Lagrangian Multiform Structures, Discrete Systems and Quantisation

Lagrangian multiforms are an important recent development in the study of integrable variational problems. In this thesis, we develop two simple examples of the discrete Lagrangian one-form and two-form structures. These linear models still display all the features of the discrete Lagrangian multiform; in particular, the property of Lagrangian closure. That is, the sum of Lagrangians around a closed loop or surface, on solutions, is zero. We study the behaviour of these Lagrangian multiform structures under path integral quantisation and uncover a quantum analogue to the Lagrangian closure property. For the one-form, the quantum mechanical propagator in multiple times is found to be independent of the time-path, depending only on the endpoints. Similarly, for the two-form we define a propagator over a surface in discrete space-time and show that this is independent of the surface geometry, depending only on the boundary. It is not yet clear how to extend these quantised Lagrangian multiforms to non-linear or continuous time models, but by examining two such examples, the generalised McMillan maps and the Degasperis-Ruijsenaars model, we are able to make some steps towards that goal. For the generalised McMillan maps we find a novel formulation of the r-matrix for the dual Lax pair as a normally ordered fraction in elementary shift matrices, which offers a new perspective on the structure. The dual Lax pair may ultimately lead to commuting flows and a one-form structure. We establish the relation between the Degasperis-Ruijsenaars model and the integrable Ruijsenaars-Schneider model, leading to a Lax pair and two particle Lagrangian, as well as finding the quantum mechanical propagator. The link between these results is still needed. A quantum theory of Lagrangian multiforms offers a new paradigm for path integral quantisation of integrable systems; this thesis offers some first steps towards this theory