Nonlocal diagrammatics and the linked cluster theorems

F. Patras

Laboratoire J.-A. Dieudonné, 
Université de Nice-Sophia Antipolis,
Parc Valrose, Nice, F-06100.

Abstract

Recent developments in quantum chemistry, perturbative quantum field theory, statistical physics or stochastic differential equations require the introduction of new families of Feynman-type diagrams. These new families arise in various ways. In some generalizations of the classical diagrams, the notion of Feynman propagator is extended to generalized propagators connecting more than two vertices of the graphs. In some others (that we will introduce), the diagrams, associated to non commuting product of operators inherit from the non commutativity of the products extra graphical properties. 

The purpose of the present talk is to introduce a general way of dealing with such diagrams. We prove in particular a "universal" linked cluster theorem and introduce, in the process, a Feynman-type "diagrammatics" that allows to handle simultaneously nonlocal (Coulomb-type) interactions, the generalized diagrams arising from the study of interacting systems (such as the ones where the ground state is not the vacuum but e.g. a vacuum perturbed by a magnetic or electric field, by impurities...) or Wightman fields (that is, expectation values of products of interacting fields). Our diagrammatics seems to be the first attempt to encode in a unified algebraic framework such a wide variety of situations. In the process, we promote two ideas. First, Feynman-type diagrammatics belong mathematically to the theory of linear forms on combinatorial Hopf algebras. Second, linked cluster-type theorems rely ultimately on Möbius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly. Based on joint works with Ch. Brouder.