Towards an Anomaly-Free Quantum Dynamics for a Weak Coupling Limit of Euclidean Gravity

Understand

The G -->0 limit of Euclidean gravity introduced by Smolin is described by a generally covariant U(1)xU(1)xU(1) gauge theory.

• The Poisson bracket algebra of its Hamiltonian and diffeomorphism constraints is isomorphic to that of gravity.
• Motivated by recent results in Parameterized Field Theory and by the search for an anomaly-free quantum dynamics for Loop Quantum Gravity (LQG), the quantum Hamiltonian constraint of density weight 4/3 for this U(1)xU(1)xU(1) theory is constructed so as to produce a non-trivial LQG-type representation of its Poisson brackets through the following steps.
• First, the constraint at finite triangulation, as well as the commutator between a pair of such constraints, are constructed as operators on the `charge' network basis.