arXiv 0901.31272009-01-20math-phhep-thmath.MP

量子场论中自同构群的谱理论与粒子结构

Spectral Theory of Automorphism Groups and Particle Structures in Quantum Field Theory

Wojciech Dybalski

备注：: PhD thesis, University of Goettingen, 2008 (advisor D. Buchholz) 156 pages

摘要：

本文给出了在给定的量子场论中粒子和亚粒子存在的一些物理判据。它是基于一个完善的谱理论的自同构群描述的能量动量转移的本地观测。特别地，构造了一种新的分解的代数的局部可观的谱子空间。除了算子的谱理论中所熟悉的纯点子空间和绝对连续子空间之外，还出现了一个新的“点连续”子空间。它属于分解的奇异-连续部分，但在一个大类的模型中是有限维的;它的维数携带了关于理论的红外结构的信息。它表明，这个点连续的子空间是平凡的，在所有的理论符合在这项工作中提出的正则性条件。此外，如果理论允许应力-能量张量，那么这个条件就意味着粒子的存在。通过证明自同构群的遍历定理，证明了观测量代数分解为纯点子空间和连续子空间的唯一性。证明是基于物理动机的相空间条件，有一些有趣的后果有关的真空结构，如收敛到一个独特的真空下大类时翻译的物理状态。

英文摘要：

This Thesis presents some physically motivated criteria for the existence of particles and infra-particles in a given quantum field theory. It is based on a refined spectral theory of automorphism groups describing the energy-momentum transfer of local observables. In particular, a novel decomposition of the algebra of local observables into spectral subspaces is constructed. Apart from the counterparts of the pure-point and absolutely continuous subspaces, familiar from the spectral theory of operators, there appears a new 'point-continuous' subspace. It belongs to the singular-continuous part of the decomposition, but is finite-dimensional in a large class of models; its dimension carries information about the infrared structure of a theory. It is shown that this point-continuous subspace is trivial in all theories complying with a regularity condition proposed in this work. Moreover, this condition entails the existence of particles if the theory admits a stress-energy tensor. The uniqueness of the decomposition of the algebra of observables into the pure-point and continuous subspace is established by proving an ergodic theorem for automorphism groups. The proof is based on physically motivated phase space conditions which have a number of interesting consequences pertaining to the vacuum structure such as the convergence of physical states to a unique vacuum under large timelike translations.

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