A section of the first paper on loop quantum gravity by Rovelli and Smolin [184] was devoted to a detailed study of ``transformation theory'' (in the sense of Dirac) on the state space of quantum gravity, and in particular on the relations between the loop states

and the states
giving the amplitude for a connection field configuration
*A*, and defined by

Here are ``eigenstates of the connection operator'', or, more precisely (since the operator corresponding to the connection is ill defined in the theory) the generalized states that satisfy

However, at the time of [184] the lack of a scalar product made transformation theory quite involved.

On the other hand, the introduction of the scalar product (14) gives a rigorous meaning to the loop transform. In fact, we can
write, for every spin network
*S*, and every state

This equation defines a unitary mapping between the two presentations of : the ``loop representation'', in which one works in terms of the basis ; and the ``connection representation'', in which one uses wave functionals .

The development of the connection representation followed a winding path through -algebraic [12] and measure theoretical [14, 16, 15] methods. The work of Ashtekar, Isham, Lewandowski, Marolf, Mourao and Thiemann has finally put the connection representation on a firm ground, and, indirectly has much clarified the mathematics underlying the original loop approach. In the course of this development, doubts were raised about the precise relations between the connection and the loop formalisms. Today, the complete equivalence of these two approaches (always suspected) has been firmly established. In particular, the work of Roberto DePietri [76] has proven the unitary equivalence of the two formalisms. For a recent discussion see also [139].

Loop Quantum Gravity
Carlo Rovelli
http://www.livingreviews.org/lrr-1998-1
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