A new development of the study for the system of PDE

偏微分方程系统研究的新进展

基本信息

批准号：
08304010
负责人：
INOUE Atsushi
金额：
$ 3.97万
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 (A)
财政年份：
1996
资助国家：
日本
起止时间：
1996 至 1997
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-08304010/
关键词：
Path integral Weyl equation Superspace Hamilton-Jacobi equation First order system of PDEs Dirac equation Method of characteristics Random Matrix Theory 偏微分方程式系 Fourier積分作用素

项目摘要

Feynman, just after his introduction of path-integral, 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 non-commutativity of the basic field yields another difficulty.With Maeda, Inoue introduced a superspace using the Frechet-Grassmann 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 time-dependent external electro-magnetic 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 … More acting on superfunctions, By this procedure, he can identify the Weyl equation as the non-commutative but scalar equation on the superspace and he may associate the non-commutative but scalar symbol function.Therefore, he may quantize the classical mechanics corresponding to that symbol after Feynman. After constructing a solution of Hamilton-Jacobi 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 super-version of the Weyl equation with time-dependent electro-magnetic potential. Using the identification of spin and superfunction, we get the desired evolution operator for the Weyl equation in the ordinary matrix-valued 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

费曼（Feynman）在引入路径融合后，就提出了一个问题，即是否有旋转量子系统的类似推导。此外，他提出，尽管基本领域的非交换性产生了另一个困难，但季度数量的使用很有帮助。借助MAEDA，Inoue使用具有无限数量的Grassmann Generator的Frechet-Grassmann代数引入了一个超空间，并且在该空间上，他们开发了基本分析，包括隐式函数定理。在研究了超级空间的真实分析之后，Inoue提供了解决Feynman问题的线索。也就是说，以时间依赖性的外部电磁潜力以weyl方程为例，他通过修改Feynman的程序来构建其进化操作员。更准确地说，他首先在超空间上以超级功能来识别旋转场，然后他代表的保利遗迹出现在Weyl方程中为差分操作员……更多地在超级功能上，他可以通过此过程确定weyl方程为非交通方程式，但他可能会在超级方程式上进行符号，但可能会导致超级范围，但可能会导致sum Moysive sumigation sumigation sumigation sumbortiatiatiation sumbortiatiation sumpartiatiate scommintal nibor scommintal nibor scommintal in Mosalther，但要范围范围范围范围，但要量身定向，但要范围范围。 Feynman之后与该符号相对应的力学。在通过Jacobi方法构造了与该符号相对应的汉密尔顿 - 雅各比方程的解决方案后，他用该解和该解决方案给出的相位定义了傅立叶积分运算符，以及由连续性方程的解给出的放大器。该操作员在超空间上提供了Weyl方程的良好参数。通过Fujiwara的时间切片方法，他通过时间依赖性的电磁潜力获得了Weyl方程超级反之的所需进化算子。使用自旋和超级函数的识别，我们在普通矩阵值式的感觉中获得了Weyl方程的所需的进化算子。此过程具有将宇宙应用于其他方程式。不仅如此，Inoue发现了对随机系统的大量使用中的大量用法在凝聚态物质领域理论中已经存在。这为我们提供了另一个可以澄清的对象。较少的