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

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、可解格点模型和二维共形场论在整个空间中的有趣和有前途的推广和扩展，这些理论已经得到了相当好的理解。首先，对于一般问题：给定一个在整个空间中精确可解的理论，“通过施加适当的边界条件，它在半空间中是否可解？“我们对一般的仿射户田场论给出了肯定的回答。下一个问题是：“半空间上的所有经典可积场论都是量子可积的吗？“我们回答说，只有非常有限的一部分是允许的稳定。将稳定性判据应用于具有“纯虚耦合常数”的仿射户田场论中的“孤子”。连同厄米性，我们得出结论，“量子修正孤子质量”是不合理的。讨论了半空间非线性sigma模型的可积性，结果表明，与仿射户田模型相比，在半空间自由边界条件下，无限多的非局域电荷不守恒.系统而全面地研究了仿射户田场论运动方程的约化问题。我们研究了相对简单的有限自由度物理系统，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测不准态的量子代数表示。

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.