Berry–Tabor conjecture in quantum chaos

Demonstrate that quantum systems with integrable classical dynamics exhibit Poisson eigenvalue statistics, while those with chaotic classical dynamics follow random matrix statistics in the appropriate universality class.

Background

The paper places the zeta-zeros/GUE connection within the broader context of quantum chaos, contrasting integrable and chaotic behavior via spectral statistics.

The Berry–Tabor conjecture remains a central unproven principle explaining when Poisson versus random-matrix statistics should appear.

References

The Berry-Tabor conjecture states that quantum systems with integrable classical limits have Poisson statistics for their eigenvalues, while those with chaotic classical limits follow random matrix statistics (GUE for time-reversal invariant systems).

— The Riemann Hypothesis: Past, Present and a Letter Through Time  (2602.04022 - Connes, 3 Feb 2026) in Subsubsection Quantum chaos