ON THE RELATION BETWEEN THE CONNECTION AND THE LOOP REPRESENTATION OFQUANTUM-GRAVITY

Using Penrose's binor calculus for SU(2) (SL(2, C)) tensor expressions , a graphical method for the connection representation of Euclidean qu antum gravity (real connection) is constructed. It is explicitly shown that: (i) the recently proposed scalar product in the loop-representa tion coincides with the Ashtekar-Lewandowski cylindrical measure in th e space of connections; (ii) it is possible to establish a corresponde nce between the operators in the connection representation and those i n the loop representation. The construction is based on an embedded sp in network, the Penrose's graphical method of SU(2) calculus, and the existence of a generalized measure on the space of connections module gauge transformations.