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A physical protocol for observers near the boundary to obtain bulk information in quantum gravity
by Chandramouli Chowdhury, Olga Papadoulaki, and Suvrat Raju
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|Authors (as registered SciPost users):
|Olga Papadoulaki · Suvrat Raju
We consider a set of observers who live near the boundary of global AdS, and are allowed to act only with simple low-energy unitaries and make measurements in a small interval of time. The observers are not allowed to leave the near-boundary region. We describe a physical protocol that nevertheless allows these observers to obtain detailed information about the bulk state. This protocol utilizes the leading gravitational back-reaction of a bulk excitation on the metric, and also relies on the entanglement-structure of the vacuum. For low-energy states, we show how the near-boundary observers can use this protocol to completely identify the bulk state. We explain why the protocol fails completely in theories without gravity, including non-gravitational gauge theories. This provides perturbative evidence for the claim that one of the signatures of holography --- the fact that information about the bulk is also available near the boundary --- is already visible in the low-energy theory of gravity.
Published as SciPost Phys. 10, 106 (2021)
List of changes
The list of changes is also provided in the reply to referee 2 and the changes are explained there.
1) The word "semiclassical" has been changed to "low-energy" at various points in the text so as to be more precise about what we mean.
2) We have elaborated on the relationship of the various cutoffs in the subsection on "Tasks of the Observers"
3) We have inserted text in the subsection on "Abilities and Limitations.." emphasizing that the need for multiple copies is not special to our protocol and follows from the inherently probabilistic nature of quantum mechanics.
4) We have inserted text in the section on "Allowed manipulations" explaining why the second order term is not displayed. (Its contribution vanishes.)
5) In the section on "Allowed measurements", we have inserted a remark and a reference to a gauge-invariant expression for the energy.
6) We have inserted a footnote in the section on "allowed measurements" explaining why the energy-spectrum is expected to be discrete about the vacuum of global AdS.
7) In the section on "implications for black holes" we have noted that, within effective field theory in a black-hole background, the quantization of energy is no-longer visible because the gaps between energy-levels are too small to seen perturbatively.
Submission & Refereeing History
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Reports on this Submission
Report 2 by Samir Mathur on 2021-4-8 (Invited Report)
- Cite as: Samir Mathur, Report on arXiv:scipost_202010_00002v2, delivered 2021-04-08, doi: 10.21468/SciPost.Report.2767
I remain puzzled about the results of this paper. However I have had a chance to understand the arguments better, so perhaps I can focus my concerns better as well. Notwithstanding these concerns, I think this paper should be published, since the authors have spent time thinking about these issues, and it would be useful to have the arguments available to the community to spur debate on these interesting issues.
From what I understand, the authors claim that in any theory with gravity, one can find the state inside some region $r < R$ by doing measurements only in a region $r>>R$; thus in particular light has not had time to bring any signals from the inner region to where the measurements are being done. Gravity can remain weak throughout, so no novel features of string theory or nonperturbative quantum gravity are being used.
The essence of the argument seems to be the following. The gravitational field h couples to the matter energy E. By doing careful measurements of the weak gravitational field near infinity, we can project the overall state to an eigenstate of the energy E. But energy eigenstates are typically travelling waves, which will keep oscillating all the way to infinity (rather than being localized wavepackets). Then we can observe this state near infinity, and thus infer what it must have been in the interior.
In particular, let the theory have two identical scale fields $\phi_1, \phi_2$. We make a localized wavepacket of one of these fields in the region $r<R$. We wish to detect from infinity which of the two fields we excited. We measure h at infinity, projecting to an energy eigenstate E for the scalar fields; then looking at the corresponding travelling wave of $\phi_i$ that reaches infinity in such an eigenstate, we will know which scalar field was excited in the interior.
One can worry about such an argument because of the simpler situation with just a scalar field in flat space. The vacuum has correlations at points that are spatial separated; this makes the Feynman propagator nonzero at spacelike separations. But an excitation at a point x cannot be detected at a spacelike - separated point y. To prove this one can do one of two things: (i) Check the vanishing of commutators of operators at these two points; this can be difficult in the gravity theory since one will need a good definition of a local operator first (ii) Put a source at one position and an Unruh -de Wit detector at the other, checking that the latter is not excited; this however is a complicated computation due to cancellations between several terms in the amplitude, and the local operators would have to be made very carefully just as in (i).
So I tried to make a counterexample to their claim as follows. Suppose we can make two solutions of the wavefunctional, satisfying the gauge constraints, with the following property: In the region $r<R$, the two are different, while in the region $r>2R$, they are identical. In this case I believe that no experiment at $r>2R$ can detect the difference between the two solutions at $r<R$. I believe the authors would claim that two such solutions cannot be made, but it was not clear to me that such was the case.
In the attached pdf file I sketch how I would make such a counterexample for the simpler case of the electromagnetic theory; I am guessing that the gravitational case is more messy to construct but similar. Thus my question would be
(i) Would such a counterexample, if it exists, disprove the authors' claim? Or have I completely misunderstood the problem?
(ii) If yes to (i), is there an error in the gauge theory example I have tried to make?
(iii) If the gauge theory counterexample is correct, is the gravity case very different? (The gravity case does have an extra spatial Laplacian term, but it is not clear why this should change the existence of such a construction)
To summarize, I remain confused by the claim of this paper. But as mentioned above, I think the paper should be published. In a complicated field like quantum gravity, it cannot be the authors' responsibility to convince another person (the referee) of their ideas; the correctness of the ideas should emerge by further investigations in the general gravity community.
- Cite as: Anonymous, Report on arXiv:scipost_202010_00002v2, delivered 2021-03-21, doi: 10.21468/SciPost.Report.2722
see my first report
see my first report
I would like to thank the authors for replying to my comments and questions and for introducing modifications to their manuscript where necessary. I recommend publishing the paper in the present form.