= Solution
Use <natural units> $\hbar=k_B=1$ for the action conventions of this question. Insert the <coherent-state resolution of identity> between imaginary-time steps and take the thermal trace. <Coherent-state time slicing> gives the normally ordered interaction and first-order time term:
$$
\boxed{Z=\int\mathcal D(\bar\psi,\psi)e^{-S},\qquad
S=\int_0^\beta d\tau\int_{[0,L]^d}d^dr\left[\bar\psi\partial_\tau\psi+\frac{\nabla\bar\psi\cdot\nabla\psi}{2m}-\mu\bar\psi\psi+\frac g2(\bar\psi\psi)^2\right].}
$$
The fields obey the <coherent-state thermal boundary conditions> \b[$\psi(\beta,\mathbf r)=\psi(0,\mathbf r)$ and $\bar\psi(\beta,\mathbf r)=\bar\psi(0,\mathbf r)$]. Bosonic <Matsubara frequencies> are therefore $2\pi n/\beta$. A periodic spatial box can be chosen for the homogeneous bulk calculation; temporal periodicity follows from the trace independently of that spatial choice. The barred and unbarred labels belong to the coherent-state integral prescription, with the usual conjugate contour for bosonic <Gaussian integral>.
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