Can quantum dynamics emerge from classical chaos?

TL;DR

This survey explains how quantum dynamics can emerge from classical chaos in Anosov geodesic flows via the Pollicott Ruelle spectrum, revealing a connection between classical correlation functions and quantum evolution.

Contribution

It uncovers a mechanism where quantum behavior arises from classical geodesic flows through spectral analysis, a novel insight into quantum-classical correspondence.

Findings

Quantum dynamics emerge from classical correlation functions.

The Pollicott Ruelle spectrum governs effective quantum-like evolution.

Spectral gaps determine the emergence of quantum behavior.

Abstract

Anosov geodesic flows are among the simplest mathematical models of deterministic chaos. In this survey we explain how, quite unexpectedly, quantum dynamics emerges from purely classical correlation functions. The underlying mechanism is the discrete Pollicott Ruelle spectrum of the geodesic flow, revealed through microlocal analysis. This spectrum naturally arranges into vertical bands; when the rightmost band is separated from the rest by a gap, it governs an effective dynamics that mirrors quantum evolution.