Formal commutators of the gravitational constraints are not well defined: A translation of Ashtekar’s ordering to the Schrödinger representation

Ashtekar's factor ordering of the gravitational constraint equations, using spinorial variables, formally implies (in his {σ} representation) a certain symmetric factor ordering of the Hamiltonian constraint in the Schr\"odinger (metric) representation. This ordering is among those for which the constraints were believed not to close, and a straightforward formal computation of the commutator fails to give closure. However, with an alternative formal computation (equivalent, in the Schr\"odinger representation, to Ashtekar's spinorial computation) the constraints close. More generally, there is no well-defined formal factor-ordering problem in quantum gravity: different formal computations of the same commutator yield different results. If the constraints are regularized by point splitting, using a fixed flat background metric, their commutators have, in general, no well-defined coincidence limit. An alternative, covariant, point-splitting prescription that uses the exponential map is available, but adopting it implies that all orderings of the momentum constraint are equivalent; all orderings of the Hamiltonian constraint would then lead to a closed commutator algebra.