Hamiltonian description of the parametrized scalar field in bounded spatial regions

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.

Hamiltonian dynamics of the parametrized electromagnetic field

“…, where b n X is the future directed g-unit normal vector field over the X map (see [15] for details). We denote the Lie derivative along the vector field ( ) a a e V X a…”

“…This, of course, can be explicitly checked by computing the Hamiltonian H according to the standard definition (see, for instance, [15]). In this situation the complete determination of the dynamics of the system just amounts to solving the equation…”

“…We will use the ideas developed in [15] to study the parametrized scalar field in bounded regions. As we will see, parametrized electromagnetism is, to a certain extent, simpler than those scalar models and is, in fact, quite helpful to understand them.…”

Constrained field theories on spherically symmetric spacetimes with horizons

“…Boundaries lead to the inclusion of surface terms which are essential in the formula-tion of the action in gravity [30][31][32] and other field theories [33][34][35][36][37][38]. The generalization to cases where the boundaries are null [16,39] as well as non-orthogonal [40], have also been considered in the literature.…”

Functional evolution of scalar fields in bounded one-dimensional regions

“…In fact, some of the difficulties of dealing with the higher dimensional cases are already present in the Robin case (which is quite interesting in its own right [6]), hence, their satisfactory resolution suggests that a complete characterization of the embeddings capable of supporting unitary evolution for n-dimensional tori is possible.…”