Application of the pseudodifferential operotous to the Feynmon path
Project/Area Number  10640176 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Basic analysis

Research Institution  Shinshu University 
Principal Investigator 
ICHINOSE Wataru Shishu Univ.Math.Sci., Prof., 理学部, 教授 (80144690)

CoInvestigator(Kenkyūbuntansha) 
千原 浩之 信州大学, 理学部, 助手 (70273068)
KUMANOGO Naoto Kogakuin Univ.Egeneering, Lecturer, 工学部, 講師 (40296778)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥2,900,000 (Direct Cost : ¥2,900,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥1,000,000 (Direct Cost : ¥1,000,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)

Keywords  pendodiffenntial / fouri intogral oporotoo / Schrodinsyr eguotion / path integral / Feynman / 擬微分作用素 / フーリエ積分作用素 / シュレディンガー方程式 / 経路積分 / Feynman(ファインマン) 
Research Abstract 
The aim of our project was to give the rigorous meaning to the Feynman path integrals used in physics. In detail, we study : (1) We give the rigorous meaning to the Feynman path integrals in phase space.(2) We give the rigorous meaning to the path integral representations of correlation functions and generating functions Z (J) in quantum mechanics and also quantum free field. Doc.N.Kumanogo was made responsible for a part of the study of the path integral by means of Fourier integral operators. We can say about our reserch result that our aim was completed except for the study of quntum free field as will be stated. In the paper [1] we proved convergence of the Feynman path integral in confuguration space independently of the choice of electromagnetic potentials, which implies the gauge invariance of the Feynman path integrals and generalizes the results proved in the author's paper in C.M.P (1997). From this result we can get the path integral representation of Z (J) in quantumm mechanics. In [2] we modified the definition of the path integral in phase space used in physics and showed its gauge invariance, convergence and the fact that this path integral in phase space is equal to the path integral in confuguration space familiar in physics. In [3] we gave the rigorous meaning to the path integral representation of correlation functions including not only position operators but also momentum operators. Here the path integrals in phase space in [2] was essentially used. In [4,5] Kumanogo gave a simple proof of convergence of the kernel of the path integral defined through piecewise classical paths by using the Fourier integral operators. In [6], usnig the same method, Kumanogo gave a simple proof of the path integral defined through broken line paths.

Report
(5results)
Research Output
(16results)