= 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 $\hbar=1$ convention,
$$
\boxed{S=\int_0^\beta d\tau\int_Vd^dr\left[
\bar\psi\left(\partial_\tau-\frac{\nabla^2}{2m}-\mu\right)\psi
+\frac g2(\bar\psi\psi)^2\right].}
$$
The <trace> imposes \b[periodic imaginary-time <boundary conditions>],
$$
\psi(\mathbf r,\beta)=\psi(\mathbf r,0),\qquad
\bar\psi(\mathbf r,\beta)=\bar\psi(\mathbf r,0),
$$
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.
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