Equivalence between the Arqu\`es-Walsh sequence formula and the number of connected Feynman diagrams for every perturbation order in the fermionic many-body problem

TL;DR

This paper establishes a precise mathematical link between the enumeration of connected Feynman diagrams in fermionic many-body problems and the Arquès-Walsh sequence, revealing a topological connection via rooted map theory.

Contribution

It derives an exact counting formula for connected Feynman diagrams that matches the Arquès-Walsh sequence, bridging quantum field theory and combinatorial topology.

Findings

The counting formula coincides with the Arquès-Walsh sequence.

Supports the topological connection between Feynman diagrams and rooted maps.

Uses a classificatory summing-terms approach linked to discrete mathematics.

Abstract

From the perturbative expansion of the exact Green function, an exact counting formula is derived to determine the number of different types of connected Feynman diagrams. This formula coincides with the Arqu\`es-Walsh sequence formula in the rooted map theory, supporting the topological connection between Feynman diagrams and rooted maps. A classificatory summing-terms approach is used, in connection to discrete mathematical theory.