System of partial differential equations and non-commutative analysis

Feynman的路径积分公式可以看作是薛定谔方程基本解的积分表示（对于某个薛定谔方程，Fujiwara给出了傅里叶积分算子型参数的数学严密构造）。在为薛定谔方程推导路径积分公式时，费南问自己是否也可以对自旋方程进行类似的推导，例如狄拉克方程。独立于Martin的试验，Berezin尝试用Grassmann变量平等对待光子和电子（这对应于Feynman用四元数作为基本场用路径积分法处理Dirac方程的建议）。10年前，我不是在Banach-Grassmann代数上构造初等分析，而是与Maeda一起在超空间R^<m/n>上不仅构造初等分析，而且构造了实分析的一部分，其中R是具有可数矩阵的Frechet-Grassmann代数。利用这一超空间，我们将R^3上值为C^4的自由狄拉克方程重新表述为R^<3/3>上值为C的自由狄拉克方程。通过这一重新表述，我们可以将余切超空间R^<6/6>上的哈密顿函数联系起来，由此我们可以构造出满足相应哈密顿-雅可比方程的相函数。我们也可以给出所谓的齐特比空的“经典对应”（就像R^6上的薛定谔粒子，R^<6/6>上的狄拉克粒子）。为了将这些扩展到具有时变外部电磁势的Weyl方程，我们不仅使用了R中继承的分级，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。对于野村，我给出了用超矩阵积分表示平均量的数学严格处理。虽然我认识Airy函数的出现，但我不知道随机矩阵理论和完全可积系统之间的真正关系。少

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