The frame bundle picture of Gaussian wave packet dynamics in semiclassical mechanics

TL;DR

This paper introduces a novel geometric interpretation of Gaussian wave packet dynamics in semiclassical mechanics using the frame bundle of phase space, offering new insights and potential extensions to general symplectic manifolds.

Contribution

It proposes viewing a subset of the unreduced space as the frame bundle of phase space, providing advantages and extending the framework to broader symplectic manifolds.

Findings

Interprets the unreduced space as the frame bundle of phase space.

Connects Gaussian wave packet parametrizations with geometric structures.

Extends the interpretation to general symplectic manifolds using diagonal lift.

Abstract

Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the manifold $T^*\mathbb{R}^n\times T^*\mathbb{R}^{2n^2}$ under a $\operatorname{O}(2n)$-symmetry, and has used this quotient to describe the relationship between two different parametrisations of Gaussian wave packet dynamics commonly used in semiclassical mechanics. In this paper we suggest a new interpretation of (a subset of) the unreduced space as being the frame bundle $\mathcal{F}(T^*\mathbb{R}^n)$ of $T^*\mathbb{R}^n$. We outline some advantages of this interpretation, and explain how it can be extended to more general symplectic manifolds using the notion of the diagonal lift of a symplectic form due to Cordero and de Le\'on.