Project/Area Number  08304010 
Research Category 
GrantinAid for Scientific Research (A)

Section  一般 
Research Field 
解析学

Research Institution  Tokyo Institute of Technology 
Principal Investigator 
INOUE Atsushi Tokyo Institute of Technology Prof.Faculty of Science,, 理学部, 教授 (40011613)

CoInvestigator(Kenkyūbuntansha) 
平良 和昭 広島大学, 理学部, 教授 (90016163)
MIYAKAWA Tetsuo Kobe University Prof.Faculty of Science,, 理学部, 教授 (10033929)
GIGA Yoshikazu Hokkaido University Prof.Guraduate School of Science,, 大学院・理学研究科, 教授 (70144110)
SUZUKI Takashi Osaka Unversity Prof.Guraduate School of Engineering Science,, 大学院・理学研究科, 教授 (40114516)
増田 久弥 東北大学, 理学系研究科, 教授 (10090523)
UKAI Seiji Tokyo Institute of Technology Prof.Guraduate School of Information Science and E, 大学院・情報理工学研究科, 教授 (30047170)
NISHIDA Takaaki Kyoto University Prof.Guraduate School of Science,, 大学院・理学研究科, 教授 (70026110)

Project Fiscal Year 
1996 – 1997

Project Status 
Completed(Fiscal Year 1997)

Budget Amount *help 
¥6,200,000 (Direct Cost : ¥6,200,000)
Fiscal Year 1997 : ¥3,900,000 (Direct Cost : ¥3,900,000)
Fiscal Year 1996 : ¥2,300,000 (Direct Cost : ¥2,300,000)

Keywords  Path integral / Weyl equation / Superspace / HamiltonJacobi equation / First order system of PDEs / Dirac equation / Method of characteristics / Random Matrix Theory / 経路積分 / Dirac方程式 / スーパー空間 / HamiltonJacobi方程式 / 1階偏微分方程式系 / ランダム行列理論 / 基本解 / 特性方程式の方法 / 偏微分方程式系 / Fourier積分作用素 
Research Abstract 
Feynman, just after his introduction of pathintegral, posed a problem whether the analogous derivation is possible for the quantum system with spin. Moreover, he proposed that the usage of quaternion numbers is helpful though the noncommutativity of the basic field yields another difficulty. With Maeda, Inoue introduced a superspace using the FrechetGrassmann algebra with infinite numbe of Grassmann generators and over that space they developped the elementary analysis including the implicit function theorem. After studying the real analysis over the superspace, Inoue gives a clue to solve Feynman's problem. That is, taking the Weyl equation with the timedependent external electromagnetic potential as an example, he constructs an evolution operator of it by modifying Feynman's procedure. More precisely speaking, he first identifies the spin field with a superfunction on the superspace and then he represents the Pauli matrices appeared in the Weyl equation as differential operators
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acting on superfunctions, By this procedure, he can identify the Weyl equation as the noncommutative but scalar equation on the superspace and he may associate the noncommutative but scalar symbol function. Therefore, he may quantize the classical mechanics corresponding to that symbol after Feynman. After constructing a solution of HamiltonJacobi equation corresponding to that symbol by Jacobi's method, he defines the Fourier Integral Operator with the phase given by that solution and the amplitude given by the solution of the continuity equation. This operator gives a good parametrix of the Weyl equation on the superspace. By Fujiwara's time slicing method, he gets the desired evolution operator of the superversion of the Weyl equation with timedependent electromagnetic potential. Using the identification of spin and superfunction, we get the desired evolution operator for the Weyl equation in the ordinary matrixvalued sense. This procedure has the universality to be applied to other equations. Not only this, Inoue finds the vast usage of superanalysis to random systems are already existing in condensed matter field theory. This gives us another object to be clarified. Less
