Nonclassical Degrees of Freedom in the Riemann Hamiltonian

Understand

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.