Diffeomorphisms of Scalar Quantum Fields via Generating Functions

TL;DR

This paper introduces a new diagrammatic proof that certain scalar quantum field theories have vanishing interacting amplitudes, using generating functions and combinatorial methods for deeper insight.

Contribution

It provides a novel, diagram-based proof of amplitude vanishing and explores combinatorial identities related to Bell polynomials and the Legendre transform.

Findings

Interacting tree amplitudes vanish in the studied theories

New combinatorial proofs of Bell polynomial identities

Insights into the connection with the combinatorial Legendre transform

Abstract

We study the application of formal diffeomorphisms to scalar fields. We give a new proof that interacting tree amplitudes vanish in the resulting theories. Our proof is directly at the diagrammatic level, not appealing to the path integral, and proceeds via a generating function analysis so is more insightful than previous proofs. Along the way we give new combinatorial proofs of some Bell polynomial identities, and we comment on the connection with the combinatorial Legendre transform.