Algebraic Structures in the Coupling of Gravity to Gauge Theories

Abstract

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 oneloop 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.

Log expansions from combinatorial Dyson–Schwinger equations

“…Indeed, the insertion places of each K-loop primitive graph are fully dressed by infinite sums 26 and the insertion operators ensure that there is no double counting. Note that Eqs.…”

“…(3.15) coincide with this general form, but only includes the K = 1 terms. 26 Let us show this quickly: Consider a K-loop primitive graph Γ. The argument of the corresponding insertion operator B Γ + is X r X K Q , where by Eqs.…”

“…First of all, it is inappropriate to grade the Hopf algebra H of Feynman graphs with respect to the loop number in this setting. Instead, H is multi-graded as [11,16,26,27]…”

The BRST Double Complex for the Coupling of Gravity to Gauge Theories

“…In particular, we have reworked the conventions such that the quadratic gauge fixing and ghost Lagrange densities are rescaled by the inverses of the respective gauge fixing parameters, ζ and ξ: This induces that unphysical propagators are rescaled by these parameters directly, which introduces an additional grading on the algebra of Feynman diagrams, cf. [16,18,19]. Furthermore, we show that all non-constant functionals on the superalgebra of particle fields that are essentially closed with respect to the diffeomorphism BRST operator P are scalar tensor densities of weight w = 1, cf.…”

Lecture Notes on Chern-Simons Perturbation Theory

“…For this theory, no satisfactory renormalization procedure has yet been found 15 . See also the discussion in [Pri18].…”