Thus the metric exists only on loops. S ince 1994, an understanding of quantum geometry has emerged. Motivated by the need for geometric observables in the canonical approach to quantum gravity and placed on mathematically rigorous foundations, a new framework for the structure of space has been introduced. Echoing an older, combinatorial definition of spacetime advocated by Penrose, the new structure is called spin networks. One tiny piece might look like this Spin networks are labeled, embedded graphs.
Edges carry spin excitations of geometry.
No longer subject to linear relations among the states as in the Rovelli-Smolin loop representation, these networks label a basis for states of spatial geometry. Using spin network techniques one may show that the spectra of area and volume operators are discrete. In this basis one also has many calculational methods, including a graphical calculus which enjoys close ties to knot theory and recoupling theory. There is also a readling list to the new variables in tex and in postscript . Seth's 'net pages at ITP-Wien.