Hilbert–Pólya Conjecture: Spectral realization of the zeta zeros

Construct a quantum Hamiltonian whose eigenvalues correspond to the nontrivial zeros of the Riemann zeta function, thereby realizing the Hilbert–Pólya conjecture.

Background

The Hilbert–Pólya conjecture asserts a spectral interpretation of the nontrivial zeros of the Riemann zeta function, motivating the search for a quantum Hamiltonian whose eigenvalues match those zeros. The authors discuss this conjecture as a key historical approach linking physics and number theory, which has inspired searches for such an operator and related signatures across multiple physical settings.

While this work does not construct the Hilbert–Pólya Hamiltonian, it provides an alternative physical correspondence: engineered many-body dynamics whose critical points align with the zeta zeros. This offers complementary insights toward understanding the potential physical origin of the zeros and informs ongoing efforts motivated by the conjecture.

References

The most influential idea in this direction is the Hilbert–Pólya conjecture, which suggests that the nontrivial zeros of the zeta function might correspond to the eigenvalues of an yet-unknown quantum Hamiltonian, motivating searches for such an operator and for signatures in diverse physical contexts, including relativistic amplitudes, chaotic quantum scattering, quantum field theory, and random matrix theory.

— The Riemann Hypothesis Emerges in Dynamical Quantum Phase Transitions  (2511.11199 - Wei et al., 14 Nov 2025) in Introduction