COMBINATORIAL SPACE FROM LOOP QUANTUM-GRAVITY

The canonical quantization of diffeomorphism invariant theories of con nections in terms of loop variables is revisited. Such theories includ e general relativity described in terms of Ashtekar-Barbero variables and extension to Yang-Mills fields (with or without fermions) coupled to gravity. It is argued that the operators induced by classical diffe omorphism invariant or covariant functions are respectively invariant or covariant under a suitable completion of the diffeomorphism group. The canonical quantization in terms of loop variables described here, yields a representation of the algebra of observables in a separable H ilbert space. Furthermore, the resulting quantum theory is equivalent to a model for diffeomorphism invariant gauge theories which replaces space with a manifestly combinatorial object.