A classical treatment of scalar electrodynamics : Gauge theory in the Port-Hamiltonian framework

The charged, massive Klein-Gordon field is a scalar field with a global U(1)-gauge symmetry. It is coupled to the Maxwellian electromagnetic field to get a system that has a local U(1)-gauge symmetry. We study the classical field theory of the Klein- Gordon field coupled to the electromagnetic field on a curved spacetime. The spacetime is foliated into spacelike hypersurfaces, each corresponding to a single “time”, such that a Hamiltonian theory may be formulated. Hamilton’s equations of motion and the gauge symmetries are worked out for the Klein-Gordon-Maxwell system under the assumption that the total energy of the system is constant. Port-Hamiltonian theory is used to reformulate the dynamics in a way that allows for the treatment of energy exchange with the environment. In the port-Hamiltonian formulation, however, the gauge symmetries are obscured.