Quantum Gravity coupled to Matter via Noncommutative Geometry

We show that the principal part of the Dirac Hamiltonian in 3+1 dimensions emerges in a semi-classical approximation from a construction which encodes the kinematics of quantum gravity. The construction is a spectral triple over a configuration space of connections. It involves an algebra of holonomy loops represented as bounded operators on a separable Hilbert space and a Dirac type operator. Semi-classical states, which involve an averaging over points at which the product between loops is defined, are constructed and it is shown that the Dirac Hamiltonian emerges as the expectation value of the Dirac type operator on these states in a semi-classical approximation.Comment: 15 pages, 1 figur

On Semi-Classical States of Quantum Gravity and Noncommutative Geometry

We construct normalizable, semi-classical states for the previously proposed model of quantum gravity which is formulated as a spectral triple over holonomy loops. The semi-classical limit of the spectral triple gives the Dirac Hamiltonian in 3+1 dimensions. Also, time-independent lapse and shift fields emerge from the semi-classical states. Our analysis shows that the model might contain fermionic matter degrees of freedom. The semi-classical analysis presented in this paper does away with most of the ambiguities found in the initial semi-finite spectral triple construction. The cubic lattices play the role of a coordinate system and a divergent sequence of free parameters found in the Dirac type operator is identified as a certain inverse infinitesimal volume element.Comment: 31 pages, 10 figure