In section I.3 of QFT in a Nutshell, Zee examines the path integral for the Lagrangian \(\mathcal{L}(\phi) = \frac{1}{2}[(\partial \phi)^2m^2\phi^2]\) (the “free”/”Gaussian” Lagrangian) with a source \(J\). This looks like $$\begin{align} Z(J)&=\int D\phi\,e^{i\int d^4x\,\{\frac{1}{2}[(\partial \phi)^2m^2\phi^2]+J\phi\}}\\ &=\int D\phi\,e^{i\int d^4x\,\{\frac{1}{2}\phi(\partial^2+m^2)\phi+J\phi\}}. \end{align}$$ (Integration by parts allows us to replace \((\partial\phi)^2\) with \(\phi(\partial^2\phi)\) in the exponent’s integrand.) The exponent [...]

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