The Two-Loop Ladder Diagram and Representations of U(2,2)

Abstract: Feynman diagrams are a pictorial way of describing integrals predicting possible outcomes of interactions of subatomic particles in the context of quantum field physics. It is highly desirable to have an intrinsic mathematical interpretation of Feynman diagrams, and in this article we find the representation-theoretic meaning of a particular kind of Feynman diagrams called the two-loop ladder diagram. This is done in the context of representations of a Lie group U(2,2), its Lie algebra u(2,2) and quaternionic analysis. The results and techniques developed in this article are used in [L] to provide a mathematical interpretation of all conformal four-point integrals - including those described by the n-loop ladder diagrams - in the context of representations U(2,2) and quaternionic analysis. Moreover, this representation-quaternionic model produces a proof of "magic identities" in the Minkowski metric space. No prior knowledge of physics or Feynman diagrams is assumed from the reader. We provide a summary of all relevant results from quaternionic analysis to make the article self-contained.