= Polynomial interactions as source differential operators
{title2=$Z[J]=\exp\!\left(i\int\mathcal L_I\!\left[\tfrac1i\tfrac{\delta}{\delta J}\right]\right)Z_0[J]$}
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 $\int du\,e^{-u^TAu/2+b\cdot u+V(u)}=e^{V(\partial_b)}I_0(b)$. With a finite-mode regulator, a Minkowski source insertion satisfies $\phi(x)e^{i\int J\phi}=(1/i)\delta e^{i\int J\phi}/\delta J(x)$; 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>.
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