Non-Abelian observable-geometric phases and the Riemann zeros
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911818330406912 |
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| author | Chen, Zeqian |
| author_facet | Chen, Zeqian |
| contents | The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19118 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Abelian observable-geometric phases and the Riemann zeros Chen, Zeqian Quantum Physics Number Theory Operator Algebras The Hilbert-Pólya conjecture asserts that the imaginary parts of the nontrivial zeros of the Riemann zeta function (the Riemann zeros) are the eigenvalues of a self-adjoint operator (a quantum mechanical Hamiltonian, in the physical sense), as a promising approach to prove the Riemann hypothesis (cf.\cite{SH2011}). Instead of the eigenvalues, in this paper we consider observable-geometric phases as the realization of the Riemann zeros in a periodically driven quantum system, which were introduced in \cite{Chen2020} for the study of geometric quantum computation. To this end, we further introduce the notion of non-Abelian observable-geometric phases, involving which we give an approach to finding a physical system to study the Riemann zeros. Since the observable-geometric phases are connected with the geometry of the observable space according to the evolution of the Heisenberg equation, this sheds some light on the investigation of the Riemann hypothesis. |
| title | Non-Abelian observable-geometric phases and the Riemann zeros |
| topic | Quantum Physics Number Theory Operator Algebras |
| url | https://arxiv.org/abs/2403.19118 |