A robust generalization of the Legendre transform for QFT

TL;DR

This paper provides a rigorous combinatorial proof that the Legendre transform of the quantum effective action generates connected graphs, revealing an underlying algebraic structure in quantum field theory.

Contribution

It introduces a purely combinatorial redefinition of the Legendre transform and proves its fundamental relation to connected graphs in QFT.

Findings

Legendre transform can be redefined combinatorially

Proof that the Legendre transform yields connected graphs

Suggests path integrals may have an underlying algebraic structure

Abstract

Although perturbative quantum field theory is highly successful, it possesses a number of well-known analytic problems, from ultraviolet and infrared divergencies to the divergence of the perturbative expansion itself. As a consequence, it has been difficult, for example, to prove with full rigor that the Legendre transform of the quantum effective action is the generating functional of connected graphs. Here, we give a rigorous proof of this central fact. To this end, we show that the Legendre transform can be re-defined purely combinatorially and that it ultimately reduces to a simple homological relation, the Euler characteristic for tree graphs. This result suggests that, similarly, also the quantum field theoretic path integral, being a Fourier transform, may be reducible to an underlying purely algebraic structure.