TITLE: Complex scalar field, generating functional?
QUESTION [6 upvotes]: The Lagrangian for the complex scalar field is$$\mathcal{L} = \partial_\mu \phi^* \partial^\mu \phi - m^2 \phi^* \phi.$$Can anyone work out or provide me a reference to the computation of the generating functional$$Z[J, J^*] = \int \mathcal{D}\phi \,\exp\left\{i \int d^4x\,[\mathcal{L} + J(x)\phi^*(x) + J^*(x)\phi(x)]\right\}?$$I need this result for my research, but unfortunately, I am not a physicist by training. Thank you very much!

REPLY [3 votes]: you can find this calculation, for example, in chapter 7 of Kleinert's path integral book, I reproduce the relevant equations: