``Sum over Surfaces'' form of Loop Quantum Gravity
Abstract
We study the ``proper time'' quantum propagator of the 3-geometry and show that its perturbation expansion is finite and computable order by order. We give a graphical representation a' la Feynman of such expansion and obtain a sum over topologically inequivalent (branched, colored) 2d surfaces in 4d. The contribution of a surface is the product of one factor per each branching point of the surface. Branching points are therefore the elementary vertices of the theory. Their value is determined by the matrix elements of the hamiltonian constraint; we evaluate the vertices explicitly, using Thiemann's operator. The formulation we obtain is a continuum version of Reisenberger's simplicial quantum gravity. Also, it has the same structure as the Crane-Yetter 4d TQFT, with a few key differences that illuminate the relation between quantum gravity and TQFT. Finally, we suggests that certain new terms should be added to the hamiltonian constraint in order to implement a ``crossing'' symmetry related to 4d diff invariance.Keywords
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