Algebraic Structures in the Coupling of Gravity to Gauge Theories

This article is an extension of the authors second master thesis [1]. It aims to introduce 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 propagator 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. One of those relations suggests that it is unphysical to consider the coupling to Spinor Electrodynamics alone and instead consider the coupling to the whole Electroweak Sector