## 6.5 The representation

I now define the quantum operators, corresponding to the -variables, as linear operators on . These form a representation of the loop variables Poisson algebra. The operator acts diagonally

(Recall that products of loop states and spin network states are normalizable states). In diagrammatic notation, the operator simply adds a loop to a (linear combination of) multiloops

Higher order loop operators are expressed in terms of the elementary ``grasp'' operation. Consider first the operator , with one hand in the point . The operator annihilates all loop states that do not cross the point . Acting on a loop state , it gives

where we have introduced the elementary length by

and and are defined in section 6.1 . This action extends by linearity, by continuity and by the Leibniz rule to products and linear combinations of loop states, and to the full . In particular, it is not difficult to compute its action on a spin network state [77]. Higher order loop operators act similarly. It is a simple exercise to verify that these operators provide a representation of the classical Poisson loop algebra.

All the operators in the theory are then constructed in terms of these basic loop operators, in the same way in which in conventional QFT one constructs all operators, including the hamiltonian, in terms of creation and annihilation operators. The construction of the composite operators requires the development of regularization techniques that can be used in the absence of a background metric. These have been introduced in [196] and developed in [186, 77, 18, 23, 138, 17].

 Loop Quantum Gravity Carlo Rovelli http://www.livingreviews.org/lrr-1998-1 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de