Canonical formalism on a null surface: The scalar and the electromagnetic fields

The Hamiltonian for the scalar and electromagnetic fields are set up on an outgoing null cone plus that portion of ${\mathrm{scrI}}^{+}$ which extends back to space-like infinity. The latter portion is just the energy radiated so that the Hamiltonian is the total energy, a constant of the motion. Because the formalism is set on a characteristic surface, the momenta must satisfy certain constraints in addition to the gauge constraints. These null-surface constraints form a second-class system in the nomenclature of Dirac. Therefore, they are eliminated from the theory by the construction of Dirac brackets. With the Dirac brackets, the Hamiltonian gives the once-integrated field equations for the dynamical field variables. The usual commutation relations for the field strengths restricted to the domain of integration for H are i times the Dirac brackets.