 Solution

For a normal-ordered bosonic Hamiltonian, insert coherent-state resolutions between short imaginary-time evolution factors. The overlaps generate $\bar{\psi }{\partial }_{\tau }\psi$ and the Hamiltonian symbol is obtained by replacing creation and annihilation operators by $\bar{\psi }$ and $\psi$. Including the chemical potential gives, in the stated $\hslash =1$ convention,The trace imposes periodic imaginary-time boundary conditions,so bosonic Matsubara frequencies are ${\omega }_n=2\pi n/\beta$. The two complex coherent-state integration variables are treated independently before imposing the physical saddle contour. There is no extra zero-point ordering term: the supplied interaction is already normal ordered.

The spatial confinement is not specified beyond the volume. The uniform bulk results below use a homogeneous box with periodic spatial boundaries, or the interior of a large vessel where boundary effects can be neglected. A hard wall instead requires a vanishing field at that wall and an inhomogeneous saddle near it. Temporal periodicity follows from the thermal trace and does not by itself impose periodic spatial boundaries.