New Variables for Classical and Quantum Gravity in all Dimensions IV. Matter Coupling

We employ the techniques introduced in the companion papers to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D+1 > 2 with compact gauge group. The technique that accomplishes that is similar to the one that has been introduced in 3+1 dimensions already: First one performs a canonical analysis of Lorentzian General Relativity using the time gauge and then introduces an extension of the phase space analogous to the one employed in the first paper of this series to obtain a connection theory with SO(D+1) as the internal gauge group subject to additional constraints. The success of this method rests heavily on the strong similarity of the Lorentzian and Euclidean Clifford algebras. A quantisation of the Hamiltonian constraint is provided.