Quantum gravity on finite sets

Although finite sets do not have any nontrivial usual manifold structure, within the more general axioms of noncommuative geometry they do. A differential structure is defined by a graph on the finite set. Unlike finite lattice approximations, there are 'no truncation errors' but rather an exact finite geometry with a rich an self-consistent structure. One can then go on to define bundles, Riemannian curvature etc over the finite set. In this case funtional integration becomes ordinary integration and a sum over differentiable structures (which we have proposed before as required in quantum gravity) becomes a sum over graphs not unlike Feynman diagrams.

The talk is based on J. Math. Phys 45 (2004) 4596-4627 (with E. Raineri) as is part of a general programme of gravity on algebras.