Surface integrals associated with the canonical formalism on a null surface

Although the study of quantum gravity is moving in a number of new directions - particularly the string models, supergravity, and the path integral formulation - there is still considerable interest in understanding the canonical approach to quantum gravity. Recently, some major progress in the study of the canonical formalism on a space-like surface has been made. The work was motivated by the positive energy theorems initially formulated in terms of spinors. Using a modification of the spinor connection on a 3-surface it has been shown that the soldering form σ/sup a//sub AB/ and the associated connection coefficients A/sub a//sup AB/ define a set of phase space variables in terms of which the Hamiltonian of general relativity takes a very simple form and furthermore the commutation of the constraints close. The author's approach is based on the study of the gravitational radiation field in asymptotically flat space-times. The author constructed the Hamiltonian of general relativity on a null cone. It is not sufficient, however, to use just the outgoing cone. As he shows differentiability as well as identification with the Hamiltonian on a space-like surface requires that one consider the Hamiltonian also on a portion of null infinity which extends back to space-like infinity