Boundaries in the Instantaneous Formulation of Field Theories

Abstract

We study boundary conditions in GiMmsy's covariant and instantaneous formulations of classical field theories and show that the instantaneous state space in the presence of a constant Dirichlet boundary condition is a tangent bundle to the configuration space of fields satisfying said condition. We then study the instantaneous state space when only the velocity of the field is required to vanish at the boundary and show that this results in a sector structure, where each sector is a tangent bundle labeled by the configuration at the boundary. Taking the Legendre transform of this sectored state space yields a sectored phase space with leafwise canonical Poisson structures. We apply this to Yang-Mills theory with spatial boundary conditions and relate our results to flux superselection sectors. The sector-moving gauge transformations are not Hamiltonian because of the lack of a boundary momentum, prompting us to propose a novel definition of the asymptotic or boundary symmetry group as the quotient of the boundary-preserving Hamiltonian transformations by the trivial ones. The physical boundary symmetry group of electromagnetism is then shown to be a copy of the global gauge group even when all sectors are considered simultaneously. Conditions are discussed under which the same holds for non-Abelian Yang-Mills theory.