New variables for classical and quantum gravity in all dimensions: IV. Matter coupling

We employ the techniques introduced in the companion papers [1, 2, 3] to derive a connection formulation of Lorentzian General Relativity coupled to Dirac fermions in dimensions D + 1 ≥ 3 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 [1] 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.