Nonclassical Degrees of Freedom in the Riemann Hamiltonian

TL;DR

This paper explores the quantum Hamiltonian linked to the Riemann zeta function zeros, revealing a nonclassical two-valued degree of freedom that explains oscillatory behaviors and resolves a sign problem in the density of states.

Contribution

It identifies the nonclassical two-valued degree of freedom in the Riemann Hamiltonian, connecting number theory with quantum chaos and universality classes.

Findings

Dominant periodic orbits contribute with a phase factor of -1.

Resolves the sign problem in the oscillatory density of Riemann zeros.

Supports the conjecture that the Riemann zeros are eigenvalues of a quantum Hamiltonian.

Abstract

The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum hamiltonian. If so, conjectures by Katz and Sarnak put this hamiltonian in Altland and Zirnbauer's universality class C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.