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
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