Polynomial interactions as source differential operators ($Z[J]=\exp \left(i\int {\mathcal{L}}_I\left[\frac{1}{i}\frac{\delta }{\delta J}\right]\right)Z_0[J]$)

In a finite-dimensional integral, $u_i{e}^{b\cdot u}={\partial }_{b_i}{e}^{b\cdot u}$. Expand $e^{V(u)}$ as a formal perturbation series to obtain $\int du{e}^{-u^{T}Au/2+b\cdot u+V(u)}=e^{V({\partial }_b)}{I}_0(b)$. With a finite-mode regulator, a Minkowski source insertion satisfies $\varphi (x)e^{i\int J\varphi }=(1/i)\delta e^{i\int J\varphi }/\delta J(x)$; this gives the displayed interacting generating functional. The substitution is a perturbative differential-operator identity, not an assertion that the interaction exponential defines a convergent series. Divide the resulting functional by $Z[0]$ for normalized correlation functions.