Fundamental Problems in Quantum Field Theory

量子场理论是众所周知的，可以在微观世界的描述中扮演基本和独立的角色。即使它的历史现在已经超过了60年了，而且有很多研究，因为应用程序是由它制造出来的，但仍然有许多未解决的问题来讨论它的基本结构。在未来，这是一个非常值得期待的问题，因为微观世界中的新自然法则将被发现在未来，并将与量子场理论背后的潜在结构有密切的关系。由于这个原因，在目前的研究方案中，我们尝试通过从这个领域的各个方面研究量子场理论中的基本问题。以下是我们在1989年和1990年获得的结果的简短摘要。第一个概念量化问题是（I）-3）， 7）-10）， （II）-2）， 3）， 5），6） and （III）-1）， 3） listed in our "Research Report"（作为"RR"）。在这些文章中，研究是由操作员在温格的通信关系、随机量化方法、路径整合方法和异常通信关系中形成的。第二部分是对相对粒子和领域的研究（2+1）-维度。It is stated in （I）-2）、4） and （II）-4） in "RR"、in which beside massless particles with peculiar spins the complete results are obtained for covariant desc\and also for field quantization of （2+1）-particles including those of fractional spin。The third is a non-perturbative approach based on the Schwinger-Dyson equation in gauge theories， which is discusted in （I）-5） and 6） in "RR"。第四部分与对称性问题和（2+1）的单位表示相结合-庞加莱组是大规模调查的。更多关于伽利略集团的共同描述是在一些细节中研究的。These are stated in （I）-4）、（II）-4） and （I）-1）、（III）-1） in "RR"、respectively。

Quantum field theory is known to play a fundamental and indispensable role in a description of the microscopic world. Although its history is now beyond 60 years and a lot of studies as well as applications has been made of it, there yet remain many unsolved problems concerning its basic structure. Furthermore it is strongly expected that new natural laws holding in the microscopic world will be discovered in future and will have close relations to a profound structure hidden behind the quantum field theory. For this reason, in the present research program we have tried to investigate fundamental problems in quantum field theory by making approaches from various sides of this field.The following is a brief summary of our results obtained during academic years 1989 and 1990.The first concerns quantization problems which are (I)-3), 7)-10), (II)-2), 3), 5), 6) and (III)-1), 3) listed in our "Research Report"(referred to as "RR"). In these articles studies are made of self-adjointness of the operators in Winger's commutation relations, stochastic quantization approaches, path-integration approaches, and anomalous commutation relations. The second is a study of relativistic particles and fields in (2+1)-dimension. It is stated in (I)-2), 4) and (II)-4) in "RR", in which beside massless particles with peculiar spins the complete results are obtained for covariant description and also for field quantization of (2+1)-particles including those of fractional spin. The third is a nonーperturbative approach based on the Schwinger-Dyson equation in gauge theories, which is discussed in (I)-5) and 6) in "RR". The fourth is concerned with symmetry problems, and unitary representations of the (2+1)-Poincare group are largely investigated. Moreover covariant descriptions of the Galilei group are studied in some detail. These are stated in (I)-4), (II)-4) and (I)-1), (III)-1) in "RR", respectively.