 Solution

For a finite-dimensional integral, $u_i{e}^{b\cdot u}={\partial }_{b_i}{e}^{b\cdot u}$. Expanding a polynomial interaction $V(u)$ formally and interchanging each finite-order derivative with the regulated integral giveswhere $I_0$ is the free Gaussian integral. This is a perturbation-series identity; it does not assert convergence of the full expansion for an arbitrary potential.

For the Minkowski functional integral, each field insertion is generated byExpand $e^{i{S}_I[\varphi ]}$, replace its field factors by these functional derivatives, and take the derivatives outside the free integral. The result is polynomial interactions as source differential operators:If the field interaction term contains derivatives of the field, the corresponding spacetime derivatives act on the source-insertion operators as well. For the scalar polynomial interactions the operator in the exponent isThe free quadratic source exponential then supplies all Wick contractions, reproducing the Feynman rules. For normalized correlation functions, divide the resulting interacting functional by $Z[0]$; the interaction operator changes this denominator too, so it cannot simply be ignored.