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2000 Fiscal Year Final Research Report Summary

System of partial differential equations and non-commutative analysis

Research Project

Project/Area Number 10640201
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Global analysis
Research InstitutionTokyo 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)

Co-Investigator(Kenkyū-buntansha) 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)
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)
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)
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)
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)
Project Period (FY) 1998 – 2000
KeywordsSuperanalysis / Feynman's problem / Random Matrix Theory / matrix integral / disordered system / Grassmann variables / spin / Painleve equation
Research Abstract

Feynman's path-integral 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 path-integral 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 path-integral method).
Instead of constructing elementary analysis on Banach-Grassmann 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 Frechet-Grassmann algebra with a countably ma … More ny 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 Hamilton-Jacobi equation. We may give also the "classical correspondence" to the so-called Zitterbewegung (like a Schrodinger particle on R^6, a Dirac particle on R^<6/6>). To extend these to the Weyl equation with time-depending external electro-magnetic 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

  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] A.INOUE: "On a construction of the fundamental solution for the free Weyl equation by Hamiltonian path-integral method-an exactly solvable case with "odd variable coefficients""Tohoku J.Math.. 50. 91-118 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.INOUE: "On a construction of the fundamental solution for the free Dirac equation by Hamiltonian path-integtal method-the classical counterpart of Zitterbewegung"Japanese Journal Math.. 24. 297-334 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.INOUE: "The first term of spectral asymptotic formula related to the continuum mechanics- generalizations of Weyl's Theorem""Navier-Stokes equations : theory and numerical methods" (ed.L.Salvi) Longman.. 184-192 (1998)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A INOUE: "On a "Hamiltonian path-integral" derivation of the Schrodinger equation"Osaka J.Math.. 36. 111-150 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.INOUE: "A partial solution for Feynman's problem : a new derivation of the Weyl equation"Electro.J.Diff.Eqns.. Conf.4. 121-145 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.INOUE & Y.NOMURA: "Some refinements of Wigner's semi-circle law for Gaussian Random Matrices using superanalysis"Asymptotic Analysis. 23. 329-375 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] A.Inoue: "On a construction of the fundamental solution for the free Wey equation by Hamiltonian path-integral method-an exactly solvable case with "odd variable coefficients""Tohoku J.Math. 50. 91-118 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Inoue: "On a construction of the fundamental solution for the free Dirac equation by Hamiltonian path-integral method-the classical counterpart of Zitterbewegung"Japanese J.Math.. 24. 297-334 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Inoue: "The first term of spectral asymptotic formula related to the continuum mechanics-generalizations of Weyl's Theorem""Navier-Stokes equations : theory and numerical methods (ed. L.Salvi), Longman". 184-192 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Inoue: "On a "Hamiltonian path-integral" derivation of the Schrodinger equation"Osaka J.Math.. 36. 111-150 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Inoue: "A partial solution for Feynman's problem : a new derivation of the Weyl equation"Mathematical Physics and Quantum Field Theory, Electron. J.Diff. Eqns.. Conf.04. 121-145 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] A.Inoue and Y.Nomura: "Some refinements of Wigner's semicircle law for Gaussian Random Matrices using superanalysis"Asymptotic Analysis. 23. 329-375 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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