc.) vacuum of quantum gravity. I completely agree with that. At the level of the "general semi-vague stories in quantum gravity" avoiding the stringy language, this bunch of insights and hints may already have been depleted. More high-precision results should arrive now. You simply need some stringy or AdS/CFT-style precision.

Maldacena also proposed an interesting principle – that the action could be "equal" to the complexity (well, someone else did so before him). That's quite a deep statement if true. At least in some simple models, "complexity" should refer to the number of simple gates in something like the tensor network that you need to obtain the configuration. So far, I only see that some "totally rudimentary checks" work out fine but I don't quite understand the justification, let alone how to use it to derive the Feynman path integral or relate it to something else. Why should complexity only matter modulo \(2\pi\) in the path integral, for example? How does the continuous complexity work at all?

Answers to these questions are missing but I do feel that many of them are really deep questions that will kickstart another big or upper-medium revolution in physics when they are answered.

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