
Journal of Lie Theory 27 (2017), No. 3, 771800 Copyright Heldermann Verlag 2017 The TwoLoop Ladder Diagram and Representations of U(2,2) Matvei Libine Dept. of Mathematics, Indiana University, 831 East 3rd Street, Bloomington, IN 47405, U.S.A. mlibine@indiana.edu 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 representationtheoretic meaning of a particular kind of Feynman diagrams called the twoloop 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 a subsequent paper entitled The conformal fourpoint integrals, magic identities and representations of U(2,2) to provide a mathematical interpretation of all conformal fourpoint integrals  including those described by the nloop ladder diagrams  in the context of representations U(2,2) and quaternionic analysis. Moreover, this representationquaternionic 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 selfcontained. Keywords: Feynman diagrams, conformal fourpoint integrals, representations of U(2,2), conformal geometry, quaternionic analysis. MSC: 22E70, 81T18, 30G35, 53A30 [ Fulltextpdf (540 KB)] for subscribers only. 