Weaving a classical metric with quantum threads

Results that illuminate the physical interpretation of states of nonperturbative quantum gravity are obtained using the recently introduced loop variables. It is shown that (i) while local operators such as the metric at a point may not be well defined, there do exist nonlocal operators, such as the area of a given two-surface, which can be regulated diffeomorphism invariantly and which are finite without renormalization; (ii) there exist quantum states which approximate a given metric at large scales, but such states exhibit a discrete structure at the Planck scale.