Is Barbero’s Hamiltonian formulation a Gauge Theory of Lorentzian Gravity?

This letter is a critique of Barbero’s constrained Hamiltonian formulation of General Relativity on which current work in Loop Quantum Gravity is based. We show that if one tries to interpret Barbero’s real SO(3) connection as a space–time gauge field, the theory is not diffeomorphism invariant. In this respect, Barbero’s connection is quite different from Ashtekar’s, which does admit a space–time interpretation as a complex SU(2) gauge field. We conclude that Barbero’s formulation is not a gauge theory of gravity in the sense that Ashtekar’s Hamiltonian formulation is. The advantages of Barbero’s real connection formulation have been bought at the price of giving up the description of gravity as a gauge field.