A new development of the study for the system of PDE

点击翻译按钮获取中文摘要

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