Calculation of the single-particle Green's function of interacting fermions in arbitrary dimension via functional bosonization

The single-particle Green's function of an interacting Fermi system with dominant forward scattering is calculated by decoupling the interaction by means of a Hubbard-Stratonowich transformation involving a bosonic auxiliary field $\phi^{\alpha}$. We obtain a higher dimensional generalization of the well-known one-dimensional bosonization result for the Green's function by first calculating the Green's function for a fixed configuration of the $\phi^{\alpha}$-field and then averaging the resulting expression with respect to the probability distribution ${\cal{P}} \{ \phi^{\alpha} \} \propto \exp [ - S_{eff} \{ \phi^{\alpha} \} ]$, where $S_{eff} \{ \phi^{\alpha} \}$ is the effective action of the $\phi^{\alpha}$-field. We emphasize the approximations inherent in the higher-dimensional bosonization approach and clarify its relation with diagrammatic perturbation theory.