System of partial differential equations and non-commutative analysis

偏微分方程组和非交换分析

基本信息

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

项目摘要

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的路径综合公式可以被视为Schrodinger方程基本解决方案的整体表示（对于某些Schrodinger方程，Fujiwara给出了傅立叶积分运算符类型参数的数学严格结构）。在推导施罗宾格方程的路径综合公式时，费南（Feynamn 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 countryly ma … More ny Grassmann generators.Using this superspace, we reformulate the free Dirac equation on R^3在c^4中的价值与超空间r^<3/3>具有价值C.通过这种改革，我们可以将汉密尔顿功能关联到cotangengent Superspace r^<6/6>上，我们可以从中构建满足相应的汉密尔顿 - 雅各布方程的相位函数。我们还可以给所谓的Zitterbewegung（例如R^6上的Schrodinger粒子，R^<6/6>上的狄拉克粒子）。为了将它们扩展到Weyl方程，并具有耗时的外部电磁潜力，我们不使用R中继承的分级，而是与罗杰斯（Banach）Topolgy相比，这是非常弱的，而Rogers等人在Rogers等人引入的Banach Topolgy中，Efetov在另一方面开始将Grassmann变量应用于随机Matrix理论中的问题。对于Nomura，我给予数学上的严格处理，用于使用超级Matix积分来表示平均数量。尽管我认识到通风函数的外观，但我不知道随机矩阵理论与完全可以整合的系统之间的真实关系。较少的