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

. Using Penroses 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 coincide 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 embedded spin network, the Penroses graphical method of SU(2) calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations. PACS numbers: 04.60.-m, 02.70.-c, 04.60.Ds, 03.70+k. Short title: On the relation between the connection and the loop representation May 30, 1996 2 1. Introduction Recently progress has been made in the Ashtekar1 formulation of Canonical Quantum Gravity2 and the relation...