On a construction of the fundamental solution for the free Weyl equation by Hamiltonian path-integral (Q1917595)

Let \(\psi(t, q): \mathbb{R}\times \mathbb{R}^3\to \mathbb{C}^2\) satisfy \[ i\hbar{\partial\over \partial t} \psi(t, q)= H\psi(t, q),\;H= - ic\hbar \sigma_j {\partial\over \partial q_j},\qquad psi(0, q)= \psi(q).\tag{1} \] Here the Pauli matrices \(\{\sigma_j\}\) are for example represented by \[ \sigma_1= \begin{pmatrix} 0 & 1\\ 1 & 0\end{pmatrix},\;\sigma_2= \begin{pmatrix} 0 & - i\\ i & 0\end{pmatrix},\;\sigma_3= \begin{pmatrix} 1 & 0\\ 0 & -1\end{pmatrix}. \] We claim that there exists the classical mechanics corresponding to the Weyl equation and that a fundamental solution of (1) is constructed as a Fourier integral operator using phase and amplitude functions defined by that classical mechanics. Therefore, the Weyl equation is obtained by quantizing that classical mechanics after Feynman's procedure. Because that Hamiltonian is ``of first order both in even and odd variables'', we should modify Feynman's argument from Lagrangian to Hamiltonian formulated ``path integral''. Main Theorem [Hamilton path-integral representation]: \[ \psi(t, q)= b\Biggl((2\pi \hbar)^{- 3/2} \hbar \int_{{\mathfrak R}^{3|2}} d\underline\xi d\underline\pi \mu(t, \overline x, \overline\theta, \underline\xi, \underline\pi)\times e^{i\hbar^{- 1} {\mathcal S}(t, \overline x, \overline\theta, \underline\xi, \underline\pi)} {\mathcal F}(\# \psi) (\underline\xi,\underline \pi)\Biggr)\Biggl|_{\overline x_B= q}. \] Here, \({\mathcal S}(t, \overline x, \overline\theta, \underline\xi, \underline\pi)\) and \(\mu (t,\overline x,\overline\theta,\underline\xi,\underline\pi)\) are solutions of Hamilton-Jacobi and continuity equations, respectively.