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Black holes, quantum chaos, and the Riemann hypothesis
by Panos Betzios, Nava Gaddam, Olga Papadoulaki
|As Contributors:||Panagiotis Betzios · Nava Gaddam · Olga Papadoulaki|
|Arxiv Link:||https://arxiv.org/abs/2004.09523v3 (pdf)|
|Date submitted:||2020-07-07 02:00|
|Submitted by:||Gaddam, Nava|
|Submitted to:||SciPost Physics|
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.
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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  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.
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.