特征线法在偏微分算子组中的推广--“超Hamilton路径积分法”在外场Weyl方程中的应用
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"
摘要：
本文将外电磁势的Weyl方程作为PDO系统的最简表示，给出了一种处理系数矩阵非交换性的新方法。更准确地说，我们构造了一个具有“矩阵状相位和振幅”的傅立叶积分算子，它给出了Weyl方程的参数。为了做到这一点，我们首先减少通常的矩阵值外尔方程的欧氏空间上的一个超空间，称为超外尔方程。通过对超空间的分析，我们可以将一个称为超哈密顿函数的函数与该超外尔方程相关联。从这个超哈密顿函数出发，我们定义了相函数和振幅函数，它们分别是超空间上的哈密顿-雅可比方程和连续性方程的解。然后，我们定义了一个傅里叶积分算子与这些相位和振幅的功能，这给了一个很好的参数，该超Weyl方程的初值问题。通过对时间切片取Lie-Trotter-Kato极限，得到了超Weyl方程的演化算子.把这个结果带回矩阵公式，我们得到最终结果。因此，我们从超空间上的经典力学得到了一个具有自旋的量子力学，它部分地回答了费曼问题。
英文摘要：
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.
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