Quantum Variational Principle and Discrete Integrable Systems

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.