System of partial differential equations and non-commutative analysis

偏微分方程组和非交换分析

• 批准号: 10640201
• 负责人: INOUE Atsushi
• 金额: $ 1.98万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640201/
• 关键词: Superanalysis; Feynman's problem; Random Matrix Theory; matrix integral; disordered system; Grassmann variables; spin; Painleve equation; 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

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

Feynman路径积分公式可以看作是薛定谔方程基本解的积分表示（对于某一薛定谔方程，Fujiwara给出了Fourier积分算子型参数的数学严格构造）。在推导薛定谔方程的路径积分公式时，费南问自己，是否也可以对有自旋的方程，如狄拉克方程，进行类似的推导。Berezin试图用Grassmann变量把光子和电子同等对待（这与Feynman提出的用四元数作为基本场用路径积分方法处理Dirac方程的建议相对应）。Grassmann代数，10年前，我开始与Maeda不仅构造初等分析，而且构造超空间R^<m/n>上的部分真实的分析，其中R是具有可数个m的Frechet-Grassmann代数。 ...更多信息 利用这个超空间，我们将R^3上的自由狄拉克方程（其值为C^4）转化为R^<3/3>上的自由狄拉克方程（其值为C）。通过这种转化，我们可以在余切超空间R^<6/6>上关联一个哈密顿函数，由此我们可以构造一个满足相应的哈密顿-雅可比方程的相函数。我们也可以给所谓的Zitterbewegung（如R ^6上的薛定谔粒子，R^<6/6>上的狄拉克粒子）以“经典对应”。为了推广到具有依赖于时间的外部电磁势的Weyl方程，我们不仅使用了R中继承的分次，而且还使用了与Rogers等人引入的Banach拓扑相比非常弱的Frechet拓扑。另一方面，Efetov开始将Grassmann变量应用于随机矩阵理论中的问题。与野村，我给出了一个数学上严格的处理表示平均量使用超矩阵积分。虽然我认识到Airy函数的出现，但我不知道随机矩阵理论与完全可积系统之间的真正关系。少

项目成果

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 Journal Math.. 24. 297-334 (1998)

A.INOUE：“通过哈密顿路径积分法构造自由狄拉克方程的基本解 - Zitterbewegung 的经典对应物”Japan Journal Math.. 24. 297-334 (1998)

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)

A.INOUE：“通过哈密顿路径积分法构造自由狄拉克方程的基本解 - Zitterbewegung 的经典对应物”日本数学杂志 24. 297-334 (1998)

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)

A.INOUE：“与连续介质力学相关的谱渐近公式的第一项 - 外尔定理的推广”“纳维-斯托克斯方程：理论和数值方法”（ed.L.Salvi）Longman.. 184-192 (1998)

A.INOUE & Y.NOMURA: "Some refinements of Wigner's semi-circle law for Gaussian Random Matrices using superanalysis"Asymptotic Analysis. 23. 329-375 (2000)

井上

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)

A.Inoue：“用哈密顿路径积分法构建自由狄拉克方程的基本解 - Zitterbewegung 的经典对应物”Japan J.Math.. 24. 297-334 (1998)