5.1 Quantization of one-Killing vector reductions

The quantization of this class of midisuperspaces has been considered by some authors but it is fair to say at this point that only a very partial knowledge has been achieved. A preliminary analysis within the LQG framework (with complex variables) was carried out in [119]. The fact that these models can be interpreted as 2+1 dimensional gravity coupled to some matter fields suggests that a quantization of this lower dimensional system essentially solves the problem for one-Killing vector reductions. In fact, some claims about the perturbative renormalizability of 2+1 gravity coupled to scalar fields have appeared in the literature [47] although, in our opinion, it is rather unclear how these results can be extrapolated to symmetric gravity in 3+1 dimensions. The reason for this scepticism is the non-trivial structure of the Hamiltonian for these systems when the right asymptotic behavior is incorporated (taking into account that, as shown by Ashtekar and Varadarajan in [15], the Hamiltonian that corresponds to the generator of time translations at spatial infinity is actually bounded from above). In any case, the physical consequences of these perturbative analyses have not been explored.
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