Regularization of the Hamiltonian constraint and the closure of the constraint algebra

In canonical quantum gravity regularization is needed to define the operator products occurring in the calculations. We examine the background dependence of the action of the regulated Hamiltonian constraint on the quantum states in the Ashtekar approach, and investigate whether the regularization preserves the closure of the constraint algebra. We compute the action on states based on smooth loops, on loops with intersections, and on loops with kinks. The results in all these cases depend on the arbitrary metric used in the calculations. We also show that the regularization does not affect the closure of the constraint algebra: The commutator of the regulated Hamiltonian constraint with the gauge and the diffeomorphism constraints equals zero and a linear combination of Hamiltonian constrains, respectively. On the other hand, the simple point-splitting regularization does not make the commutator of two Hamiltonian constraints expressible as a combination of constraints.