Hi marcus!
It seems to me that there are many different approaches, from around the world, that are converging at an accelerated pace.
Maybe not? … It might just be my desire to understand minimum length. http://arxiv.org/PS_cache/arxiv/pdf/0704/0704.2397v1.pdf
The Quantum Configuration Space of Loop Quantum Cosmology
J. M. Velhinho
18 April 2007

I even looked up the following to try to get a better idea of what is happening. http://www.iop.org/EJ/article/0264-9381/20/1/103/q301l3.html
Polymer and Fock representations for a scalar field
Abhay Ashtekar, Jerzy Lewandowski and Hanno Sahlmann
11 Dec 2002
“Our choices will ensure that the polymer scalar field can `live' on quantum geometry.”
I also skimmed the thread “Ashtekar's "Shadow states" paper”
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Well … it’s not called the QMLS.
Can someone draw a figure of what they think it looks like. A polymer representation on a line with triads in a 3d configuration, (a lattice). All that come to my mind is a funny looking torus which I labeled “void” with the triads making the dual tetra. http://www.geocities.com/j_jall/3dspace.gif
jal

Jal, the Velhinho paper studies the fundamental definitions of LQC from a sophisticated mathematical perspective.
I think the paper could be valuable to a mathematician who wanted to understand the formal rigorous basis of LQC.

If one wants to read LQC papers intuitively and follow partly on the basis of physics-sense, then I don't think one needs the Velhinho paper. But if one specializes in high abstract math, I suspect it could be very useful.

For the first 3 or 4 years of LQC, from 2001-2004, nobody mentioned "Bohr compactification of the Reals". Bojowald was just proceeding and building it up based on his physical grasp and working by analogy with the full theory. He, and others working with him, got several interesting results (including removal of the BB singularity---quantum corrections making gravity repel at very high density---bounce---a natural inflation phase).

But in 2004 a paper appeared co-authored with Ashtekar and Lewandowski that brought in the "Bohr compactification"----a different topology on the Reals.

By an odd coincidence this was invented by Niels Bohr's brother Harald who was a mathematician. And it just happened to be what Bojowald Ashtekar Lewandowski needed to put LQC on a mathematically more precise and rigorous footing.

But it was a kind of abstract or "moral" accomplishment---it didnt seem to change any of the practical results. In other words, more interesting mathematically than physically.

Velhinho's paper is a careful and lengthy clarification of that whole business, pointing out mathematical ramifications.

He has been doing this kind of thing for 3 or 4 years now. He follows LQG and LQC and from time to time, as a highly trained abstract mathematician, he illuminates some aspect in depth to show what is going on from HIS perspective. I think it is a valuable contribution. If there were some subtle inconsistency, his spotlight might find it. Or he might help other mathematicians get into the field. But at this point I don't expect his papers to help me personally understand better.
But