Relating short‑time expansions to chaos diagnostics

Establish a precise theoretical connection between short‑time expansions of dynamics (e.g., nested commutators or Poisson‑bracket based expansions of Heisenberg operators or classical observables) and rigorous diagnostics of chaos in both quantum and classical systems, clarifying criteria by which such short‑time trajectory expansions can definitively indicate chaotic versus integrable behavior.

Background

The paper analyzes short‑time operator growth and its connection to high‑frequency tails of spectral functions. It argues that, near the classical limit, operator growth shows similar asymptotic behavior in both integrable and chaotic systems and thus cannot reliably distinguish them. This motivates the need for a rigorous bridge between short‑time expansions and chaos diagnostics that capture long‑time sensitivity and instability.

The authors contrast these short‑time probes with their proposed long‑time, low‑frequency approach based on fidelity susceptibility and spectral functions, which does distinguish dynamical regimes. A formal connection between these two perspectives is currently lacking.