System of partial differential equations and non-commutative analysis

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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