Project/Area Number  10640201 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Global analysis

Research Institution  Tokyo Institute of Technology 
Principal Investigator 
INOUE Atsushi Graduate School of Science and Engineering Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (40011613)

CoInvestigator(Kenkyūbuntansha) 
ITO Hidekazu Graduate School of Science and Engineering Tokyo Institute of Technology Associate Professor, 大学院・理工学研究科, 助教授 (90159905)
MURATA Minoru Graduate School of Science and Engineering Tokyo Institute of Technology Professor, 大学院・理工学研究科, 教授 (50087079)
NOMURA Yuji Graduate School of Science and Engineering Tokyo Institute of Technology Assistant, 大学院・理工学研究科, 助手 (40282818)
ISOBE Takushi Graduate School of Science and Engineering Tokyo Institute of Technology Assistant, 大学院・理工学研究科, 助手 (10262255)
小澤 真 東京工業大学, 大学院・理工学研究科, 助教授 (00126020)
MORITA Takehiko Graduate School of Science and Engineering Tokyo Institute of Technology Associate Professor, 大学院・理工学研究科, 助教授 (00192782)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,100,000 (Direct Cost : ¥3,100,000)
Fiscal Year 2000 : ¥500,000 (Direct Cost : ¥500,000)
Fiscal Year 1999 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Superanalysis / Feynman's problem / Random Matrix Theory / matrix integral / disordered system / Grassmann variables / spin / Painleve equation / スーパー解析 / ファインマン問題 / ランダム行列理論 / 行列積分 / 無秩序系 / グラスマン変数 / スピン / パンルベ方程式 / 血秩序系 / パンルべ方程式 / 非可換代数 / 偏微分方程式系 / 超対称性 / ワイル方程式 / スーパー空間 / 量子化 / Dirac方程式 / Weyl方程式 
Research Abstract 
Feynman's pathintegral formula may be regarded as an integral representation of the fundamental solution of Schrodinger equation (for a certain Schrodinger equation, a mathematical rigorous construction of a parametrix of Fourier integral operator type is given by Fujiwara). At the time of deriving pathintegral formula for Schrodinger equation, Feynamn asked himself whether it is also possible to do analogously for the equation with spin, for example, Dirac equation. Independent of Martin's trial, Berezin tried to treat photon and electron on equal footing by using Grassmann variables (this corresponds to a proposal of Feynman using quaternion as the fundamental field to treat Dirac equation by pathintegral method). Instead of constructing elementary analysis on BanachGrassmann algebra, 10 years before, I begun with Maeda to construct not only elementary analysis but also a part of real analysis over the superspace R^<m/n>, where R is the FrechetGrassmann algebra with a countably
… More
many Grassmann generators. Using this superspace, we reformulate the free Dirac equation on R^3 with value in C^4 to that on superspace R^<3/3> with value C.By this reformulation, we may associate a Hamiltonian function on the cotangent superspace R^<6/6> from which we may construct a phase function satisfying corresponding HamiltonJacobi equation. We may give also the "classical correspondence" to the socalled Zitterbewegung (like a Schrodinger particle on R^6, a Dirac particle on R^<6/6>). To extend these to the Weyl equation with timedepending external electromagnetic potential, we use not ony the grading inherited in R but also the Frechet topology which is very weak compared with Banach topolgy introducd by Rogers etc. On the other hand, Efetov begun to apply the Grassmann variables to the problem in Random matrix theory. With Nomura, I give a mathematical rigorous treatment for representing the averaged quantity using super matix integrals. Though I recognize the appearance of Airy function, but I don't know the true relation between Random matrix theory and completely integrable system. Less
