Fundamental Problems in Quantum Field Theory

Quantum field theory is known to play a fundamental and indispensable role in a description of the microscopic world.Although its history is now beyond60 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#china_person0#and1990.The first concerns quantization problems which are(I)-3)，7)-10)，(II)-2)，3)，5)，6)and(III)-1)，3)listed in our which are(I)-3)。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-perturbative approach based on the Schwinger-Dyson@equation in gauge theories，which is discussed in(I)-5)and6)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。

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.