1997 Fiscal Year Final Research Report Summary
A new development of the study for the system of PDE
Grant-in-Aid for Scientific Research (A)
|Allocation Type||Single-year Grants |
|Research Institution||Tokyo Institute of Technology |
INOUE Atsushi Tokyo Institute of Technology Prof.Faculty of Science, -> 東京工業大学, 理学部, 教授 (40011613)
UKAI Seiji Tokyo Institute of Technology Prof.Guraduate School of Information Science and E, 大学院・情報理工学研究科, 教授 (30047170)
NISHIDA Takaaki Kyoto University Prof.Guraduate School of Science,, 大学院・理学研究科, 教授 (70026110)
GIGA Yoshikazu Hokkaido University Prof.Guraduate School of Science,, 大学院・理学研究科, 教授 (70144110)
SUZUKI Takashi Osaka Unversity Prof.Guraduate School of Engineering Science,, 大学院・理学研究科, 教授 (40114516)
MIYAKAWA Tetsuo Kobe University Prof.Faculty of Science,, 理学部, 教授 (10033929)
|Project Period (FY)
1996 – 1997
|Keywords||Path integral / Weyl equation / Superspace / Hamilton-Jacobi equation / First order system of PDEs / Dirac equation / Method of characteristics / Random Matrix Theory|
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
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
Research Products (14 results)