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Solvable System with non-trivial Boundary Conditions : Quantum Group and Excahnge Algebra

具有非平凡边界条件的可解系统：量子群和交换代数

基本信息

批准号：
06640395
负责人：
SASAKI Ryu
金额：
$ 0.7万
依托单位：
KYOTO UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1994
资助国家：
日本
起止时间：
1994 至 1995
项目状态：
已结题

项目摘要

In this project, we chose "quantum field theory with non-trivial boundary condition" as an interesting and promising generalisation and extension of integrable quantum field theory, solvable lattice models and two dimensional conformal field theory in the whole space, which have been rather well understood. At first, for the general question : given an exactly solvable theory in the whole space, " is it solvable in a half space by imposing appropriate boundary conditions? " we gave a positive answer for general affine Toda field theories. The next question was : "are all the classically integrable field theories on a half space quantum integrable? " We answered that only a very limited part was allowed by the stability. The criterion for stability was applied to the "solitons" in the affine Toda field theory with "pure imaginary coupling constant". Together with the hermiticity we concluded that "quantum corrections to the soliton masses" was not justifiable. The integrability of the half-space non-linear sigma models was addressed and it was shown the infinite set of non-local charges was not conserved for the free boundary condition in half space, in sharp contrast to the affine Toda case. The reduction of the equation of motion of affine Toda field theory was investigated systematically and comprehensively. Many new reduction relations were found. We investigated relatively simple physical systems with finite degrees of freedom and discussed the effects of the quantum group (algebra), which was another big element of the current research. The representation of the minimum uncertainty states in quantum optics, like the coherent and the squeezed states were given in terms of various quantum algebras.

在这个项目中，我们选择了“具有非平凡边界条件的量子场论”作为可积量子场论、可解晶格模型和二维共形场论在整个空间中的一个有趣且有前途的推广和扩展，这些理论已经得到了很好的理解。首先，对于一般问题：给定一个在整个空间中完全可解的理论，“通过施加适当的边界条件，它在半空间中是否可解？”“我们对一般仿射户田场论给出了肯定的答案。下一个问题是：“在半空间量子上的所有经典可积场论都是可积的吗？”我们回答说，稳定性只允许一个非常有限的部分。将稳定性判据应用于具有“纯虚耦合常数”的仿射Toda场论中的“孤子”。与厄米性一起，我们得出结论，“对孤子质量的量子修正”是不合理的。研究了半空间非线性sigma模型的可积性，证明了半空间自由边界条件下无限非局部电荷集不守恒，与仿射Toda情况形成鲜明对比。系统、全面地研究了仿射Toda场论运动方程的约简问题。发现了许多新的约简关系。我们研究了具有有限自由度的相对简单的物理系统，并讨论了量子群（代数）的影响，这是当前研究的另一个重要元素。给出了量子光学中相干态和压缩态等最小不确定性态的各种量子代数表示。