Referring to Why is Standard Model + Loop Quantum Gravity usually not listed as a theory of everything
Underlying papers are: J. W. Barrett, “Holonomy and path structures in general relativity and Yang-Mills theory”. Int. J. Theor. Phys., 30(9):1171–1215, 1991 & arxiv.org/0705.0452
Details of the LQG quantization: http://www.hbni.ac.in/phdthesis/phys/PHYS10200904004.pdf
The difference with canonical quantization is discussed at https://arxiv.org/abs/gr-qc/0211012 and does not seem (of course earlier paper) to address the issue raised above.
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As I read the Physics.SE post, I see that Barrett shows (for YM) a requirement for smooth mapping of loops on smooth manifolds to smooth curves to use these curves as representation of the original holonomies. Smoothness seems critical.
LQG does it in a configuration space (Hilbert pre quantization) and repeats the process to represent holonomies and create conjugate variables: holonomy of connections on phase space (i.e. on Hilbert space) and fluxes of tetrads. The constraints that generate spatial diffeomorphisms are not suitable operators... So, in order to generate the Hamiltonian, the quantization relies on these holonomies and unitary transforms of the diffeomorphisms. The latter mapping is not continuous nor smooth. Such quantization is known as the Polymer quantization (e.g. https://arxiv.org/pdf/gr-qc/0211012.pdf).
For the LQG variables, it seems that the condition for this to work (Barrett's paper) are lost, and it is argued that 1) it is an issue (as the equivalence is lost by violating the smoothness requirements) 2) it is why IR fails (no macroscopic spacetime can be recovered). I was asking if here is LQG answer/point of view on that. Indeed, as it is so fundamental to the quantization (not UV first then It considerations), even the resulting discrete spacetime (for UV), i.e., spin foam, would be a result of this loss of smoothness when recovering spacetime.
I am wondering if there has ever been an answer / fix from the LQG community to address that concern?
