A Conspectus on Group Averaging

[selfAdjoint]

In a new paper on the arxiv,

Group averaging, positive definiteness and
superselection sectors1
Jorma Louko


http://www.arxiv.org/PS_cache/gr-qc/pdf/0512/0512076.pdf

the technique of quantizing with a non-compact Lie group called Group Averaging is discussed with regard to when (so far as is now known) it leads to meaningful results or otherwise. As will be recalled from several threads, GA is a sometimes controversial tool in attempts to quantize gravity, used by Thiemann, Rovelli, and others.

From Louko's conclusion:

From the gravitational viewpoint, systems whose gauge group is a Lie group tend to arise in symmetry reductions of gravity, as is the case with spatially homogeneous cosmologies, or in systems that have been constructed by hand to mimic certain aspects of gravity, as is the case with all the systems discussed in this contribution. In gravity proper, however, the gauge group is infinite dimensional, and the Poisson bracket algebra of the constraints closes not by structure constants but by structure functions. While group averaging with nonunimodular Lie groups may give some insight into structure functions [10], and while a formalism that ties group averaging to BRST techniques has been developed [19], an extension of group averaging to systems with structure functions remains yet to be developed to a level that would allow a precise discussion of convergence properties and the observables in the ensuing quantum theory.