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

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).

A NEW SEMICLASSICAL DYNAMICS FROM THE INTERACTION REPRESENTATION

This paper develops a new semiclassical mechanics from an exact quantum prescription. In this formulation, a zeroth order propagation, rather than Hamiltonian, is specified. The exact, full evolution operator is then given from a specific interaction representation of the evolved "perturbation" Hamiltonian. We then investigate a variety of approximate, semiclassical, and mixed Quantum/Classical methods, along with exact methodologies to evaluate this time dependent interaction Hamiltonian. The approximate full evolution operator can be described in a variety of ways including an iterated Lippman-Schwinger like equation, and an expansion of the perturbation propagator generated from the time evolved Hamiltonian.