W [J] = N \int_{periodic} D x exp [- S_{0} [x] - \int_{0}^{β} d τ (\frac{λ}{4 !} x^{4} + \frac{μ}{6 !} x^{6}) + \int_{0}^{β} d τ J x] .

Here

W

is the source-dependent partition function, despite the common alternative convention in which that letter denotes its logarithm. A functional derivative of

e^{\int J x}

brings down one factor

x (τ)

at the differentiated point. Expanding the interaction exponential and replacing each such factor by

δ / δ J (τ)

therefore gives

W [J] = exp [- \int_{0}^{β} d τ {\frac{λ}{4 !} (\frac{δ}{δ J ( τ )})^{4} + \frac{μ}{6 !} (\frac{δ}{δ J ( τ )})^{6}}] W_{0} [J] .

W_{0} [J] = W_{0} [0] exp (\frac{1}{2} \int J G^{(0)} J)

because

G^{(0)} = D^{- 1}

with periodic boundary conditions. Restoring

ℏ

replaces the upper limit by

ℏ β

and the interaction coefficients by

λ / (4! ℏ)

and

μ / (6! ℏ)

, exactly as in the source.

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