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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'

机译：部分系统特征方法的扩展   微分算子 - 一个带外部的Weyl方程的应用   “超级哈密顿路径积分法”

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摘要

By taking the Weyl equation with external electro-magnetic potentials as thesimplest representative for a system of PDOs, we give a new method of treatingnon-commutativity of coefficients matrices. More precisely, we construct aFourier Integral Operator with``matrix-like phase and amplitude'' which gives aparametrix for that Weyl equation. To do this, we first reduce the usual matrixvalued Weyl equation on the Euclidian space to the one on the superspace,called the super Weyl equation. Using analysis on superspace, we may associatea function, called the super Hamiltonian function, corresponding to that superWeyl equation. Starting from this super Hamiltonian function, we define phaseand amplitude functions which are solutions of the Hamilton-Jacobi equation andthe continuity equation on the superspace, respectively. Then, we define aFourier integral operator with these phase and amplitude functions which givesa 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, weget the desired evolutional operator of the super Weyl equation. Bringing backthis result to the matrix formulation, we have the final result. Therefore, weget a quantum mechanics with spin from a classical mechanics on the superspacewhich answers partly the problem of Feynman.
机译：通过采用具有外部电磁势的Weyl方程作为PDO系统的最简单代表，我们给出了一种处理系数矩阵的非对易性的新方法。更精确地讲，我们构造了一个傅立叶积分算子，该算子具有``类似于矩阵的相位和幅度''，从而给出了该Weyl方程的参数。为此，我们首先将欧几里得空间上通常的矩阵值Weyl方程简化为超空间上的一个矩阵，即超级Weyl方程。通过对超空间的分析，我们可以关联一个函数，称为超哈密顿函数，对应于该超维方程。从这个超级哈密顿函数开始，我们定义相位和振幅函数，分别是超空间上的Hamilton-Jacobi方程和连续性方程的解。然后，我们利用这些相位和幅度函数定义一个傅里叶积分算子，该算子为该超级Weyl方程的初值问题提供了很好的参数。在对时间切片采取Lie-Trotter-Kato极限之后，我们得到了所需的演化算子超级Weyl方程。将这个结果带回到矩阵公式中，我们得到了最终结果。因此，我们从超空间上的经典力学中得到了具有自旋的量子力学，部分地解决了费曼的问题。

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    Inoue, Atsushi;

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  • 年度 2002
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  • 原文格式 PDF
  • 正文语种 {"code":"en","name":"English","id":9}
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