Nonlocal, noncommutative diagrammatics and the linked cluster Theorems

TL;DR

This paper introduces a unified algebraic framework for nonlocal, noncommutative Feynman-type diagrams, extending classical diagrammatics to handle generalized propagators and interactions in quantum and statistical physics.

Contribution

It develops a universal linked cluster theorem and a diagrammatic approach that incorporates noncommutative products and nonlocal interactions within a Hopf algebraic structure.

Findings

Proves a universal linked cluster theorem for generalized diagrams

Establishes a Feynman-type diagrammatics for noncommutative operators

Unifies various diagrammatic approaches in quantum physics and statistical mechanics

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 (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 M\"obius inversion on the partition lattice. The two theories should therefore be introduced and presented accordingly