Coupling fermions to quantum gravity via spectral triples

In my talk I will first review the construction of a semi-finite spectral triple over a configuration space of connections. This spectral triple involves a noncommutative algebra of holonomy loops and a Dirac type operator, which resembles a functional derivation operator. The interaction between the algebra and the Dirac type operator reproduces the Poisson structure of general relativity when formulated in terms of Ashtekar variables. This means that the spectral triple captures the kinematics of quantum gravity. I will show that within this spectral triple construction natural, semi-classical states exist on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in 3+1 dimensions in a semi-classical limit. These states have, thus, a natural interpretation as one-particle states. Finally, I will show how also many-particle states might also emerge from the construction.