Weaving a classical geometry 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 {\it non-local} operators, such as the area of a given 2-surface, which can be regulated diffeomorphism invariantly and which are finite {\it without} renormalization; ii)there exist quantum states which approximate a given flat geometry at large scales, but such states exhibit a discrete structure at the Planck scale; iii) these results are tied together by the fact that the spectra of the operators that measure the areas of surfaces are quantized in integral units of the Planck area. 
Abhay Ashtekar, Carlo Rovelli and Lee Smolin