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

Abstract

Abstract Recently Ohsawa (Lett Math Phys 105:1301–1320, 2015) has studied the Marsden–Weinstein–Meyer quotient of the manifold $$T^*\mathbb {R}^n\times T^*\mathbb {R}^{2n^2}$$T∗Rn×T∗R2n2 under a $$\mathrm {O}(2n)$$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 $${\mathcal {F}}(T^*\mathbb {R}^n)$$F(T∗Rn) of $$T^*\mathbb {R}^n$$T∗Rn. 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 León (Rend Circ Mat Palermo 32:236–271, 1983).