REGULARIZATION OF THE HAMILTONIAN CONSTRAINT AND THE CLOSURE OF THE CONSTRAINT ALGEBRA

In canonical quantum gravity regularization is needed to define the op erator products occurring in the calculations. We examine the backgrou nd 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 int ersections, and on loops with kinks. The results in all these cases de pend on the arbitrary metric used in the calculations. We also show th at the regularization does not affect the closure of the constraint al gebra: The commutator of the regulated Hamiltonian constraint with the gauge and the diffeomorphism constraints equals zero and a linear com bination of Hamiltonian constrains, respectively. On the other hand, t he simple point-splitting regularization does not make the commutator of two Hamiltonian constraints expressible as a combination of constra ints.