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Black holes, quantum chaos, and the Riemann hypothesis

by Panos Betzios, Nava Gaddam, Olga Papadoulaki

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Submission summary

Authors (as Contributors): Panagiotis Betzios · Nava Gaddam · Olga Papadoulaki
Submission information
Arxiv Link: (pdf)
Date accepted: 2021-10-28
Date submitted: 2021-06-14 15:21
Submitted by: Gaddam, Nava
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Gravitation, Cosmology and Astroparticle Physics
  • High-Energy Physics - Theory
Approach: Theoretical


Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this quantum Hamiltonian captures the near horizon dynamics of the scattering matrix of the Schwarzschild black hole, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix.

Published as SciPost Phys. Core 4, 032 (2021)

List of changes

Dear editor,

As per your suggestions, we have modified the draft to address the recommendations of the referee:

Requested change 1) The definition of region II we use is indeed what we write, it is intended to be the second exterior of the Kruskal diagram. Sometimes the interior is called region II, perhaps leading to the referee's confusion (this interior is U>0, V>0). So no change has been made here. We have also added the figures we used in our response to the referee to the paper, for clarification.

Requested change 2) We have removed the confusing sentence the referee points to; it is no significance to the rest of the paper.

Requested change 3) In equation 8, the identification written is for classical functions (as we now clarified), which indeed does not take the conjugate nature of the variables into account, as the referee points out. The appropriate identifications are what we present in equations (14) and (15) in the new version. As the referee asks, these are indeed written as two equations, one for \psi(U) and another for \psi(V). We have also added references to Connes' work.

We hope that this suffices and that you would accept it for publication. Thank you for your consideration.
The authors.

Reports on this Submission

Anonymous Report 2 on 2021-8-12 (Invited Report)


1) the paper deals with a very interesting problem connecting several areas on theoretical physics and mathematics
2) the derivation of results is rigorous and can be easily followed
3) it contains interesting suggestions concerning the realization of the so called Riemann zeros with the emission or absortion spectrum of a Hamiltonian
4) The latter Hamiltonian is given by the Berry Keating xp model supplemented by certain boundary conditions


1) the term "quantum chaos" appears in the title but it is not clear, nor described in some detail, how it may appears in the results presented except for some general considerations


This paper is worth to be published in this journal. It contains very interesting results that may motivate the research in a fundamental problem such as relating black holes and the Riemann hypothesis

Requested changes

The new version answers positively the comments made in my previous report

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: perfect

Anonymous Report 1 on 2021-6-22 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2004.09523v4, delivered 2021-06-22, doi: 10.21468/SciPost.Report.3098


Dear Editor,

In this paper, authors demanded a CPT invariance of the wavefunction, that in turn gives certain condition on U and V operators. Then they have considered the generators of Dilatation (the Hamiltonian mentioned in equation (1) ) of the paper. Using these conditions, one can show that after solving the spectrum of the Hamiltonian, the spectrum consists of even and odd modes. Even modes are given by the Riemann zeta function and the odd modes are given by zeroes Dirichlet beta function. Then authors comment that using the idea put forward by Berry and Keating may put forward an interesting playground to study the Riemann hypothesis.

This may be an interesting connection but I do not see its obvious utility. Maybe the authors can elaborate or review the idea of Berry and Keating a bit more and make a concrete connection or at least a bit more pointers on how their analysis will provide me with an interesting insight into the Riemann hypothesis. So far the far merely points out a mathematical relation that may be interesting.

Also, the authors claim in the abstract "It strengthens the
proposal that this quantum Hamiltonian captures the near horizon dynamics of
the scattering matrix of the Schwarzschild black hole, given the rich chaotic spectrum upon discretisation. " Except for some passing comment about the phase space in some places of the paper (eg the last paragraph on page 5) I do not see any concrete calculation which supports the statement mentioned in the abstract. There are other studies of the spectrum which points to the act the there is a rich chaotic structure, are the authors simply referring to those? or have they done some more analysis in this paper? (which is not obvious at least to me).

This paper in its current form may be considered for sci-post physics core if the editor thinks it is appropriate.

  • validity: ok
  • significance: low
  • originality: ok
  • clarity: good
  • formatting: reasonable
  • grammar: good

Author:  Nava Gaddam  on 2021-06-30  [id 1536]

(in reply to Report 1 on 2021-06-22)

We are thankful to the referee for reviewing our article. We would like to respond to the two broad comments they have raised.

  1. Since the work of BK, there has been a long program of attempting to construct similar Hamiltonians that may produce the spectrum of non-trivial Riemann zeros (see ref. 14 of the present article and references therein). Connes has also proposed that these might instead be seen as 'missing states' in the spectrum. BK's original work considered an inverted harmonic oscillator (but has some shortcomings and doesn't yet prove the Riemann hypothesis). In our article, the inverted harmonic oscillator arises from the near-horizon dynamics of 4d Schwarzschild black holes and therefore carries angular degrees of freedom arising from the partial waves. These naturally give rise to additional ingredients to the BK proposal inspired by black hole dynamics. We have clearly not carried out a detailed mathematical study of self-adjointness and associated Maslov indices, etc. Without such further careful study, it is unclear exactly what our proposal means for a proof of the Riemann hypothesis. Nevertheless our paper posits that it is intimately tied to black hole physics.

  2. 't Hooft has long argued that gravitational interactions near the horizon have an important role to play in the information paradox. In our previous work in 2016 (ref. 17 of the current article), we showed that these interactions can be recast in the form of an infinite tower of inverted harmonic oscillators. The natural question is whether such simple inverted harmonic oscillators can capture the rich, chaotic dynamics expected of black holes. In the present article, we have shown the existence of a physically motivated boundary condition on our proposed quantum Hamiltonian that gives rise to a spectrum of non-trivial Riemann zeros (eqs 16 and 17, and the discussion thereafter). This is the main result of the calculation we presented in this article. Therefore, the statement, from the abstract, quoted by the referee is justified by the main calculation we have performed in this article. 

Thank you for your consideration, The authors.

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