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，离散可积系统的研究取得了很大进展，特别是高阶四边形格上的二维偏差分方程所描述的系统（格KP系统）。量子理论仍然落后，即使一些早期的工作在20世纪90年代建立了一些代数结构背后的“不可积量子映射”（Nijhoff/卡佩尔）和“量子场论的时空晶格”（Faddeev/Volkov），提供了一个结构的量子不变量和量子酉算子。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。正则方法的一个主要问题是建立相应的Hilbert空间，而路径积分方法的一个重要问题是奇异性的作用。博士项目将努力在量子水平上进一步发展拉格朗日多形式理论（Lobb/Nijhoff，2009）的同时回答这些问题。该项目将集中在一些关键的例子，如量子晶格KdV系统及其约化，以推动理论向前发展，然后分支到可积量子系统的其他例子。