Hamiltonian approach to Ehrenfest expectation values and Gaussian quantum states

TL;DR

This paper develops a geometric Hamiltonian framework for quantum expectation values, especially Gaussian states, revealing their Lie-Poisson structures and unifying classical and quantum dynamics.

Contribution

It introduces a Hamiltonian geometric approach to Ehrenfest expectation values and Gaussian states, identifying their Lie-Poisson structures and momentum map properties.

Findings

Expectation values are equivariant momentum maps for the Heisenberg group.

Ehrenfest's theorem has a Lie-Poisson Hamiltonian structure.

Gaussian states' dynamics couple to second-order moments with a Lie-Poisson structure.

Abstract

The dynamics of quantum expectation values is considered in a geometric setting. First, expectation values of the canonical operators are shown to be equivariant momentum maps for the action of the Heisenberg group on quantum states. Then, the Hamiltonian structure of Ehrenfest's theorem is shown to be Lie-Poisson for a semidirect-product Lie group, named the `Ehrenfest group'. The underlying Poisson structure produces classical and quantum mechanics as special limit cases. In addition, quantum dynamics is expressed in the frame of the expectation values, in which the latter undergo canonical Hamiltonian motion. In the case of Gaussian states, expectation values dynamics couples to second-order moments, which also enjoy a momentum map structure. Eventually, Gaussian states are shown to possess a Lie-Poisson structure associated to another semidirect-product group, which is called the Jacobi group. This structure produces the energy-conserving variant of a class of Gaussian moment models previously appeared in the chemical physics literature.