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 Incoming links: Generating functional

The logarithm of a generating functional generates connected correlation functions because its cumulant expansion removes products of disconnected vacuum components.

ϕ (x) e^{i \int J ϕ} = \frac{1}{i} \frac{δ}{δ J ( x )} e^{i \int J ϕ} = - i \frac{δ}{δ J ( x )} e^{i \int J ϕ} .

Expanding the interaction exponential, making this replacement in every term, and resumming gives

Z_{n} [J] = exp [- i g \frac{( - i ) ^{n}}{n !} \int d^{d} x \frac{δ ^{n}}{δ J ( x ) ^{n}}] Z_{0} [J] .

This formal identity assumes a common regulator, a source-independent normalization, and permission to interchange the path integral, power series, and functional derivatives. A normalized functional with

Z_{n} [0] = 1

requires division by the same expression evaluated at

J = 0

, which removes connected Vacuum Feynman diagrams.

The disjoint Fourier supports make the quadratic cross term

\int \partial ϕ \cdot \partial χ

vanish. Therefore

S [ϕ + χ] = S [ϕ] + S_{0, χ} [χ] + S_{int} [ϕ, χ],

where

S_{int} [ϕ, χ] = \frac{g}{3 !} \int d^{d} x (3 ϕ^{2} χ + 3 ϕ χ^{2} + χ^{3}) .

Define the high-mode free generating functional with source convention

Z_{χ} [J] = \int D χ e^{- S_{0, χ} [χ] - \int J χ} .

Then inserting

χ = - δ / δ J

reproduces every high-field factor, and

e^{- W [ϕ]} = e^{- S [ϕ]} e^{- S_{int} [ϕ, - δ / δ J]} Z_{χ} [J]_{J = 0} .

Expanding the interaction exponential and applying Wick theorem evaluates the Wilsonian effective action as a sum of connected diagrams whose internal lines are restricted to the high-momentum shell.