On the Enumerative Structures in Quantum Field Theory

TL;DR

This thesis explores enumerative combinatorics in quantum field theory, focusing on chord diagrams in Dyson-Schwinger equations and the effects of field transformations on amplitudes, with new applications in QED and Yukawa theories.

Contribution

It introduces new applications of rooted connected chord diagrams in quenched QED and Yukawa theories, and provides a novel proof regarding the vanishing of certain tree-level amplitudes.

Findings

Enumeration of rooted connected chord diagrams is extended.

Chord diagrams are used to express solutions to Dyson-Schwinger equations.

A new proof shows vanishing of tree-level amplitudes under field diffeomorphisms.

Abstract

This thesis addresses a number of enumerative problems that arise in the context of quantum field theory and in the process of renormalization. In particular, the enumeration of rooted connected chord diagrams is further studied and new applications in quenched QED and Yukawa theories are introduced. Chord diagrams appear in quantum field theory in the context of Dyson-Schwinger equations, where, according to recent results, they are used to express the solutions. In another direction, we study the action of point field diffeomorphisms on a free theory. We give a new proof of a vanishing phenomenon for tree-level amplitudes of the transformed theories.