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

Maurice A De Gosson

doi:10.1063/5.0011113

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 ℏ and represent the ellipsoids of minimum uncertainty, which we called “quantum blobs” in previous work.

Highlights

Let us consider a bipartite physical system A ∪ B consisting of two subsystems A and B with phase spaces R2nA and R2nB, respectively
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 contains a symplectic ball with the same

Summary

INTRODUCTION

Let us consider a bipartite physical system A ∪ B consisting of two subsystems A and B with phase spaces R2nA and R2nB , respectively We assume that both A and B are Hamiltonians with respective Hamiltonian functions HA(xA, pA) and HB(xB, pB). The step involves replacing the total Hamiltonian H in (3) with a local approximation H0 for each z0 This local Hamiltonian is obtained as follows: let zt be the solution of the Hamilton equations for H passing through z0 at time t = 0. Where SA,t is a symplectic matrix in the smaller phase space R2nA and zA,t is the projection of zt This striking (and non-trivial) result follows from a generalization (in the linear case) of Gromov’s famous symplectic non-squeezing theorem.. By Liouville’s theorem, S(B2Rn(z0)) and B2Rn(z0) have the same volume

Symplectic terminology

Statement and discussion

Entropy and symplectic capacities

A generalization of the thawed Gaussian approximation

Gaussian mixed states and the Wigner ellipsoid

PERSPECTIVES AND SPECULATIONS