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Black holes, quantum chaos, and the Riemann hypothesis

by Panos Betzios, Nava Gaddam, Olga Papadoulaki

Submission summary

As Contributors: Panagiotis Betzios · Nava Gaddam · Olga Papadoulaki
Arxiv Link: (pdf)
Date submitted: 2020-07-07 02:00
Submitted by: Gaddam, Nava
Submitted to: SciPost Physics
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 dynamics of the scattering matrix on a Schwarzschild black hole background, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix.

Current status:
Editor-in-charge assigned

Submission & Refereeing History

Reports on this Submission

Anonymous Report 1 on 2020-10-11 Invited Report


1) the paper addresses an important problem in quantum gravity
2) it proposes an interesting connection with a seemingly unrelated problem as the Riemann hypothesis
3) the authors have a good knowledge of the literature
4) the paper is generally well written


1) the paper lacks clarity in the sections that support the main claim


The authors analyze in this paper the quantization of two Hamiltonians that describe the near-horizon dynamics of a Schwarzschild blackhole. The corresponding S-matrices were derived in a previous paper [17] by the same authors showing that they are equal to the one found by ‘t Hooft that describes the gravitational back reaction of the blackhole [15,16]. The Hamiltonians, given in eqs. (1) and (2), are written in terms of the Kruskal variables U and V whose commutator is the Heisenberg one for x and p, up to some constants for the Hamiltonian (2). The U-V variables also carry the orbital quantum numbers l,m thanks to the rotational symmetry of the Schwarzschild metric.

To analyze this model the authors drop the l, m dependence obtaining the Hamiltonian H= (U V+ VU)/2 that Berry and Keating conjectured to be related to Riemann hypothesis (this is the famous xp model) [10,11]. More explicitely, these authors proposed that a certain quantization of H= xp would yield a discrete spectrum given by the non trivial zeros of the Riemann zeta function. To do so, Berry and Keating imposed certain boundary conditions on the wave function but they were not consistent. The aim of the authors in this paper is to find consistent boundary conditions for the UV (i.e. xp) Hamiltonian that would yield the Riemann zeros in the spectrum. Those boundary conditions are derived from the gauging of the CPT symmetry in quantum gravity, that several authors have proposed in the past.

I find very appealing the ideas proposed in this paper, specially the connection between the near-horizon dynamics and the xp Hamiltonian. However, I do not
agree with the conclusion, namely that imposing the CPT symmetries yield boundary conditions that discretize the spectrum that then corresponds to the zeros of the Riemann zeta function or the Dirichlet beta function.

Let me explain this statement in detail. The problem is how the discrete symmetry T^+_2 is implemented. The authors say that it leaves the center (U,V)=(0,0) invariant, but using eq. (10) one obtains (0, infinity) under the transformation. I understand that the goal of the authors is to adapt somehow, in the blackhole geometry, the Berry-Keating idea of the Planck cell, but this is not well explained in the manuscript. I suggest the authors to clarify this part. Let me assume for the time being that there are indeed consistent boundary conditions based on CPT. These conditions lead to eq.(14) that the authors arguee is satisfied provided the energy E is the imaginary part of a Riemann zero. It is not clear that this is the case. First of all, the lhs of this equation depends on V while the rhs depends on U. I guess, one has to impose that |UV| = h and then use equations (12) and (13). If so, please do it explictely. But the main problem here is that for a Riemann zero the lhs and rhs of eq.(14) actually vanish, so is not clear that this is an eigenvalue equation.
Can one write (14) as zeta(1/2 + i E)=0? Something like this also appears in the Berry-Keating paper, where a condition was derived so that the wave function vanishes (see eq.(25) and below of [10}). The vanishing of the lhs and rhs could rather be interpreted, not as bound states, but as missing states along the lines of Connes spectral realization ot the Riemann zeros. If this were the case it would be a very interesting result too.
The paper satisfies the general criteria for publication in this journal, but before acceptance the authors have to answer the questions posed above.

Requested changes

1) After eq.(5) it is said that region II in the Penrose diagram corresponds to U > 0 and V < 0. I think there is V >0.
2) At the top of page 5 it is written:
“Demanding complete conformal invariance on the wavefunctions trivialises them.”
What is meant by that?
3) In page 5 it is given the CPT transformation of the wave function. They should be given both for psi(U) and psi(V) independently since they are conjugate variables.

  • validity: good
  • significance: high
  • originality: high
  • clarity: low
  • formatting: good
  • grammar: excellent

Author Nava Gaddam on 2020-10-12
(in reply to Report 1 on 2020-10-11)
answer to question

We would like to first thank the referee for the careful reading of our paper, and for the encouraging feedback. We understand the referee's concerns as being twofold:

a) Clarification of the boundary condition. In retrospect, the discussion of the boundary condition in the paper is perhaps unclear, and distracted by the discussion of the Dihedral group. The operator T_2^+ is a symmetry of the Hamiltonian. |UV| = h defines a hyperbolae and any point on such a hyperbola is time translated in these coordinates by dilation. Now, consider the central causal diamond (centered around U=0=V) of area 4h. This diamond is made of four squares of area h (one in each quadrant). One of the four vertices of the central causal diamond lies in Region I (with coordinates U= - \sqrt{h} and V = \sqrt{h}) and another lies in Region II (with coordinates U = \sqrt{h} and V = - \sqrt{h}). The effect of dilations is such that each of the squares making up the central causal diamond is elongated into a rectangle with the same area as the square, namely h.

The boundary condition then identifies hyperbolae passing through these vertices. This is our concrete proposal for how to adapt the boundary condition of Berry-Keating. This ensures that the Planckian uncertainty in going to the ultra local points within Planckian distance of the horizon is accounted for.

Please see the attached images for a picture of the central causal diamond, and the hyperbolae being referred to.

b) The eigenvalue equation. Equation (14) is arrived at, by using s = 1/2 + i E/\hbar, and combining the S-matrix relations (12) and (13) together with the unnumbered equations defining the transformation of the wavefunctions under CPT. Now, equation (14) is an equivalence relation that must hold for all U and V on the hyperbolae |UV| = h (therefore, it is an infinite number of conditions on the hyperbola). Of course, one trivial way to satisfy this equivalence relation is to demand that all fourier coefficients of the wavefunctions be zero. This trivialises the wavefunctions. So, the only way to satisfy this equivalence relation for non-trivial wavefunctions is by independently setting \zeta(1/2+i E/\hbar) = 0 *and* \zeta(1/2 - i E/\hbar) = 0. Therefore, we believe that this is precisely what the referee desires.

--- The authors

P.S: Subject to the Editor's approval, we are happy to clarify these points and also attend to the changes requested by the referee.



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