N-fold supersymmetry and its extension to multi-particle system and field theorie

N重超对称性及其在多粒子系统和场论中的推广

基本信息

批准号：
14540257
负责人：
AOYAMA Hideaki
金额：
$ 2.24万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2002
资助国家：
日本
起止时间：
2002 至 2005
项目状态：
已结题

项目摘要

In this research, extention of the N-fold supersymmetry in supersymmetric quantum mechanics was investigated. N-fold supersymmetry was found from the asymptotic behaviour of the perturbative coefficients in 1-dimensional quantum mechanics by the present investigator. Since this symmetry shares many features of the ordinary supersymmetry, its extension to the multi-particle system and field theories is apparently important. This project was focused on this point. Many trials were made in several directions during the project period. The main result of this project, however, is the finding of the 2-fold supersymmetry in 3-dimensional quantum mechanics, through a long-series of calculation for finding the solution to the supersymmetry algebra.In the said construction, some ansatz were made for the form of the supercharge and the Hamiltonian. The 2-fold supersymmetry algebra induces a set of non-linear partial differential equations for the functions in the ansatz. There are about 15 functions to be obtained and thus solution is rather difficult to come by. We, however, have managed to show that the solution exists and identified several of them, thereby enabling the construction of the 3-dimensional model for the first time.The importance of the solvable model in all categories is evident. In this project, the shape-invatiant models are solvable. Therefore, we have been searching for them in our construction described above. We are close to concluding that such a model does not exist, although this is somewhat preliminary.

在本研究中，研究了超对称量子力学中 N 重超对称性的扩展。本研究人员从一维量子力学中微扰系数的渐近行为中发现了 N 重超对称性。由于这种对称性具有普通超对称性的许多特征，因此将其扩展到多粒子系统和场论显然很重要。本项目正是围绕这一点展开的。项目期间在多个方向进行了多次尝试。然而，这个项目的主要成果是通过一系列寻找超对称代数解的计算，发现了3维量子力学中的2重超对称性。在上述构造中，提出了一些设想对于增压和哈密顿量的形式。 2 重超对称代数导出了 ansatz 中函数的一组非线性偏微分方程。需要获取的函数大约有15个，求解起来相当困难。然而，我们已经成功地证明了解决方案的存在并确定了其中的几个，从而首次能够构建 3 维模型。可解模型在所有类别中的重要性是显而易见的。在这个项目中，形状不变模型是可解的。因此，我们在上述构建中一直在寻找它们。我们即将得出这样的模型不存在的结论，尽管这在某种程度上是初步的。