Holonomy loops, spectral triples and quantum gravity**Based on talks given by JMG at the QG2 Conference, Nottingham, July 2008; at the QSTNG Conference in Rome in Sept/Oct 2008; at the AONCG Conference, Canberra, Dec 2008.

Abstract

We review the motivation, construction and physical interpretation of a semi-finite spectral triple obtained through a rearrangement of central elements of loop quantum gravity. The triple is based on a countable set of oriented graphs and the algebra consists of generalized holonomy loops in this set. The Dirac-type operator resembles a global functional derivation operator and the interaction between the algebra of holonomy loops and the Dirac-type operator reproduces the structure of a quantized Poisson bracket of general relativity. Finally, we give a heuristic argument as to how a natural candidate for a quantized Hamiltonian might emerge from this spectral triple construction.