Hamiltonian treatment of linear field theories in the presence of boundaries: a geometric approach

TL;DR

This paper provides a detailed geometric analysis of the constraint structure for scalar and electromagnetic fields with boundaries, advancing the understanding of gauge theories in bounded regions.

Contribution

It develops a geometric constraint algorithm tailored for field theories with boundaries, addressing functional analytic challenges and clarifying physical degrees of freedom.

Findings

Clarified the constraint structure in boundary field theories

Enhanced understanding of gauge symmetries with boundaries

Provided a rigorous geometric framework for boundary conditions

Abstract

The purpose of this paper is to study in detail the constraint structure of the Hamiltonian and symplectic-Lagrangian descriptions for the scalar and electromagnetic fields in the presence of spatial boundaries. We carefully discuss the implementation of the geometric constraint algorithm of Gotay, Nester and Hinds with special emphasis on the relevant functional analytic aspects of the problem. This is an important step towards the rigorous understanding of general field theories in the presence of boundaries, very especially when these fail to be regular. The geometric approach developed in the paper is also useful with regard to the interpretation of the physical degrees of freedom and the nature of the constraints when both gauge symmetries and boundaries are present.