## Monthly Archives: April 2011

### nutshell notes: free propagators

In section I.3 of QFT in a Nutshell, Zee examines the path integral for the Lagrangian $$\mathcal{L}(\phi) = \frac{1}{2}[(\partial \phi)^2-m^2\phi^2]$$ (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)^2-m^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 [...]