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Non-Abelian observable-geometric phases and the Riemann zeros

Zeqian Chen

Abstract

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.

Non-Abelian observable-geometric phases and the Riemann zeros

Abstract

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.
Paper Structure (11 sections, 5 theorems, 85 equations)

This paper contains 11 sections, 5 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Non-Abelian observable-geometric phase
  3. An example
  4. The Riemann zeros
  5. Conclusion
  6. Appendix: Geometry over non-Abelian quantum connection
  7. Operator-principal fiber bundles
  8. Non-Abelian quantum connection
  9. Quantum parallel transportation
  10. Quantum holonomy
  11. Geometric interpretation of non-Abelian observable-geometric phases

Key Result

Proposition 2.1

With the above notations, for $1 \le j \le n$, This shows that $\tilde{\psi}^{(j)}_m (t)$'s are "the parallel transportation" in some sense (see Proposition prop:GeoInterNaOGP below).

Theorems & Definitions (25)

  • Proposition 2.1
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.1
  • Definition 6.1
  • Proposition 6.1
  • proof
  • Definition 6.2
  • ...and 15 more