The Hamiltonian constraint in quantum gravity

Abstract

We develop further the Rovelli-Smolin loop variable formulation of canonical quantum gravity. The main result is the construction of a hamiltonian operator which is well defined on all of loop state space. The hamiltonian operator acts as a generator of displacements which depend on the curvature of the loop and also reroutes the parametrisation of the loops at their intersections. Both a regularisation and a renormalisation are used in the construction. The quantum constraint algebra is worked out in some detail. We find that the diffeomorphism-hamiltonian commutator does not close. This can only be acceptable provided the physical state space is “sufficiently large”. We find a sector of state space annihilated by the diffeomorphism and hamiltonian operators which is also the full sector of a topological quantum field theory of the type studied by Horowitz. The topological sector itself is not large enough to allow non-closure of the quantum constraint algebra. We therefore make some initial attempts to find out whether the “large”, formal Rovelli-Smolin sector of physical states, namely the set of loop wavefunctionals having compact support on diffeomorphism equivalence classes of smooth, non-intersecting loops, is annihilated by the hamiltonian operator. In the final part of the paper we discuss the Bengtsson and Witten formulations of D = 2+1 quantum gravity in terms of the loop variables. It is argued that the Bengtsson formulation can be used as a toy model to study the loop variable approach to D = 3+1 gravity. In the conclusion we discuss further directions for research.