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，它们结合在一起形成了量子场论，即相对论量子力学。量子场论以其怀特曼函数为特征。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的缓和分布。超函数被认为是具有局部性质的广义函数中最一般的一种。因此，与超函数相对应的量子场（更准确地说，其Wightman函数为傅立叶超函数的量子场）被认为是最普遍的，作者自1976年以来一直在研究。由于傅里叶超函数的测试函数空间是由解析函数组成的，因此它不包含紧支撑函数。这使得研究超功能量子场论（HFQFT）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紧支持的函数，所以我们不能在HFQFT的框架中定义限定在有界区域内的域算子。然而，我们可以证明我们可以定义与有界区域内的可观测值对应的算子和算子代数（1998）。(2)证明了具有非平凡算子代数的HFQFT模型的存在性，该算子代数对应于有界区域内的可观测集合。

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.