Emergent Dirac Hamiltonians in Quantum Gravity

We modify the construction of the spectral triple over an algebra of holonomy loops by introducing additional parameters in form of fam-ilies of matrices. These matrices generalize the already constructed Euler-Dirac type operator over a space of connections. We show that these families of matrices can naturally be interpreted as parameter-izing foliations of 4-manifolds. The corresponding Euler-Dirac type operators then induce Dirac Hamiltonians associated to the corre-sponding foliation, in the previously constructed semi-classical states