Analytic normal-modes propagator for bosonic free ring-polymer Hamiltonian

Derive an analytic expression for the normal-modes propagator of the free ring polymer part of the Hamiltonian for bosonic path integral molecular dynamics, so that integration can be performed in normal modes rather than restricted to Cartesian coordinates.

Background

i-PI 3.0 implements bosonic path integral molecular dynamics using an algorithm that includes exchange effects with quadratic scaling, enabling simulations of large bosonic systems. However, the current implementation must use Cartesian-coordinate integration because the corresponding normal-modes propagator for the free ring-polymer Hamiltonian for bosons is unavailable.

Having an analytic expression for this propagator would enable the use of normal-modes integrators (as commonly done for distinguishable particles), potentially improving stability and efficiency for bosonic simulations and facilitating multiple-time stepping strategies.