A new development of the study for the system of PDE

Feynman在他引入路径积分之后，提出了一个问题，即对于具有自旋的量子系统，类似的推导是否可能。此外，他提出，尽管基本场的不可交换性带来了另一个困难，但四元数的使用是有帮助的。与前田一起，井上利用具有无限数量的Grassmann生成元的Frechet-Grassmann代数引入了一个超空间，并在该空间上发展了包括隐函数定理在内的初等分析。在研究了超空间的真实分析之后，井上给出了解决费曼问题的线索。也就是说，以具有含时外电磁场的Weyl方程为例，通过改进Feynman过程，构造了它的演化算符。更准确地说，他首先用超空间上的超函数来确定自旋场，然后他将出现在韦尔方程中的泡利矩阵表示为微分算子…更多地作用于超函数，通过这个过程，他可以将Weyl方程识别为超空间上的非对易标量方程，并可以将非对易标量符号函数联系起来，从而可以量子化与Feynman之后的符号相对应的经典力学。在用Jacobi方法构造了对应于该符号的Hamilton-Jacobi方程的解之后，他定义了由该解给出的相位和由连续性方程的解给出的幅度的傅里叶积分算子。这个算符给出了超空间上Weyl方程的一个很好的参数线。通过Fujiwara的时间切片方法，他得到了具有含时电磁场的Weyl方程的超型演化算子。通过对自旋和超函数的识别，得到了Weyl方程在一般矩阵值意义下所需的演化算子，该方法具有普适性，适用于其他方程。不仅如此，井上还发现，在凝聚态物质场理论中，对随机系统进行超分析的大量应用已经存在。这给了我们另一个需要澄清的对象。较少

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