On the relation between the connection and the loop representation of quantum gravity

Abstract

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