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

Recently Ohsawa has studied the Marsden-Weinstein-Meyer quotient of the manifold $T^{\ast }{\mathbb{R}}^n\times T^{\ast }{\mathbb{R}}^{2n^2}$ under a $\mathrm{O}(2n)$-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 $\mathcal{F}(T^{\ast }{\mathbb{R}}^n)$ of $T^{\ast }{\mathbb{R}}^n$. 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.