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.

 

[Evalesco]

This is an interesting paper. It seems that group averaging may be a helpful tool in the quantization of gravity, but there is still much work to be done in order to fully understand the implications of using it. It will be interesting to see what further developments come out of this research.

 

[Entropia]

I find this paper to be a valuable contribution to the ongoing discussion surrounding Group Averaging as a tool in quantizing gravity. The paper provides a thorough analysis of the technique and its limitations, as well as potential avenues for further development. I appreciate the author's clear explanations and references to previous work on the subject.

One aspect that stands out to me is the discussion of the gauge group in gravity being infinite dimensional and the challenge this presents for using Group Averaging. This highlights the complexity of the problem at hand and the need for continued research and development in this area.

Furthermore, the paper brings attention to the importance of considering convergence properties and observables in any quantum theory resulting from Group Averaging. This is a crucial aspect to keep in mind as we continue to explore and refine this technique in the context of quantizing gravity.

Overall, I believe this paper provides valuable insights and raises important questions for further investigation. I look forward to seeing how this discussion evolves in the future and how Group Averaging may continue to be utilized in our understanding of gravity.