Note on the phase space of asymptotically flat gravity in Ashtekar–Barbero variables

We describe the canonical phase space of asymptotically flat gravity in Ashtekar–Barbero (AB) variables. We show that the Gauss constraint multiplier must fall off slower than previously considered in order to recover ADM phase space. The generators of the asymptotic Poincare group are derived within the AB phase space without reference to the ADM generators. The resulting expressions are shown to agree, modulo Gauss constraint terms, with those obtained from the ADM generators. A payoff of this procedure is a new expression for the generator of asymptotic rotations, which is polynomial in the triad and hence better suited for quantum theory. Our treatment complements an earlier description by Thiemann in the context of self-dual variables. (paper)