费米变量与超对称性量子力学(Ⅰ)

FERMIONIC VARIABLES AND SUPERSYMMETRY IN QUANTUM MECHANICS

本文描述了具有一个Bose坐标和一个Fermion坐标的单粒子量子力学系统。在文(Ⅰ)中,用了Fermion坐标的距阵表示法,也用了Fermion坐标的反对易C数实现;用了Schrodinger波函数和算符表示,也用了Feynman的路径积分的方法来研究该系统。特别,导出了两个严格等价的(0+1)维的路径积分表达式。在文(Ⅱ)中,讨论了该系统显示超寻称性的条件。作为应用,详尽地分析了被Witten所讨论过的一些模型中超对称的破缺是如何发生的。通过迭代法,找到了Schrodinger方程的近似基态波函数,得到了测定非微扰的超对称破缺的基态能与超势的关系。

This paper describes the quantum mechanics of a single particle system with one bosonic and one fermionie coordinate, under the matrix representation of the fermions and under the anticommuting c—number representation, in the Schrodinger wave function and in Feynman path integral formalism. In particular, two exact and equivalent expressions for fermionic path integrals in(0+1) dimensions are derived. Under suitable circumstances the system considered can exhibit supersymmetry. As an application it is analyzed in detail how the spontaneous breaking of supersymmerry, occurring in certain modeles discussed by Witten, can take place. By developing an iterative scheme to find the approximate ground state solutions to the Schrodinger equation, the relation between the ground state energy measuring the breaking of supersymmetry with non—perturbation and supersymmetric potentials is ot obtained.