指数理论与QFT的绝热极限
Index Theory and Adiabatic Limit in QFT
摘要：
本文以提案的形式讨论了量子场论的三种严格的数学方法之间的关系：1）Haag的局部代数公式，2）Wightman公式和3）基于微局部重整化方法的微扰公式。在这个项目中，我们研究1）和3）之间的关系，并利用1）和2）之间的已知关系。该提案的主要目标在于获得绝热极限存在的障碍（在现象学标准模型方法中的限制问题）。我们扩展的方法变形的Dütsch和Fredenyst（在Bordeman-Waldmann意义上）和应用Fedosov建设的正式指数-一个类似的指数变形辛流形，推广Atiyah-Singer指数。我们提出了实现该建议的一些初步步骤。
英文摘要：
The paper has the form of a proposal concerned with the relationship between the three mathematically rigorous approaches to quantum field theory: 1) local algebraic formulation of Haag, 2) Wightman formulation and 3) the perturbative formulation based on the microlocal renormalization method. In this project we investigate the relationship between 1) and 3) and utilize the known relationships between 1) and 2). The main goal of the proposal lies in obtaining obstructions for the existence of the adiabatic limit (confinement problem in the phenomenological standard model approach). We extend the method of deformation of Dütsch and Fredenhagen (in the Bordeman-Waldmann sense) and apply Fedosov construction of the formal index -- an analog of the index for deformed symplectic manifolds, generalizing the Atiyah-Singer index. We present some first steps in realization of the proposal.
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