The Egorov Theorem for Transverse Dirac Type Operators on Foliated Manifolds

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. Introduction The Egorov theorem is a fundamental fact in microlocal analysis and quantum mechanics. It relates the evolution of pseudodifferential operators on a compact manifold (quantum observables) determined by a first order elliptic operator with the corresponding evolution of classical observables — the bicharacteristic flow on the space of symbols. More precisely, let M be a compact manifold and let P be a positive, self-adjoint, elliptic, first order pseudodifferential operator on M with the positive principal symbol p ∈ S1(T ∗M \ 0). Let ft be the bicharacteristic flow of the operator P , that is, the Hamiltonian flow of p on T ∗M . Egorov’s theorem [8] states that, for any pseudodifferential operator A of order 0 with the principal symbol a ∈ S0(T ∗M \ 0), the operator A(t) = eitPAe−itP is a pseudodifferential operator of order 0. The principal symbol at ∈ S0(T ∗M \0) of this operator is given by the formula at(x, ξ) = a(ft(x, ξ)), (x, ξ) ∈ T M \ 0. In the particular case P = √ ∆g, where ∆g is the Laplace-Beltrami operator of a Riemannian metric g on M , the corresponding bicharacteristic flow is the geodesic flow of g on T ∗M . 2000 Mathematics Subject Classification. 58J40, 58J42, 58B34.