Quantum discrete integrable systems

量子离散可积系统

• 批准号: 2704447
• 金额: --
• 依托单位: University of Leeds
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2704447
• 关键词: Quantum; discrete; integrable; systems

In recent decades a lot of progress has been made in discrete integrable systems, in particular systems described by 2D partial difference equations on quadrilateral lattices of higher order and in 3 dimensions as well (lattice KP systems). The quantum theory is still lagging behind even though some early work in the 1990s established some of the algebraic stuctures behind 'inegrable quantum mappings' (Nijhoff/Capel) and 'quantum field theory on the space-time lattice' (Faddeev/Volkov), providing a construction of quantum invariants and quantum unitary operators. More recent work with Field (2005) and King (2018) formed first step via a Lagrangian approach and a Feynman path integral. A main issue in the canonical approach is to establish the relevant Hilbert spaces of the theory, while in the path integral approach the role of singularities is an impotant question for nonlinear quantum integrable models. The PhD project will endeavour to give answers to these questions alongside the further development of the Lagrangian multiform theory (Lobb/Nijhoff, 2009) at the quantum level. The project will focus on some key examples, such as the quantum lattice KdV system and its reductions, in order to push the theory forward, and then branch out to other examples of integrable quantum systems.

近几十年来，离散可积系统取得了很多进展，特别是由高阶四边形晶格和三维四边形晶格上的二维偏微分方程描述的系统（晶格 KP 系统）。尽管 20 世纪 90 年代的一些早期工作建立了“不可忽略的量子映射”（Nijhoff/Capel）和“时空晶格的量子场论”（Faddeev/Volkov）背后的一些代数结构，但量子理论仍然落后，提供量子不变量和量子酉算子的构造。 Field (2005) 和 King (2018) 最近的工作通过拉格朗日方法和费曼路径积分形成了第一步。规范方法的一个主要问题是建立理论的相关希尔伯特空间，而在路径积分方法中，奇点的作用是非线性量子可积模型的一个重要问题。该博士项目将努力回答这些问题，同时在量子层面进一步发展拉格朗日多形式理论（Lobb/Nijhoff，2009）。该项目将重点关注一些关键示例，例如量子晶格 KdV 系统及其简化，以推动理论发展，然后扩展到可积量子系统的其他示例。