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Some people's ideas about quantum gravity are vaguely overlapping

LUX: a new LUX paper finds nothing and improves (lowers) the upper limits on the nucleon-dark-matter cross section by a factor of 4 relatively to the best constraints in the past (also LUX, 2015).
String/M-theory and quantum gravity are two faces of the same beast. String/M-theory is the honest, well-defined face that allows you to calculate everything accurately, to any precision, with a perfect predictive power, at least in principle. The predictions are constructed in the top-down fashion and this fact is explicit. It's also a face where the spacetime isn't guaranteed to exist, may be absent, or its geometry may be ambiguous due to dualities. And because of the top-down approach, you don't know in which vacuum you should start to get the desired long-distance phenomena.

Quantum gravity is ultimately the same thing because all consistent theories of quantum gravity are some solutions to string/M-theory. However, quantum gravity is the face in which the spacetime and the well-known phenomena located in it are among the first aspects of the theory we notice. We basically construct our expectations about the phenomena from our experience. They take place in a spacetime we automatically associate with the spacetime of our experience and whenever some detailed laws of dynamics are found or guessed, they are basically extracted in the bottom-up way, as the phenomenologists normally do. The absence of fundamentally exact calculations is the most obvious bug of this bottom-up approach.

For those reasons, ideas in papers about quantum gravity are unavoidably more vague than those in papers about string theory proper, they are intuitive in character, and even when you feel that some claims must be right, it's sometimes hard to say whether two authors or groups of authors are saying the same thing or whether an apparent contradiction is really there.

A recent paper reminded me of this vagueness and overlapping cultures of quantum gravity intensely. After dozens of Boltzmann Brain-related and similar papers that make no sense whatsoever or that are either popular rehashes of trivialities or repetitions of a pop-science crackpottery (about misinterpretations of quantum mechanics, the links of the arrow of time to cosmology, and similar junk), Sean Carroll co-authored a paper that contains some good ideas relevant for the current research of quantum gravity.

On his blog, in the text
Space emerging from quantum mechanics,
he promotes the June 2016 Caltech paper
Space from Hilbert Space: Recovering Geometry from Bulk Entanglement
that he wrote with graduate student ChunJun Cao and Spyridon Michalakis, an outreach manager and staff researcher. After some annoyingly confusing explanations whether gravity comes from quantum mechanics or vice versa and obnoxiously elementary explanations e.g. of the concept of the Hilbert space apparently addressed to someone who has never heard about it (sorry, if you're one of those, please stop reading because the chance that your reading will lead to anything useful is basically zero), he gets to some basic points of their paper.

The paper outlines the basic strategy as follows:
  • Imagine that the Hilbert space of the whole quantum gravitational theory is some tensor product of many mutually isomorphic factors
  • Focus on special states in the whole Hilbert space that respect some area-entropy law
  • Reconstruct areas between the regions from the entanglement entropy
  • Find a way to see that a nearly flat metric is hiding in the Hilbert space
  • Try to linearize the perturbation of the entropy or entanglement entropy in perturbations of the states
  • Translate the perturbations of the entropy to perturbations of the geometry and interpret it as an Einstein-equation-induced curvature of the metric by the matter inside
Now, this is surely very similar to efforts to "construct the spacetime out of the quantum" that many other people have previously tried – and one may ask the question whether the amount of originality in the paper is nonzero at all. The strategy is vague and not guaranteed to succeed but at the same moment, it makes some sense.

At least one devil is certainly hiding in the details and the details in the paper by Cao et al. don't seem to work too well. The equations they derive aren't really Einstein's equations but some purely spatial and otherwise mutated cousins. Also, they claim to see "some" version of ER=EPR except that it's not the right thing and it only applies to the "highly quantum" wormholes – while their setup should seemingly be perfect for the nearly smooth wormholes if the setup really worked.

