Symplectic Coarse-Grained Classical and Semi-Classical Evolution of Subsystems: New Theoretical Aspects

TL;DR

This paper explores the classical and semiclassical evolution of subsystems within Hamiltonian systems using symplectic geometry, extending theoretical principles to better understand quantum and classical dynamics.

Contribution

It introduces a new extension of Gromov's symplectic camel principle applicable to subsystem projections, linking classical and quantum phase space analysis.

Findings

Extended Gromov's principle to subsystem projections.

Connected symplectic geometry with quantum uncertainty ellipsoids.

Provided a framework for semiclassical evolution analysis.

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 N. Dias and J. Prata. 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 sqrt(h_bar) and represent ellipsoids of minimum uncertainty, which we have called "quantum blobs" in previous work.