Introduction

From the previous chapters it is clear that the ultimate challenge in a pQFT is the determination of all its primitive elements, that is, the calculation of all Feynman diagrams without subdivergences. They provide the letters of an alphabet on which the combinatorial Hopf algebra of Chapter 3 acts, which then delivers any renormalized Feynman graph.

However, naively writing down all Feynman diagrams without subdivergences overcounts the number of letters in which QFT is formulated. We have seen that the topology of Feynman diagrams relates to knot theory, and Feynman diagrams with a similar topology evaluate to similar numbers. But then, there are relations between such numbers, and relations between Feynman graphs. To know all of them would enable us to determine the alphabet in which the language of QFT is effectively expressed.

In this chapter we investigate if counterterms of Feynman diagrams realize a four-term relation (4TR) amongst them, in the hope that this might explain the relations between them, and their relation to topology. This is a very bold idea. The four-term relation relates naturally to Chern–Simons theory [Bar-Natan 1995], and we cannot expect that four-term relations hold in general. Nevertheless, it is of interest to see how far this idea can be pushed, and though a final answer to the problem can not yet be presented, the results allow interesting insights into the differences between a mere topological field theory and the UV-divergent sector of a full-fledged QFT.