QUANTUM-GRAVITY HAMILTONIAN FOR MANIFOLDS WITH BOUNDARY

In canonical quantum gravity when space is a compact manifold with bou ndary there is a Hamiltonian given by an integral over the boundary. H ere we compute the action of this ''boundary Hamiltonian'' on observab les corresponding to open Wilson lines in the new variables formulatio n of quantum gravity. In cases where the boundary conditions fix the m etric on the boundary (e.g., in the asymptotically Minkowskian case) o ne can obtain a finite result, given by a ''shift operator'' generatin g translations of the Wilson line in the direction of its tangent vect or. A similar shift operator serves as the Hamiltonian constraint in t he work of Morales-Tecotl and Rovelli on quantum gravity coupled to We yl 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 hol e.