Holonomy Loops, Spectral Triples & Quantum Gravity

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. based on talk given by J.M.G. at the QG2 conference, Nottingham, juli 2008; at the QSTNG conference in Rome in sept/oct 2008; at the AONCG conference, Canberra, december 2008. 1email: [email protected] 2email: [email protected] 3email: [email protected] 1 ar X iv :0 90 2. 41 91 v1 [ he pth ] 2 4 Fe b 20 09