## 6.1 Loop algebra

Certain classical quantities play a very important role in the quantum theory. These are: the trace of the holonomy of the connection, which is labeled by loops on the three manifold; and the higher order loop variables, obtained by inserting the E field (in n distinct points, or ``hands'') into the holonomy trace. More precisely, given a loop in M and the points we define:

and, in general

where is the parallel propagator of along , defined by

(See [77] for more details.) These are the loop observables, introduced in Yang Mills theories in [95, 96], and in gravity in [183, 184].

The loop observables coordinatize the phase space and have a closed Poisson algebra, denoted by the loop algebra. This algebra has a remarkable geometrical flavor. For instance, the Poisson bracket between and is non vanishing only if lies over ; if it does, the result is proportional to the holonomy of the Wilson loops obtained by joining and at their intersection (by rerouting the 4 legs at the intersection). More precisely

Here

is a vector distribution with support on and is the loop obtained starting at the intersection between and , and following first and then . is with reversed orientation.

A (non-SU(2) gauge invariant) quantity that plays a role in certain aspects of the theory, particularly in the regularization of certain operators, is obtained by integrating the E field over a two dimensional surface S

where f is a function on the surface S, taking values in the Lie algebra of SU (2). As an alternative to the full loop observables (5, 6, 7), one can also take the holonomies and E [S, f] as elementary variables [15, 17]; this is more natural to do, for instance, in the C*-algebric approach [12].

 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