Canonical analysis of field theories in the presence of boundaries: Maxwell + Pontryagin

Abstract

We study the canonical Hamiltonian analysis of gauge theories in the presence of boundaries. While the implementation of Dirac’s programme in the presence of boundaries, as put forward by Regge and Teitelboim, is not new, there are some instances in which this formalism is incomplete. Here we propose an extension to the Dirac formalism—together with the Regge–Teitelboim strategy—that includes generic cases of field theories. We see that there are two possible scenarios, one where there is no contribution from the boundary to the symplectic structure and the other case in which there is one, depending on the dynamical details of the starting action principle. As a concrete system that exemplifies both cases, we consider a theory that can be seen both as defined on a four dimensional spacetime region with boundaries—the bulk theory—or as a theory defined both on the bulk and the boundary of the region—the mixed theory—the bulk theory is given by the four-dimensional Maxwell + U(1) Pontryagin action while the mixed one is defined by the four-dimensional Maxwell + three-dimensional U(1) Chern–Simons action on the boundary. Finally, we show how these two descriptions of the same system are connected through a canonical transformation that provides a third description. The focus here is in defining a consistent formulation of all three descriptions, for which we rely on the geometric formulation of constrained systems, together with the extension of the Dirac–Regge–Teitelboim formalism put forward in the manuscript.