Bold Feynman Diagrams and the Luttinger–Ward Formalism Via Gibbs Measures: Non-perturbative Analysis

Many-body perturbation theory (MBPT) is widely used in quantum physics, chemistry, and materials science. At the heart of MBPT is the Feynman diagrammatic expansion, which is, simply speaking, an elegant way of organizing the combinatorially growing number of terms of a certain Taylor expansion. In particular, the construction of the ‘bold Feynman diagrammatic expansion’ involves the partial resummation to infinite order of possibly divergent series of diagrams. This procedure demands investigation from both the combinatorial (perturbative) and the analytical (non-perturbative) viewpoints. In this paper, we approach the analytical investigation of the bold diagrammatic expansion in the simplified setting of Gibbs measures (known as the Euclidean lattice field theory in the physics literature). Using non-perturbative methods, we rigorously construct the Luttinger–Ward formalism for the first time, and we prove that the bold diagrammatic series can be obtained directly via an asymptotic expansion of the Luttinger–Ward functional, circumventing the partial resummation technique. Moreover we prove that the Dyson equation can be derived as the Euler–Lagrange equation associated with a variational problem involving the Luttinger–Ward functional. We also establish a number of key facts about the Luttinger–Ward functional, such as its transformation rule, its form in the setting of the impurity problem, and its continuous extension to the boundary of the domain of physical Green’s functions.