On Spectral Triples in Quantum Gravity

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This talk is concerned with the existence of spectral triples in quantum gravity. I will review the construction of a spectral triple over a functional space of connections. Here, the *-algebra is generated by holonomy loops and the Dirac type operator has the form of a global functional derivation operator. The spectral triple encodes the Poisson structure of General Relativity when formulated in terms of Ashtekars variables. Finally I will argue that the Hamiltonian of General Relativity may emerge from the construction via the requirement that inner automorphisms vanish on the vacuum sector.