Part of it depends on what you think the goal of QG is. Some people would say it is to understand the structure of geometry at very small scale. I would say it is to understand geometry at
very high energy density. The two goals seem identical but there may be a subtle difference.
In terms of a standard cutoff k (standing for energy, wavenumber, inverse length...) energy density is simply the fourth power k
4 so what's the difference between taking k to infinity and taking k
4?
For me there is a difference because I think of a nonperturbative background independent QG achieving three things:
to recover GR in appropriate limit
to be tested against early universe observation
to resolve the GR glitch at the start of expansion, probably with a bounce.
So a successful theory of quantum geometry would have to model the bounce, an evolution through extremely high energy density
that you can't put in a box. And the results of which we can observe after the fact.
The paradigm is slightly different from what one automatically thinks of as a quantum experiment---a box with a cat in it, or some other such boxed system with the experimenter and his classical paraphernalia outside.
I think if you look at footnote 5 on page 5 of the paper you will see that Bonzom and Smerlak are being told a reservation about their results which goes more or less as follows. The aim of QG is to put some geometry in a box and study it as a quantum system so that we can understand the *microscopic structure* of geometry. What you, Bon and Smer, are doing is not directly relevant to that, because a box and its contents are topologically TRIVIAL and a 2004 paper of Freidel Louapre already told how to deal with that situation.
http://arxiv.org/abs/hep-th/0401076 .
The reservation is that we don't need what they are doing (with this cellular quantization paper) because our main business is to learn about the microscopic degrees of freedom of geometry in a box and this doesn't advance that particular program.
But I think this may be the wrong perspective. It does not reflect what I think is the most interesting thing, namely the behavior of geometry at extreme energy density at the start of expansion, which we can actually OBSERVE because we see the ancient light that came from it---the microwave background---and maybe other stuff too.
Of course it could be the right perspective. In footnote 5 they are reporting comment by a very smart expert, A. Perez, so I feel a bit odd being in disagreement. I could easily be wrong about this. I hope some other people have thoughts on the matter.
