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

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.

Hamiltonian description of the parametrized scalar field in bounded spatial regions

“…This points out to the existence of additional requirements necessary to have consistent dynamics. This situation exactly mimics the one found in the Hamiltonian treatment of the scalar field with Dirichlet boundary conditions (see [21]) and is in perfect agreement with the known results for the scalar field in the smooth case [22]. From here on, the determination of the infinite chain of conditions necessary to have well defined dynamics for smooth field and embeddings, follows exactly the steps of the GNH algorithm as the main geometric issue involved is the tangency of the Hamiltonian vector fields to the submanifolds defined by the successive conditions.…”

“…where ϕ ∶ M → R is a real scalar field on M, B ∶ ∂ Σ M → R is a fixed smooth function and we are using the metric volumes on M and ∂ Σ M. This action is the obvious generalization of the one used in [21] to discuss the Hamiltonian formulation for the scalar field with Robin boundary conditions (it can be essentially found in [22, page 227]). Our results can be trivially extended to the case where a potential term V (ϕ) is included as in [3].…”

“…This condition is found in the Hamiltonian treatment of the scalar field with Dirichlet boundary conditions [21], where a special foliation with z ⊥ X = 1 and z ⊺ X = 0 is used.…”

“…Several comments are in order now. First, just mention that the non-homogeneous case is straightforward once the homogeneous one is understood (see [21]), so we will only consider the latter. A second comment refers to the way the variational principle works.…”

“…In the first case one has to restrict oneself to field configurations (and, hence, also variations) that vanish at the boundary ∂ Σ M, whereas in the second the boundary conditions themselves appear as a consequence of the requirement that the action is stationary under arbitrary variations. The Hamiltonian description of these models based on the use of the GNH method can be found in [21].…”