Real alternative to quantum gravity in loop space

We show that the Hamiltonian constraint of four-dimensional Lorentzian gravity, defined on a space of real, SU(2)-valued connections, in spite of its nonpolynomiality possesses a natural quantum analogue in a lattice-discretized formulation of the theory. This opens the way for a systematic search of its zero eigenvectors. The unambiguous and well-defined kinematical scalar product is that of the SU(2)-gauge theory. We also comment on various aspects of the continuum theory.