“…This formulation has been used in the context of continuum and lattice gauge theory 3], and it has found a particularly e ective application in quantum gravity 2,4], because it allows a description of the di eomorphism invariant quantum states in terms of knot theory 2,5], and, at the same time, because it partially diagonalizes the quantum dynamics of the theory, leading to the discovery of solutions of the dynamical constraints 2,6]. Recent results in quantum gravity based on the loop representation include the construction of a nite physical Hamiltonian operator for pure gravity 7] and fermions 8], the computation of the physical spectra of area 9] and volume 10], and the developement of a perturbation scheme that may allow transition amplitudes to be explicitely computed 7,11,12]. A mathematically rigorous formulation of quantum eld theories whose con guration space is a space of connections, inspired by the loop representation, has been recently developed 13,14] and the kinematics of the theory is now on a level of rigor comparable to that of constructive quantum eld theory 15].…”