Towards an anomaly-free quantum dynamics for a weak coupling limit of Euclidean gravity

Abstract

The ${G}_{\mathrm{Newton}}\ensuremath{\rightarrow}0$ limit of Euclidean gravity introduced by Smolin is described by a generally covariant $U(1{)}^{3}$ gauge theory. The Poisson-bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity. Motivated by recent results in parametrized field theory and by the search for an anomaly-free quantum dynamics for loop quantum gravity, the quantum Hamiltonian constraint of density weight $4/3$ for this $U(1{)}^{3}$ theory is constructed so as to produce a nontrivial loop quantum gravity type representation of its Poisson brackets through the following steps. First, the constraint at finite triangulation and the commutator between a pair of such constraints are constructed as operators on the ``charge'' network basis. Next, the continuum limit of the commutator is evaluated with respect to an operator topology defined by a certain space of ``vertex smooth'' distributions. Finally, the operator corresponding to the Poisson bracket between a pair of Hamiltonian constraints is constructed at finite triangulation in such a way as to generate a ``generalized'' diffeomorphism and its continuum limit is shown to agree with that of the commutator between a pair of finite-triangulation Hamiltonian constraints. Our results, in conjunction with the recent work of Henderson, Laddha and Tomlin in a ($2+1$)-dimensional context, constitute the necessary first steps toward a satisfactory treatment of the quantum dynamics of this model.