Comment on "On the canonical approach to quantum gravity"

In the model discussed by Ashtekar and Horowitz, it is shown that the constraint operator applied to the state functional allows support outside the classically allowed region only if no Hermiticity requirements are placed on the fundamental operators. If one requires that the constraint condition be written as the square of the Hermitian operator $p_{\theta }$ on the state function, then the classically forbidden region is also forbidden quantum mechanically, and, furthermore, within the classically allowed region, only a discrete set of points are allowed quantum mechanically. The latter restriction arises because the original manifold is compact, thereby forcing a quantization of the conjugate momenta which, in the model under consideration, forces a quantization of the allowed points. These features are also exhibited in the path-integral quantization of a similar system quantized on a torus.