The method of the Riemann-Hilbert problem for singular phenomenas in mathematical physics

利用Riemann-Hilbert问题的方法研究了奇异现象。讨论了以下三个问题：（1）讨论了量子场论中的发散问题。Riemann-Hilbert问题的方法给出了局部发散的标准形。因此，我们可以用标准散度除散度，我们可以得到与散度相关联的向量丛。这种反常可以用丛的特征类来描述，而这些特征类可以通过Fuchs关系直接得到。(2)夸克禁闭可以用Riemann-Hilbert问题来讨论。在给出阿贝尔化方法之后，我们可以得到奇异场和奇异规范变换。然后，我们可以制定的黎曼-希尔伯特问题的奇异性，我们可以得到的非平凡的异常，它给出了胶子的质量。通过对迈斯纳效应的讨论，我们可以得到夸克禁闭。(3)关于非微扰场论的黎曼-希尔伯特问题。预计非可拓方法将是真实的场论所必需的。在这里，我们为此目的准备基本材料。本文用代数方法处理分形几何，讨论奇异性。即考虑Cuntz代数的表示，讨论无限维Clifford代数及其周期性定理。

Singular phenomenas are investigated by use of the method of the Riemann-Hilbert problems. The following three topics are treated :(1) The problems of divergence in quantum field theory are discussed. The method of the Riemann-Hilbert problems supply the canonical forms of the divergences in local. Hence we can divide the divergence by the canonical one and we can get the vector bundle associated to the divergences. The anomaly can be described in terms of the characteristic classes of the bundle, which can be imeadiatlly obtained through the Fuchs relations.(2) The quarkconfinement can be discussed by use of the Riemann-Hilbert problems. After the formulation of the A belianization method, we can obtain the singular fieilds and the singular gauge trasformations. Then we can formulate the Riemann-Hilbert problems for the singularities and we can get the non-trivial anomaly which gives the mass of the gluons. Following the discussions in the Meissner effects, we can gent the quarkconfinement.(3) Toward the Riemann-Hilbert problem for non pertubative field theory. It is expected that non-pertabative method will be necessary for the real field theory. Here we prepare fundamental materials for this purpose. We treat the fractal geometry and discuss the sigularities by use of the algebraic method. Namely we consider the representations of the Cuntz algebras and discuss the infinite dimensional (Clifford algebras and the periodicity theorems in them.