Wave relations

TL;DR

This paper investigates the properties of boundary relations in the wave equation, revealing new insights into their Lagrangian nature in various spacetime settings, including Minkowski space and beyond.

Contribution

It demonstrates that boundary relations for the wave equation are Lagrangian in Minkowski space and extends this result to a broad class of metrics, with conjectures for higher dimensions.

Findings

Relations are Lagrangian in 2D Minkowski space.

Boundary relations remain meaningful without Hamiltonian evolution.

Counterexample in Misner space shows relation is not always Lagrangian.

Abstract

The wave equation (free boson) problem is studied from the viewpoint of the relations on the symplectic manifolds associated to the boundary induced by solutions. Unexpectedly there is still something to say on this simple, well-studied problem. In particular, boundaries which do not allow for a meaningful Hamiltonian evolution are not problematic from the viewpoint of relations. In the two-dimensional Minkowski case, these relations are shown to be Lagrangian. This result is then extended to a wide class of metrics and is conjectured to be true also in higher dimensions for nice enough metrics. A counterexample where the relation is not Lagrangian is provided by the Misner space.