Black holes, quantum chaos, and the Riemann hypothesis

Betzios, Panagiotis; Gaddam, Nava; Papadoulaki, Olga

doi:10.21468/SciPostPhysCore.4.4.032

SciPost Physics Core

Black holes, quantum chaos, and the Riemann hypothesis

Panagiotis Betzios, Nava Gaddam, Olga Papadoulaki

SciPost Phys. Core 4, 032 (2021) · published 10 December 2021

doi: 10.21468/SciPostPhysCore.4.4.032
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Abstract

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.

TY  - JOUR
PB  - SciPost Foundation
DO  - 10.21468/SciPostPhysCore.4.4.032
TI  - Black holes, quantum chaos, and the Riemann hypothesis
PY  - 2021/12/10
UR  - https://scipost.org/SciPostPhysCore.4.4.032
JF  - SciPost Physics Core
JA  - SciPost Phys. Core
VL  - 4
IS  - 4
SP  - 032
A1  - Betzios, Panagiotis
AU  - Gaddam, Nava
AU  - Papadoulaki, Olga
AB  - 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.
ER  -

@Article{10.21468/SciPostPhysCore.4.4.032,
	title={{Black holes, quantum chaos, and the Riemann hypothesis}},
	author={Panagiotis Betzios and Nava Gaddam and Olga Papadoulaki},
	journal={SciPost Phys. Core},
	volume={4},
	pages={032},
	year={2021},
	publisher={SciPost},
	doi={10.21468/SciPostPhysCore.4.4.032},
	url={https://scipost.org/10.21468/SciPostPhysCore.4.4.032},
}

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