Quantum Variational Principle and Discrete Integrable Systems

由于Lobb&Nijhoff(2009)，该项目涉及一种被称为拉格朗日多形理论的可积系统的变分描述的新形式。这种新的方法已经被成功地证明是对具有所谓多维一致性的可积系统的适当描述，并且已经被证明对大量可积系统都是成立的，无论是在连续的偏微分方程组的情况下还是在时空格子上的离散系统的情况下。主要目的是在量子水平上考虑这一理论，King&Nijhoff(2017)已经朝着这个方向迈出了第一步。虽然这些结果主要涉及二次拉格朗日量的线性情况，但它对费曼传播子形式的量子变分原理的见解预计也适用于可积拉格朗日量的非线性情况。有几个模型有资格作为量子变分原理的试验场，其中一类是Calogero-Moser类型的模型，申请人一直在从传统量子理论的角度在他的MmathPhys项目中进行研究。因此，这些模型具有作为扩展多形理论思想的实验室的所有良好特征：它们拥有众所周知的传统量子理论，具有已知的特殊函数类作为哈密顿量的本征函数，而经典多形结构是由Yoo-Kong，Lobb&Nijhoff(2011)建立的，它们允许在离散和连续时间的经典水平上的精确解。该项目将寻求建立费曼传播子的量子多态结构，从而探索更具挑战性的问题，例如关于费曼路径积分测量的问题。如果成功，这些结果将被推广到其他量子模型，例如作为可积晶格系统的有限维约化而产生的可积量子映射(Nijhoff，Capel&Papageorgiou，1992)。

The project deals with a novel formulation of the variational description of integrable systems known by the name of Lagrangian multiform theory, due to Lobb & Nijhoff (2009). This new approach has been successfully shown to be the pertinent description of integrable systems exhibiting the so-called multidimensional consistency property, and has been demonstrated to hold for a large number of integrable systems both in the continuous case of PDEs as well as the discrete case of systems on the space-time lattice. The main aim is to consider this theory on the quantum level, and first steps in this direction have already been undertaken by King & Nijhoff (2017). Whereas those results pertain mostly to the linear case of quadratic Lagrangians, the insights it gave into the quantum variational principle in terms of Feynman propagators are expected to hold for the nonlinear case of integrable Lagrangians as well. There are several models that qualify as a testing ground for the quantum variational principle, one class of which are the Calogero-Moser type models which the applicant has been investigating in his MmathPhys project from a conventional quantum theory point of view. Thus, these models have all the good signatures as a laboratory to expand the ideas of the multiform theory: they possess a well understood conventional quantum theory, with known class of special functions as eigenfunctions of the Hamiltonian, while the classical multiform structure was established by Yoo-Kong, Lobb & Nijhoff (2011), and they allow exact solutions on the classical level both in discrete as well as continuous time. The project will seek to establish the quantum multiform structure for the Feynman propagators, and thus probe into the more challenging issues, such as the ones regarding the Feynman path integral measure. If successful these results will be expanded to other quantum models such as the integrable quantum mappings (Nijhoff, Capel & Papageorgiou, 1992) arising as finite-dimensional reductions of integrable lattice systems.