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

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

1996-05-28

arxiv.org/abs/gr-qc/9605064v1

Class.Quant.Grav. 14 (1997) 53-70

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

19 pages, ioplppt.sty and epsfig.sty

Scientific paper

10.1088/0264-9381/14/1/009

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.

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