On the constraints of quantum gravity in the loop representation

Based on Ashtekar\u27s new variables there are two different representation for canonical quantum gravity, the connection representation and the loop representation. In contrast to the situation in the connection representation, it is not straightforward to obtain the constraints of quantum gravity in the loop representation. Classically, the constraints expressed in terms of the Ashtekar variables can be obtained from the loop variables in the limit where loops are shrunk to a point. We give precise meaning to this shrinking process at the classical and quantum level and find as the corresponding operator the area derivative on the space of loop functionals. This enables us to derive the diffeomorphism and hamiltonian constraints of quantum gravity in the loop representation in terms of the area derivative. In particular, the hamiltonian constraint arises in exactly the form already obtained by Gambini, but we do not resort to a formal transform between the connection and the loop representation. We interpret the hamiltonian constraint in terms of a generalized version of the shift operator introduced by Rovelli and Smolin. © 1993