Symplectic coarse-grained classical and semclassical evolution of subsystems: New theoretical approach

Abstract

We study the classical and semiclassical time evolutions of subsystems of a Hamiltonian system; this is done using a generalization of Heller's thawed Gaussian approximation introduced by Littlejohn. The key tool in our study is an extension of Gromov's "principle of the symplectic camel"obtained in collaboration with Dias, de Gosson, and Prata [arXiv:1911.03763v1 [math.SG] (2019)]. This extension says that the orthogonal projection of a symplectic phase space ball on a phase space with a smaller dimension also contains a symplectic ball with the same radius. In the quantum case, the radii of these symplectic balls are taken equal to √h and represent the ellipsoids of minimum uncertainty, which we called "quantum blobs"in previous work.