Non-local Lagrangians: a variational approach to non-local conservation laws for wave mechanics

TL;DR

This paper develops a new variational framework using non-local Lagrangians to derive conservation laws in wave mechanics, extending previous methods to more general symmetries and higher dimensions.

Contribution

It introduces a generalized class of non-local Lagrangians that enable derivation of non-local conservation laws for complex wave systems, surpassing prior approaches.

Findings

Generalizes previous non-local conservation law derivations

Valid for a broader set of symmetry transformations

Applicable to higher-dimensional wave systems

Abstract

We introduce a class of non-local Lagrangians which allow for the variational derivation of non-local conser- vation laws in a self-consistent manner. The formalism developed here generalizes previous approaches, used in the context of $\mathcal{PT}$ symmetric quantum mechanics and optics in the paraxial approximation in a twofold way: firstly it is valid for a larger set of linear symmetry transforms and secondly it enables the derivation of additional non-local conservation laws for general higher dimensional wave mechanical systems.