Semiclassical approximation and noncommutative geometry

TL;DR

This paper investigates the semiclassical evolution of quantum observables in chaotic systems, showing that their symbols evolve via a push-forward mechanism within a noncommutative algebra framework for long times.

Contribution

It introduces a Toeplitz quantization approach to describe the symbol evolution of observables in chaotic quantum systems over long timescales.

Findings

Symbol of evolved observable approximates push-forward of initial symbol

Uses noncommutative algebra of unstable foliation

Applicable for times up to ^{-2+\u03b5}

Abstract

We consider the long time semiclassical evolution for the linear Schr\"odinger equation. We show that, in the case of chaotic underlying classical dynamics and for times up to $\hbar^{-2+\epsilon},\ \epsilon>0$, the symbol of a propagated observable by the corresponding von Neumann-Heisenberg equation is, in a sense made precise below, precisely obtained by the push-forward of the symbol of the observable at time $t=0$. The corresponding definition of the symbol calls upon a kind of Toeplitz quantization framework, and the symbol itself is an element of the noncommutative algebra of the (strong) unstable foliation of the underlying dynamics.