Egorov's theorem in the Weyl--H\"ormander calculus

We prove a general version of Egorov's theorem for evolution propagators in the Euclidean space, in the Weyl--H\"ormander framework of metrics on the phase space. Mild assumptions on the Hamiltonian allow for a wide range of applications that we describe in the paper, including Schr\"odinger, wave and transport evolutions. We also quantify an Ehrenfest time and describe the full symbol of the conjugated operator. Our main result is a consequence of a stronger theorem on the propagation of quantum partitions of unity.