Beyond the single-particle picture of density functional theory, many of the most widely used computational methods in quantum many-body physics are based on the formalism of Green’s functions. These include the GW method and the dynamical mean-field theory (DMFT), among many others. Many-body perturbation theory (MBPT) defines a major category of approaches to computing Green’s functions based on expansions in terms of the so-called bold Feynman diagrams. This theory, as well as Green's function methods generally, can be understood formally in terms of a construction called the Luttinger-Ward (LW) functional. In the pedagogical context of Euclidean lattice field theories, we give an overview of these developments and explain how they can be justified rigorously. We explain how the methods transfer to the setting of interacting fermions and discuss what can be proved about the mathematical structure of certain Green's function methods in this setting. Finally, we highlight open problems in the fermionic setting.
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