The Study of Hyperfunction Quantum Field Theory

相对论和量子力学是20世纪物理学的重大发现，它被结合到量子场论中，这就是相对论量子力学。量子场论的特点是它的怀特曼函数。Wightman函数是广义函数，通常我们考虑所谓的标准量子场论，其中Wightman函数是Schwartz的回火分布。在具有局部性质的广义函数中，超函数被认为是最一般的函数。因此，与超函数(更准确地说，怀特曼函数为傅立叶超函数的量子场)相对应的量子场被认为是自1976年以来最普遍的，并被作者研究过。由于傅里叶超函数的测试函数空间由解析函数组成，所以它不包含紧支撑函数。这就使得超函数量子场论的研究变得困难。这一次，我们得到了以下结果：(1)由于在傅里叶超函数的测试函数空间中不存在具有紧支撑的函数，所以我们不能在超函数量子场论的框架内定义限制在有界区域内的场算符。(2)证明了HFQFT模型的非平凡算子代数对应于有界区域上的可观测集合。

The relativity and quantum mechanics, which are big discoveries in physics in the 20-th century, are combined into the quantum field theory, which is relativistic quantum mechanics. The quantum field theory is characterized by its Wightman functions. The Wightman functions are generalized functions and usually we consider the so called the standard quantum field theory, where the Wightman functions are Schwartz's tempered distributions. The hyperfunction is considered to be the most general one among generalized functions that has the local properties. Therefore the quantum fields which correspond to hyperfunctions (more precisely, the quantum field whose Wightman functions are Fourier hyperfunctions) are considered to be the most general and studied since 1976 by the author. Since the test-function space of Fourier hyperfunctions consists of analytic functions, it does not contain functions of compact supports. This fact makes the study of hyperfunction quantum field theory (HFQFT) difficult.This time, we have the following results: (1) Since there are no functions with compact supports in the test-function space of Fourier hyperfunctions, we cannot define the field operator restricted in the bounded region in the framework of HFQFT. Nevertheless we could prove that we can define the operators and operator algebras which correspond to the observables in the bounded region (1998).(2) We proved the existence of the model of HFQFT which has the nontrivial operator algebras which correspond to the set of observables in the bounded region.