ℏ = 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:

Z = \int D (\overset{ˉ}{ψ}, ψ) e^{- S}, S = \int_{0}^{β} d τ \int_{[0, L]^{d}} d^{d} r [\overset{ˉ}{ψ} \partial_{τ} ψ + \frac{\nabla ψ ˉ \cdot \nabla ψ}{2 m} - μ \overset{ˉ}{ψ} ψ + \frac{g}{2} (\overset{ˉ}{ψ} ψ)^{2}] .

The fields obey the coherent-state thermal boundary conditions $ψ (β, r) = ψ (0, r)$ and $\overset{ˉ}{ψ} (β, r) = \overset{ˉ}{ψ} (0, r)$ . Bosonic Matsubara frequencies are therefore

2 πn / β

. 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.