 Solution

Use natural units $\hslash =k_B=1$ 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:The fields obey the coherent-state thermal boundary conditions $\psi (\beta ,\mathbf{r})=\psi (0,\mathbf{r})$ 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.