Quantum discrete integrable systems

近几十年来，离散可积系统的研究取得了很大的进展，特别是高阶四边形格子上的二维偏差分方程组所描述的系统(格子KP系统)。尽管20世纪90年代的一些早期工作建立了一些代数结构，落后于不可移植的量子映射(Nijhoff/Capel)和时空格子上的量子场论(Faddeev/Volkov)，提供了量子不变量和量子酉算符的构造，但量子理论仍然落后于此。最近与菲尔德(2005)和King(2018)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理论的相关Hilbert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。博士项目将努力在量子水平上进一步发展拉格朗日多态理论(Lobb/Nijhoff，2009)的同时，给出这些问题的答案。这个项目将集中在一些关键的例子上，例如量子晶格KdV系统及其约化，以推动理论的发展，然后扩展到其他可积量子系统的例子。

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.