# Algebraic Structures in the Coupling of Gravity to Gauge Theories

## TL;DR

This paper explores the algebraic structures underlying the perturbative quantization of gravity coupled with gauge theories, focusing on renormalization techniques and Hopf algebraic methods to address divergences.

## Contribution

It introduces a Hopf algebraic framework for renormalization in quantum gravity coupled to gauge theories, extending Furry's Theorem and analyzing two-loop propagators.

## Key findings

- Generalized Furry's Theorem for quantum gravity and electrodynamics
- Explicit two-loop propagator calculations
- Relations for multiplicative renormalization

## Abstract

This article is an extension of the author's second master thesis [1]. It aims to introduce to the theory of perturbatively quantized General Relativity coupled to Spinor Electrodynamics, provide the results thereof and set the notation to serve as a starting point for further research in this direction. It includes the differential geometric and Hopf algebraic background, as well as the corresponding Lagrange density and some renormalization theory. Then, a particular problem in the renormalization of Quantum General Relativity coupled to Quantum Electrodynamics is addressed and solved by a generalization of Furry's Theorem. Next, the restricted combinatorial Green's functions for all two-loop propagators and all one-loop divergent subgraphs thereof are presented. Finally, relations between these one-loop restricted combinatorial Green's functions necessary for multiplicative renormalization are discussed. Keywords: Quantum Field Theory; Quantum Gravity; Quantum General Relativity; Quantum Electrodynamics; Perturbative Quantization; Hopf Algebraic Renormalization