So much like in the case of many previous papers, it isn't possible to immediately see that the paper represents any significant positive progress in comparison with philosophically similar and comparably incomplete papers in the past. Many expert readers will surely conclude that they're stuck in a loop and the details don't make more sense than the details in previous papers that didn't quite succeed.

At the same moment, I tend to believe that someone will ultimately find very clear, nearly rigorous ideas that sound similar, that actually work, and that will be shown mathematically equivalent to the string/M-theory top-down definitions of the beast. I am surely trying to find the clear rules and sometimes it feels that I am very close to the final answer. What will be found in the future will represent a clear progress and like Juan Maldacena, I do think that the clear progress won't be possible without a particular string/M-theory-based working example of the ideas. It's likely that the degree of confusion similar to that in the paper by Cao et al. is unavoidable in all similar papers that avoid string theory.

The first step in their strategy is the assumption that the Hilbert space may be written as a tensor product of many factors. This corresponds to the assumption of "locality" if not "ultralocality", something that is surely questionable in quantum gravity. Degrees of freedom in two regions aren't independent of each other because each region must be careful not to create a black hole that would destroy the other region. Equivalently, creation operators for objects in two regions should be "dressed" by the operators guaranteeing the gravitational backreaction (see e.g. a new paper by Giddings and Donnelly) and the latter effectively break the independence of the regions, anyway. Also, the black holes seem to require some acausal transmission of information for them to be able to shed the information in the Hawking radiation.

But maybe in some attitude to the spacetime geometry and/or in some gauge, this assumption of the factorization is legitimate. I am sure that many researchers who are emotionally attached to GR want this kind of independence to be right in some definition of the theory. Quantum field theories at fixed spacetime geometrical backgrounds largely do respect the tensor factorization. Every point in the spacetime contributes some fields and their canonical momenta. It's "only" the Hamiltonians that aren't simply additive (they include an interaction part – and even the spatial derivatives are "interactions between adjacent regions", something that breaks the ultralocality) and that force the content of one region to "adapt" to the forces exerted by another region.

One new technical reason that leads me to believe that the tensor factorization could be legitimate, after all, is the exploitation of ER=EPR in the "backwards" way. ER=EPR says that the spacetime of a non-traversable wormhole may be cut into seemingly independent pieces, two black holes. So maybe one can cut the spacetime into lots of regions at regular places as well. The small Hilbert space factors could describe some singular Universes with a Planck time life expectancy but their strong entanglement of particular forms – cooperation – can turn them into long-lived, non-singular, nearly flat spacetimes.

However, in quantum gravity, the Hamiltonian is something that is curving the spacetime geometry and, as a consequence, modifying the causal structure of the background spacetime, too. So the very definition of the independence of two sets of degrees of freedom – in two regions – becomes dependent on the state of these two regions or objects. This dependence isn't self-evidently the same thing as the Papadodimas-Raju state dependence but I do think that the Papadodimas-Raju state dependence ultimately follows from simple considerations of this sort, anyway.

If the factorization into the smaller factors is possible at all, the main "unknown" that needs to be derived is something like the Hamiltonian – and perhaps the whole stress-energy tensor – for some class of states that may be considered perturbations of the same background. When all the things are done correctly and the consistency conditions are correctly imposed as constraints, we should get the same possible "set of Hamiltonians" that we may derive as the set of all Hamiltonians in all backgrounds or vacua of string/M-theory.

There are many ways to create new backgrounds out of older ones: deformation by a coherent state (condensate) of gravitons or other particles, dualities, ER=EPR that allows us change the spacetime topology "regionally", flop and conifold transitions that actually do something similar, and so on.

Perturbative string theory has clear ways to describe allowed (weakly coupled string) vacua. They're pretty much in one-to-one correspondence with some two-dimensional conformal, modular-invariant theories. But what is the generalization of this set of technical rules beyond the weakly coupled regime of string theory? The general, perhaps bootstrapy, rules of quantum gravity should take over. The conformality and modular invariance of the weakly coupled string theory should be derivable as a limit of the quantum gravity rules in an important limiting subclass of situations.

