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- LQG starts with a quantization of spacetime which is topologically M
^{3}*R; it is by no means clear if this "background-dependence" does not throw away physical relevant sectors (of different topologies) of theory space - in the most general form the new SFs can to some extent go beyond this topology (as they do not specify a topology at all - only in a kind of classical limit), but again it's not clear if all topological sectors can be considered in this context
- LQG uses SU(2)-colored graphs w/o the notation of "dimension"; up to now I see no reason how dim=4 could be singled out from these graphs
- LQG uses SU(2)-colored graphs w/o the notation of "dimension"; up to now I see no reason why especially SU(2) shall be used - why not any other gauge group? restricting to SU(2) can be justified only b/c one knows about the starting point dim=4 - which is a cat hunting its own tail; in that sense LQG suffers from a kind of "landscape problem"
- it is clear that 4-dim spacetime has some interesting properties, namely that it allows for an uncountable set of smooth structures; LQG uses exactly one smooth structure in the very beginning!
- I indicated in another thread that the very existence of uncountably many smooth structures in dim=4 could single out dim=4; my idea was to construct a measure counting all different smooth structures on top of all topological manifolds in all dimensions; that would single out non-compact M
^{4}immediately simply due to "counting" w/o any additional dynamical input; LQG has nothing to say about that - I am convinced (but of course I can be wrong) that matter (fields) do emerge from spacetime; up to now the results about framed graphs in LQG seem to be a dead end (no new results since years)
- adding matter on top of the SFs is the wrong way; it will never result in a unification of spacetime (gravity) and matter
- the asymptotic safety approach indicates that gravity may be non-perturbatively renormalizable; due to that approach all possible couplings constructed from g, R, ... should be taken into account and should be subject to renromalization group flow similar to Kadanoff's block spins; up to now LQG does not say anything about that
- LQG treats the BI parameter (or the BI field) in a rather special way; there are indication that the BI parameter is related to the theta-angle in QCD, but I think final results are still missing (this is of course of major interest as the BI parameter plays a prominent role in the spectra of certain operators; it is strange that a semi-classical calculation a la Hawking w/o
*quantum*gravity should be used to fix the value of this parameter!) - LQG treats the cosmological constant in a very special way (e.g. approaches via quantum deformations of SU(2)); it is unclear why the cc should have such a special role, whereas the AS approach indicates that all couplings can be treated in a harmonized manner
- taken together it is not clear if the cc is an algebraic input (via q-deformed SU(2)) or a dynamical output (e.g. via renromalization group) of the theory
- the LQC results are very encouraging; but afaik this theory says nothing about the initial conditions of the universe, especially regarding its entropy; up to now LQC treats the evolution of the universe near the big bounce as time-symmetrical; results using infinitly many degrees are required
- there are indications that the holographic principle is a fundamental principle of nature, but LQG does not use this principle at all, e.g. in order to define "boundary Hilbert spaces" to define the theory; the holographic principle shows up only in very special calculations, e.g. in calculating the states and the entropy of a black hole