Journal of Mathematical Chemistry

, Volume 50, Issue 3, pp 552–576 | Cite as

Nonlocal, noncommutative diagrammatics and the linked cluster theorems

  • Christian Brouder
  • Frédéric PatrasEmail author
Original Paper


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 (introduced in the present article), the diagrams, associated to noncommuting product of operators inherit from the noncommutativity of the products extra graphical properties. The purpose of the present article 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. Among others, our theorems encompass the usual versions of the theorem (although very different in nature, from Goldstone diagrams in solid state physics to Feynman diagrams in QFT or probabilistic Wick theorems).


Linked cluster theorem Feynman diagram Combinatorial Hopf algebra Cumulant Truncated moment function 


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Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut de Minéralogie et de Physique des Milieux Condensés, CNRS UMR 7590Université Pierre et Marie Curie-Paris 6Paris Cedex 05France
  2. 2.Laboratoire J.-A. Dieudonné, CNRS UMR 6621Université de NiceNice Cedex 02France

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