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首页> 外文期刊>Journal of Mathematical Chemistry >Nonlocal, noncommutative diagrammatics and the linked cluster theorems
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Nonlocal, noncommutative diagrammatics and the linked cluster theorems

机译：非局部，非可交换图解和链接簇定理

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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).
机译：量子化学，微扰量子场论，统计物理学或随机微分方程的最新发展要求引入Feynman型图的新族。这些新家庭以各种方式出现。在经典图的某些概括中，Feynman传播子的概念扩展为连接图的两个以上顶点的广义传播子。在其他一些文章（在本文中介绍）中，与运算符的非交换乘积相关的图继承于产品的非交换乘积的额外图形属性。本文的目的是介绍处理此类图表的一般方法。我们特别证明了一个“通用”链接簇定理，并在此过程中引入了费曼型“图解法”，该法允许同时处理非局部（库仑型）相互作用，这是研究相互作用系统（例如例如基态不是真空，而是例如受到磁场或电场，杂质等干扰的真空或怀特曼场（即相互作用场的乘积的期望值）的状态。我们的图表似乎是在如此广泛的情况下，在统一代数框架中进行编码的首次尝试。在此过程中，我们提倡两个想法。首先，Feynman型图表在数学上属于组合Hopf代数上的线性形式理论。第二，链接的簇类型定理最终依赖于分区格上的莫比乌斯反演。因此，应该对两种理论进行介绍​​和介绍。除其他外，我们的定理涵盖了该定理的通常形式（尽管本质上存在很大差异，从固态物理学中的戈德斯通图到QFT中的费曼图或概率Wick定理）。

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