“Sum over surfaces” form of loop quantum gravity

We derive a spacetime formulation of quantum general relativity from (Hamiltonian) loop quantum gravity. In particular, we study the quantum propagator that evolves the three-geometry in proper time. We show that the perturbation expansion of this operator is finite and computable order by order. By giving a graphical representation in the manner of Feynman of this expansion, we find that the theory can be expressed as a sum over topologically inequivalent (branched, colored) two-dimensional (2D) surfaces in 4D. The contribution of one surface to the sum is given by the product of one factor per branching point of the surface. Therefore branching points play the role of elementary vertices of the theory. Their value is determined by the matrix elements of the Hamiltonian constraint, which are known. The formulation we obtain can be viewed as a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Ooguri-Crane-Yetter 4D topological field theory, with a few key differences that illuminate the relation between quantum gravity and topological quantum field theory. Finally, we suggest that certain new terms should be added to the Hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4D diffeomorphism invariance.