Quantum gravity on finite sets
- Shahn Majid (Queen Mary, University of London, London, United Kingdom)
Abstract
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.
