R. Borissov, REGULARIZATION OF THE HAMILTONIAN CONSTRAINT AND THE CLOSURE OF THE CONSTRAINT ALGEBRA, Physical review. D. Particles and fields, 55(4), 1997, pp. 2059-2068
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.