The Egorov theorem for transverse Dirac type operators on foliated manifolds

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.

27 pages, to appear in Journal of Geometry and Physics

References in corpus (5)

• Geometry of the transport equation in multicomponent WKB approximations
• A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators
• High energy limits of Laplace-type and Dirac-type eigenfunctions and frame flows
• The trace formula for transversally elliptic operators on Riemannian foliations
• Egorov's theorem for transversally elliptic operators on foliated manifolds and noncommutative geodesic flow

Cited by in corpus (6)

• Transverse noncommutative geometry of foliations
• An index theorem on asymptotically static spacetimes with compact Cauchy surface
• Laplacians on smooth distributions
• Classical and Quantum Dynamics on Orbifolds
• Vanishing theorem for transverse Dirac operators on Riemannian foliations
• Noncommutative Hamiltonian dynamics on foliated manifolds