Abstract
A modern notion of integrability is that of multidimensional consistency (MDC), which classically implies the coexistence of (commuting) dynamical flows in several independent variables for one and the same dependent variable. This property holds for both continuous dynamical systems as well as for discrete ones defined in discrete space-time. Possibly the simplest example in the discrete case is that of a linear quadrilateral lattice equation, which can be viewed as a linearised version of the well-known lattice potential Korteweg-de Vries (KdV) equation. In spite of the linearity, the MDC property is non-trivial in terms of the parameters of the system. The Lagrangian aspects of such equations, and their nonlinear analogues, has led to the notion of Lagrangian multiform structures, where the Lagrangians are no longer scalar functions (or volume forms) but genuine p-forms in a multidimensional space of independent variables. The variational principle involves variations not only with respect to the field variables, but also with respect to the geometry in the space of independent variables. In this paper we consider a quantum analogue of this new variational principle by means of quantum propagators (or equivalently Feynman path integrals). In the case of quadratic Lagrangians these can be evaluated in terms of Gaussian integrals. We study also periodic reductions of the lattice leading to discrete multi-time dynamical commuting mappings, the simplest example of which is the discrete harmonic oscillator, which surprisingly reveals a rich integrable structure behind it. On the basis of this study we propose a new quantum variational principle in terms of multiform path integrals.
Highlights
Discrete integrable systems [1] have started to play an increasingly important role in deepening the understanding of integrability as a mathematical notion, thereby forging new perspectives in both analysis, geometry and algebra
In his seminal paper of 1933, [38], Paul Dirac expressed his credo that the Lagrangian formulation of classical dynamics, in comparison to the Hamiltonian one, was more fundamental, and he posed the question of a Lagrangian approach to quantum mechanics
The related question of what would constitute a variational point of view in quantum mechanics was partly, but not fully, answered by those approaches
Summary
Discrete integrable systems [1] have started to play an increasingly important role in deepening the understanding of integrability as a mathematical notion, thereby forging new perspectives in both analysis (e.g. the discovery of difference analogues of the Painlevé equations), geometry (the development of discrete differential geometry, [2]) and algebra (e.g. the development of cluster algebras through the so-called Laurent phenomenon). Integrable systems are important because they can be treated by exact and rigorous methods, and because they appear to be universal: they have a rare tendency of emerging in a large variety of contexts and physical situations, such as in correlations functions in scaling limits, random matrices and in energy level statistics of even chaotic systems Their intricate underlying structures gave rise to new mathematical theories, such as quantum groups and cluster algebras, revealing novel types of combinatorics. It is this very feature that forms a powerful tool in the exact solvability of such equations through techniques such as the inverse scattering transform (a nonlinear analogue of the Fourier transform), Lax pairs and Bäcklund transformations This story about the variational description of integrable systems started with the paper [5], where the Lagrangian structure of a class of 2D quadrilateral lattice equations was studied, which are integrable in the sense of the MDC property.
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