Quantum Variational Principle and Discrete Integrable Systems

该项目涉及可积系统的变分描述的新公式，称为拉格朗日多形式理论，由于Lobb & Nijhoff（2009）。这种新的方法已被成功地证明是相关的描述表现出所谓的多维一致性属性的可积系统，并已被证明持有大量的可积系统在连续的情况下的偏微分方程以及离散的情况下的系统的时空格。主要目的是在量子水平上考虑这个理论，King & Nijhoff（2017）已经在这个方向上迈出了第一步。虽然这些结果主要涉及线性情况下的二次拉格朗日，它的见解给量子变分原理的费曼传播预计持有的非线性情况下的可积拉格朗日以及。有几种模型有资格作为量子变分原理的测试场，其中一类是Calogero-Moser型模型，申请人一直在他的MmathPhys项目中从传统量子理论的角度研究该模型。因此，这些模型具有所有良好的签名作为实验室来扩展多形式理论的思想：它们具有很好理解的常规量子理论，具有已知的一类特殊函数作为哈密顿量的本征函数，而经典的多形式结构由Yoo-Kong，Lobb & Nijhoff（2011）建立，并且它们允许在离散时间和连续时间上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形式结构，从而探讨更具挑战性的问题，例如关于费曼路径积分测度的问题。如果成功，这些结果将被扩展到其他量子模型，如可积量子映射（Nijhoff，卡佩尔和Papageorgiou，1992）所产生的有限维约化的可积晶格系统。

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.