方程(γ_μ _μ+m)M=0的通解

THE GENERAL SOLUTION OF THE EQUATION(γ_μ _μ+m)M=0

F.Sauter(1930)引入了方程,(-i γ_μ ~μ-e/cγ_μA~μ+imc)M=0,其中M是4×4矩阵,以代替Dirac方程,(-i γ_μ ~μ-e/cγ_μA~μ+imc)Ψ=0,其中Ψ是4×1矩阵.F.Sauter(1930),A.Eddington(194)和M.F.Ross(1986)分别给出了这个方程当A~μ=0时的一个特解.本文则借助于广义逆矩阵的理论,求出了这个方程当A~μ=0时的通解.

F. Sauter (1930) introduced the equation(-i γ_μ ~μ-e/cγ_μA~μ+imc)M=0insted of the Dirac equation(-i γ_μ ~μ-e/cγ_μA~μ+imc)Ψ=0where M and Ψ are a 4×4 matrix and 4×1 a matrix respectively. F. Sautcr(l930), A. Eddington (1946) and M. F. Ross (1986) gave a particular solution of the equation when A~μ=0 respectively. In this paper, the general solution of the equation when A~μ=0 is found with the help of the theory of generalized inverse of a matrix.The general solution is M={ U-(ip_μγ_μ+m)^+(ip_μγ_μ+m)U } exp(ip_μx_μ) where U is an arbitrary 4×4 matrix and (ip_μγ_μ+ m)^+ is the Moore-Penrose inverse of (ip_μγ_μ+m).