Hamiltonian surface charges using external sources
Cedric Troessaert
TL;DR
This work reframes the integrability problem of Hamiltonian surface charges by treating boundary conditions as external sources and introducing external symmetries that act on these sources. It develops a consistent framework with generalized differentiable functionals and a modified Poisson bracket to handle both finite-degree-of-freedom systems and field theories, including gauge theories, and demonstrates how surface charges can be defined through canonical generators in the presence of nontrivial boundary data. The approach yields an extended representation of the external-symmetry algebra, incorporates boundary terms, and clarifies the relationship to non-integrability by shifting focus to external transformations and corresponding charges. The scalar-field example and the gauge-theory analysis illustrate how boundary data and Lagrange multipliers influence the algebra of charges and its potential extensions, with potential implications for AdS/CFT-like setups and other gauge/gravity contexts.
Abstract
In this work, we interpret part of the boundary conditions as external sources in order to solve the integrability problem present in the computation of surface charges associated to gauge symmetries in the hamiltonian formalism. We start by describing the hamiltonian structure of external symmetries preserving the action up to a transformation of the external sources of the theory. We then extend these results to the computation of surface charges for field theories with non-trivial boundary conditions.