Constraint algebra in loop quantum gravity reloaded. I. Toy model of a U(1)3 gauge theory

Abstract

We analyze the issue of anomaly-free representations of the constraint algebra in loop quantum gravity (LQG) in the context of a diffeomorphism-invariant U(1)3 theory in three spacetime dimensions. We construct a Hamiltonian constraint operator whose commutator matches with a quantization of the classical Poisson bracket involving structure functions. Our quantization scheme is based on a geometric interpretation of the Hamiltonian constraint as a generator of phase space-dependent diffeomorphisms. The resulting Hamiltonian constraint at finite triangulation has a conceptual similarity with the μ¯ scheme in loop quantum cosmology and highly intricate action on the spin-network states of the theory. We construct a subspace of non-normalizable states (distributions) on which the continuum Hamiltonian constraint is defined which leads to an anomaly-free representation of the Poisson bracket of two Hamiltonian constraints in loop quantized framework. Our work, along with the work done in [C. Tomlin and M. Varadarajan, Phys. Rev. D 87, 044039 (2013)], suggests a new approach to the construction of anomaly-free quantum dynamics in Euclidean LQG.