What do these laws of quantum gravity say in general and why are the string/M-constructions we know special examples of those? That's a big question.

I do share the excitement about the possibility that the universal links between the quantum information and geometry – the glue-entanglement duality – do tell us what all the possible Hamiltonians are, what all the possible ways to interpret the quantum information as "some things in a gravitating spacetime" are, what are all the ways to visualize the locations of all the information in the spacetime. But what are these rules?

The glue-entanglement duality that Cao et al. recycle is telling us that two tensor factors of the big Hilbert space contain objects that are "next to each other" and "sharing a big border area" if their entanglement is high. When they're close in this way, it also means that the Hamiltonian (imagine one for a quantum field theory on a precalculated background) will contain some simple kinetic terms such as \((\nabla \Phi)^2\).

It seems that quantum gravity is able to see "who is close to whom" automatically or dynamically. Take a Hilbert space without any structure. Write it as a tensor power of a finite-dimensional Hilbert space. It seems that states in the big Hilbert space that respect the high degree of entanglement between the geometrically nearby tensor factors do automatically allow you to reparameterize their vicinity as excitations of some geometric background in a theory including quantum gravity and quantum fields. A theory of quantum gravity is able to see two similar (highly entangled) pieces of information in the Hilbert space and immediately decide that they're constructed from the same fields in regions \(A,A'\) of a spacetime, just shifted by a small amount in a direction perpendicular to the areas \(A,A'\).

Can rules like that be made rigorous so that you get a fully consistent theory, i.e. a vacuum of string/M-theory? It's pretty clear that if you do things naively, you get stupid, inconsistent theories. Figure 1 of the paper by Cao et al. (see above) looks dangerously similar to the spin networks in loop quantum gravity. (A more optimistic or flattering comparison would be the AdS/MERA tensor networks.) Moreover, Cao et al. pay lip service to the tensor factors made of qubits (their dimensions are powers of two) which are particularly unnatural, stupid Ansätze fashionable among the discrete physics crackpots but not real quantum gravity experts.

Is there a way to fix these defects and turn the strategy into something that is compatible with a fully consistent theory of quantum gravity? At the end, if some tensor factors of the Hilbert space exist for a region, they have to be infinite-dimensional because there's almost certainly no natural finite dimensionality. (I am ready to believe that \(196,883\) could be a natural dimensionality of the factors in the pure \(AdS_3\) minimum-radius gravity, however, thanks to Witten's application of the monstrous moonshine.) But if the tensor factors are infinite-dimensional (or even higher-dimensional), there has to exist a "big part of the job" in specifying which states in these smaller spaces are relevant, low-energy states.

That's a difficulty because it seemed that the very goal of cutting the Hilbert space into the tensor factors was to "derive" the dynamics – how the regions influence each other, at least by gravity – from something more fundamental. But we still need to "insert" some information about the internal dynamics of each region – something that wasn't derived from anything more fundamental. We can seemingly achieve our goal for a "part of dynamics" only.

The rules that are solved by all the consistent vacua of quantum gravity must be some self-referring, bootstrapy rules that tell us that if we insert some rules for the dynamics of a region into the algorithm and derive the interactions between several adjacent regions, we get the same rules we started with, or something like that, and the world sheet conformal symmetry (and/or the state-operator correspondence of a two-dimensional CFT) will be seen a special limiting case of this ability of a consistent quantum gravity's arrangement to "reproduce itself at a higher level".

What I write obviously sounds vague because it is vague. But I am sure that there are numerous good experts who have a similar feeling about a definition of a theory of quantum gravity that will emerge in the future. In the past, I did succeed in the crystallization of a vague hunch into the form of a basically rigorous crystal-clear crystal ;-) but this task is almost certainly harder and more ambitious so I or we may need some extra time and energy...

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