Quantum Gravity Hamiltonian For Manifolds With Boundary

In canonical quantum gravity when space is a compact manifold with boundary there is a Hamiltonian given by an integral over the boundary. Here we compute the action of this ''boundary Hamiltonian'' on observables corresponding to open Wilson lines in the new variables formulation of quantum gravity. In cases where the boundary conditions fix the metric on the boundary (e.g., in the asymptotically Minkowskian case) one can obtain a finite result, given by a ''shift operator'' generating translations of the Wilson line in the direction of its tangent vector. A similar shift operator serves as the Hamiltonian constraint in the work of Morales-Tecotl and Rovelli on quantum gravity coupled to Weyl spinors. This suggests the appearance of an induced field theory of Weyl spinors on the boundary, analogous to that considered in Carlip's work on the statistical mechanics of the (2+1)-dimensional black hole.