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.

References

At the moment, the integration should be done in Cartesian coordinates (normal_modes propagator='bab'), because an analytic expression for the normal-modes propagator of the free ring polymer part of the Hamiltonian is not known for bosons.

— i-PI 3.0: a flexible and efficient framework for advanced atomistic simulations  (2405.15224 - Litman et al., 2024) in Section IV.A, Bosonic and Fermionic Path Integral Molecular Dynamics

The primary problem of calculation ESCF is that the operators of kinetic $\hat K$ and potential $\hat U$ energies do not commute; hence, an exact explicit analytical formulation for the matrix elements remains unknown.

— Decoherence of the pair entangled states at thermalization and interparticle interaction. Heisenberg--Wigner path integral Monte Carlo simulations  (2610.26033 - Filinov et al., 19 Oct 2026) in Section 2, “Heisenberg--Wigner path integral representation and ‘replica trick’”