  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.
