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A paper written in 2020 by Harlow and Shaghoulian (reference 1) proposes a connection between unitary black hole evaporation and the non-existence of global symmetries in quantum gravity. In passing (page 6), they make this statement about a different connection, namely between locality and black hole remnants (which never evaporate):

...Einstein gravity is not renormalizable in $3+1$ dimensions..., and so far the “asymptotic safety” program that looks for a strongly-coupled UV fixed point for Einstein gravity in $3 + 1$ dimensions (such as the proponents of loop quantum gravity hope to find) has been unsuccessful. Moreover even if such a program were successful, the above examples suggest that it would lead to black holes whose entropy is not consistent with the Bekenstein-Hawking formula..., basically because locality would ensure the validity of UV/IR decoupling so one would be able to explicitly construct remnants. ...these remnants would necessarily involve high-energy degrees of freedom in some essential way.

Why would the "validity of UV/IR decoupling" allow us to explicitly construct remnants? Are they merely saying that such a theory would necessarily have remnants? That I could believe. But they seem to be saying that we would automatically be able to explicitly construct those remnants. Is that really what they mean? If so, then how does that follow from the validity of UV/IR decoupling?


  1. Harlow and Shaghoulian, Global symmetry, Euclidean gravity, and the black hole information problem (https://arxiv.org/abs/2010.10539)
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    $\begingroup$ I wonder if there's a more objective way to frame this question, since it's not possible to know what the authors had in their minds when they wrote that unless one of them comments here. For example, "is it possible to explicitly construct a remnant in an asymptotically safe theory" (with a reference to the paper for context. FWIW, while I don't know what they meant, given the ratio of words per equations in many information loss papers including this one, if by "explicitly construct" they ended up meaning "exists but we can't write an explicit formula", I wouldn't be surprised. $\endgroup$
    – Andrew
    Nov 24, 2021 at 13:18
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    $\begingroup$ In fact given that they seem to be skeptical that an asymptotically safe gravity theory exists, it would be strange if they had an explicit formula for a remnant derived from an asymptotically safe gravity theory. Maybe they meant there's some procedure that could be used to generate a formula if there was such a theory, but to me that's starting to get to be a non-trivial enough statement that there would likely be a paper that proved that statement that they should have cited, if that's really what they meant. $\endgroup$
    – Andrew
    Nov 24, 2021 at 13:50
  • $\begingroup$ My read is that you are emphasizing the wrong part of the phrase. It is success of the program that is crucial for constructability. “UV/IR decoupling” (as a property of candidate QG theory) means that we have hierarchy of UV dof excitations of increasing complexity that all have the same IR description, remnants. But it is the “success of the program” (of finding UV completion) that would allow one to “explicitly construct” those remnants almost by definition (if you cannot produce those remnant states you have not yet identified the relevant UV dofs). $\endgroup$
    – A.V.S.
    Nov 25, 2021 at 19:11

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Disclaimer: Personally, I am not fully sure about this paper, since it outlines a connection between the Page curve and non-existence of global symmetries. For me, the idea of a Page curve in gravity is a bit difficult to digest since the Hilbert space of gravity doesn't factorize upon spatial partitioning due to the Gauss constraint. However the general arguments can possibly be correct, and there might be connections between unitarity and lack of global symmetries, possibly in some other form.

Now note that a local quantum field theory has the following features:

  1. There is no mixing between UV and IR degrees of freedom due to locality.
  2. There is no bound on the amount of information you can store in a region. In fact, using a naive calculation one can show that the entanglement entropy diverges even for a bulk scalar field, and one needs to appropriately regulate such quantities.

In a local QFT, the fact that there is no UV/IR mixing along with no bound on information content poses no restriction to the construction of remnant like objects with high information content which cannot be extracted from causally disconnected regions.

However, both these features are violated in a quantum theory of gravity. A drastic violation of the first feature (and locality in general) is the black hole complementarity principle, where the interior degrees of freedom are encoded in the exterior degrees of freedom, i.e., $$O(x_{in}) = P(O(x^1_{out}), O(x^1_{out}), O(x^1_{out})...),$$ where $P$ denotes a polynomial of $O(\mathcal{N})$ where $\mathcal{N}$ is the central charge given by $\mathcal{N}=N^2$, where $N$ is the gauge group of the boundary CFT. Such a description is necessary to consistently resolve no-cloning and strong subadditivity paradoxes.

The second feature is in contrast to the Bekenstein Hawking bound, which is a necessary property of any well-defined quantum theory of gravity. Analogously, there exist nice fine grained RT formula and its generalizations to calculate the entanglement entropy which predict a finite entropy.

This naively indicates that the existence of remnants is ill posed, since their effects would be visible using far-away experiments, and they violate the Bekenstein Hawking entropy. This is yet another argument against remnants, apart from the standard one involving excessive contributions to low energy scattering experiments due to their existence.

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  • $\begingroup$ I'm trying to understand the "no restriction" statement. In a local QFT, storing an arbitrary abount of information in a fixed volume requires an arbitrary amount of energy, hence the need to regulate the entanglement entropy with some kind of cutoff. It seems to me that the only way to have unlimited information capacity in a given volume is to have unlimited mass (which gravity doesn't allow, but local QFT does). Does the "no restriction" statement mean something else? $\endgroup$ Nov 25, 2021 at 3:22
  • $\begingroup$ I think I get it now. I was only considering physically sensible QFTs, namely those that satisfy the nuclearity property, which limits the entropy in a given volume with a given energy. But we could drop that requirement without abandoning locality, like in a QFT with an unlimited number of massless particle species. Such a QFT wouldn't be physically sensible, though, because it would have absurd thermodynamic properties... and that's your point. A theory with remnants whose info-content is high enough to resolve the info paradox without abandoning locality would be thermodynamically absurd. $\endgroup$ Nov 25, 2021 at 5:03
  • $\begingroup$ @ChiralAnomaly Yes I am thinking of local quantum field theories (LQFT) where the description holds upto a given UV scale. However gravity should not be thought of such a LQFT. There are many hints, both classical and quantum, which point at the same, the Hamiltonian is a boundary term, holography, black hole complementarity etc. One can squeeze in many degrees of freedom in an given bounded region in a LQFT, but gravity doesn't allow you to do so. $\endgroup$
    – Bruce Lee
    Nov 25, 2021 at 6:20
  • $\begingroup$ Right -- I fully appreciate that quantum gravity is holographic, not an LQFT, and (something like) BH complementarity seems inevitable to me. But I hadn't thought much about the remnants idea before, because I never really understood the motive for it, so here I'm thinking through it a little more carefully to give it a fair shake... and I'm realizing that my first impression was pretty much correct. The remnants idea was dead on arrival. Thanks! $\endgroup$ Nov 25, 2021 at 15:16

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