The Egorov theorem for transverse Dirac type operators on foliated manifolds

TL;DR

This paper proves an Egorov theorem for transverse Dirac operators on foliated manifolds, linking quantum and classical evolutions and describing noncommutative geodesic flow in Riemannian foliations.

Contribution

It establishes the Egorov theorem for transversally elliptic operators and characterizes the transverse bicharacteristic flow as the transverse geodesic flow for Dirac operators.

Findings

Egorov's theorem is proved for transversally elliptic operators.

Transverse bicharacteristic flow corresponds to transverse geodesic flow.

Results describe noncommutative geodesic flow in Riemannian foliations.

Abstract

Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of its symbols determined by the parallel transport along the orbits of the associated transverse bicharacteristic flow. For a particular case of a transverse Dirac operator, the transverse bicharacteristic flow is shown to be given by the transverse geodesic flow and the parallel transport by the parallel transport determined by the transverse Levi-Civita connection. These results allow us to describe the noncommutative geodesic flow in noncommutative geometry of Riemannian foliations.