Generating functional for electron-phonon systems from functional integrals with one-sided boundary conditions

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Abstract
We derive the generating functional of thermal Green’s functions for an interacting electron-phonon system through the use of functional integrals in holomorphic representations (c numbers for phonons and Grassmann numbers for electrons). These integrals are over paths obeying one-sided boundary conditions at either 0 or β=1/kT, and do not require normalization factors unlike previous derivations. We recover the Matsubara perturbation formalism without using any results from the standard operator method.