Hamiltonian description of the parametrized scalar field in bounded spatial regions

TL;DR

This paper develops a Hamiltonian framework for a parametrized scalar field in bounded regions with various boundary conditions, extending previous work to include spatial boundaries and their geometric implications.

Contribution

It generalizes Hamiltonian formulations to bounded regions with Dirichlet, Neumann, and Robin boundary conditions, incorporating the geometry of embeddings and boundary effects.

Findings

Formulation of the symplectic structure for bounded regions

Analysis of Hamiltonian vector fields with boundary conditions

Extension of parametrized field systems to include spatial boundaries

Abstract

We study the Hamiltonian formulation for a parametrized scalar field in a regular bounded spatial region subject to Dirichlet, Neumann and Robin boundary conditions. We generalize the work carried out by a number of authors on parametrized field systems to the interesting case where spatial boundaries are present. The configuration space of our models contains both smooth scalar fields defined on the spatial manifold and spacelike embeddings from the spatial manifold to a target spacetime endowed with a fixed Lorentzian background metric. We pay particular attention to the geometry of the infinite dimensional manifold of embeddings and the description of the relevant geometric objects: the symplectic form on the primary constraint submanifold and the Hamiltonian vector fields defined on it.