6.2 Loop quantum gravity6 The Formalism6 The Formalism

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 tex2html_wrap_inline2588 in M and the points tex2html_wrap_inline2592 we define:


and, in general


where tex2html_wrap_inline2594 is the parallel propagator of tex2html_wrap_inline2596 along tex2html_wrap_inline2588, defined by


(See [77Jump To The Next Citation Point In The Article] for more details.) These are the loop observables, introduced in Yang Mills theories in [95, 96], and in gravity in [183Jump To The Next Citation Point In The Article, 184Jump To The Next Citation Point In The Article].

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 tex2html_wrap_inline2600 and tex2html_wrap_inline2602 is non vanishing only if tex2html_wrap_inline2604 lies over tex2html_wrap_inline2588 ; if it does, the result is proportional to the holonomy of the Wilson loops obtained by joining tex2html_wrap_inline2588 and tex2html_wrap_inline2610 at their intersection (by rerouting the 4 legs at the intersection). More precisely




is a vector distribution with support on tex2html_wrap_inline2588 and tex2html_wrap_inline2614 is the loop obtained starting at the intersection between tex2html_wrap_inline2588 and tex2html_wrap_inline2610, and following first tex2html_wrap_inline2588 and then tex2html_wrap_inline2610 . tex2html_wrap_inline2624 is tex2html_wrap_inline2610 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, 6Popup Equation, 7Popup Equation), one can also take the holonomies and E [S, f] as elementary variables [15Jump To The Next Citation Point In The Article, 17Jump To The Next Citation Point In The Article]; this is more natural to do, for instance, in the C*-algebric approach [12Jump To The Next Citation Point In The Article].

6.2 Loop quantum gravity6 The Formalism6 The Formalism

image Loop Quantum Gravity
Carlo Rovelli
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