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

TL;DR

This paper develops a graphical method using Penrose calculus to relate the connection and loop representations in Euclidean Quantum Gravity, establishing measure equivalences and operator correspondences.

Contribution

It introduces a new graphical approach to connect the connection and loop representations in quantum gravity, utilizing Penrose calculus and generalized measures.

Findings

Scalar product in loop representation matches Ashtekar-Lewandowski measure

Operators in connection and loop representations are related

Graphical method simplifies the understanding of quantum gravity representations

Abstract

Using Penrose 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: {\it (i)} the recently proposed scalar product in the loop-representation coincide with the Ashtekar-Lewandoski cylindrical measure in the space of connections; {\it (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 Penrose graphical method of $SU(2)$ calculus, and the existence of a generalized measure on the space of connections modulo gauge transformations.