An extension of the method of characteristic to a system of Partial Differential Operators-- an application to the Weyl equation with external field by "Super Hamiltonian path-integral method"

By taking the Weyl equation with external electro-magnetic potentials as the simplest representative for a system of PDOs, we give a new method of treating non-commutativity of coefficients matrices. More precisely, we construct a Fourier Integral Operator with``matrix-like phase and amplitude'' which gives a parametrix for that Weyl equation. To do this, we first reduce the usual matrix valued Weyl equation on the Euclidian space to the one on the superspace, called the super Weyl equation. Using analysis on superspace, we may associate a function, called the super Hamiltonian function, corresponding to that super Weyl equation. Starting from this super Hamiltonian function, we define phase and amplitude functions which are solutions of the Hamilton-Jacobi equation and the continuity equation on the superspace, respectively. Then, we define a Fourier integral operator with these phase and amplitude functions which gives a good parametrix for the initial value problem of that super Weyl equation. After taking the Lie-Trotter-Kato limit with respect to the time slicing, we get the desired evolutional operator of the super Weyl equation. Bringing back this result to the matrix formulation, we have the final result. Therefore, we get a quantum mechanics with spin from a classical mechanics on the superspace which answers partly the problem of Feynman.