一个模型量子场论的Hopf代数结构
Hopf Algebra Structure of a Model Quantum Field Theory
摘要：
最近关于微扰量子场论（PQFT）结构的出色工作揭示了分析（黎曼Zeta函数）、拓扑（纽结理论）、组合图论（费曼图）和代数（霍普夫结构）之间惊人的相互作用。像PQFT这样成熟的场论的复杂性所固有的困难意味着这些领域之间关系的本质美可能会有些模糊。我们的目的是在一个简单的零维场论的背景下，即一个不依赖于时空的非对易算符的量子理论中，展示这些结构中的一部分（尽管不是全部）。这些玻色子创建和湮灭运营商的组合特性，这是我们选择的例子，可以描述的图形，类似于费曼图的PQFT，我们显示具有一个霍普夫代数结构。我们的方法是基于玻色子气体的配分函数。在本论文集的后续注释中，我们简要描述了我们简单模型的Hopf代数与PQFT代数之间的关系。
英文摘要：
Recent elegant work on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis(Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra.
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